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MILNE'S RATIONALITY CONJECTURE FOR ABELIAN VARIETIES
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A rewritten, self-contained exposition.
The mathematical content follows the original paper; wording, structure and
signposting have been reworked for clarity and accessibility.
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PART 0. PLAIN-LANGUAGE GUIDE (read this first)
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This part has no proofs. It says what the paper is about, in ordinary words,
so that a curious reader can follow the story before meeting the machinery.
0.1 What is an abelian variety?
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Picture the surface of a donut. It is a torus: you can travel around it in two
independent directions and come back to where you started. An abelian variety
of dimension d is the higher-dimensional version of this: a compact shape with
d complex "wrap-around directions" that (i) also carries a rule for adding
points, so it is a group, and (ii) can be cut out by polynomial equations with
algebraic-number coefficients. d is its dimension.
To each abelian variety of dimension d one attaches a rational vector space
H^1(A,Q) of dimension 2d. It records the basic "1-dimensional loop" data. All
the richer shape information is built from it by exterior powers: the r-th
exterior power gives the space H^{2r}(A,Q) of "2r-dimensional" measurements.
0.2 What is a Hodge class?
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Inside H^{2r}(A,Q) there is a distinguished piece, the classes "of type (r,r)".
These are the Hodge classes. The famous Hodge conjecture predicts that every
Hodge class is the cohomology class of a codimension-r subvariety (an algebraic
"cycle"). That conjecture is wide open in general.
For abelian varieties, though, one crucial thing is known: Deligne proved that
every Hodge class on an abelian variety is ABSOLUTE HODGE. In plain terms this
means the class has a canonical image in every cohomology theory at once -- in
every ell-adic theory (ell a prime different from the residue characteristic p)
and in the crystalline theory at p -- and all these images are compatible with
each other. This compatibility is the quiet engine behind everything below.
0.3 Good reduction, and what specialization does
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Fix a prime p and a p-adic place w of the algebraic numbers Q-bar. "A has good
reduction at w" means we can reduce the defining equations mod p and still get
a smooth abelian variety A_0 of the same dimension, now over F = the algebraic
closure of F_p.
A cohomology class gamma on A then specializes to a class gamma_0 on A_0, and
Deligne's absolute-Hodge property guarantees a specialization in EVERY theory
at once: ell-adic specializations for all ell != p, and a crystalline one at p.
Here is the catch that makes the whole question nontrivial. The reduced variety
A_0 can possess divisors (codimension-1 subvarieties) that do NOT lift back to
divisors of A. Upstairs these objects simply do not exist; downstairs they do,
and gamma_0 can interact with them.
0.4 Divisors and intersection numbers
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A divisor D on A_0 has a cohomology class c_1(D). On a d-dimensional A_0, take
a class gamma_0 of codimension r and a product of d-r divisors. Since r + (d-r)
= d, their product is a codimension-d class, i.e. a single number: its degree
(or "trace"). This number is what the paper is about.
Milne's question, roughly: for each product of d-r divisors on A_0, is this
trace a RATIONAL number, and is it the SAME rational number in every cohomology
theory (every ell != p, and crystalline at p)?
Why is this not obvious? If gamma were the class of an actual algebraic cycle
on A, then yes: intersection numbers are rational and do not depend on which
cohomology theory you compute them in. But gamma is merely a Hodge class, and
the divisors live only on the reduction A_0. The theorem says the conclusion
still holds anyway.
0.5 The main theorem, informally
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Let A/Q-bar be an abelian variety of dimension d with good reduction A_0/F at
the place w over p. Let gamma be a rational Hodge class of codimension r on A.
Then for every list D_1,...,D_{d-r} of divisors on A_0 there is a single
rational number q such that:
* the ell-adic trace of gamma_0 times the divisor product equals q, for every
ell != p; and
* the crystalline trace of gamma_0 times the divisor product equals q as well.
One number q, common to all realizations, for the fixed reduction data. This is
Milne's rationality conjecture for abelian varieties, and this paper proves it
in every residue characteristic (including characteristic 2).
0.6 Two consequences worth naming
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The proof also yields two extra results, each interesting on its own:
* Every rational Hodge class on a good-reduction abelian variety specializes
into ONE FIXED "good theory of rational Tate classes" (a bookkeeping device
of Milne), independently of the class. See Corollary 5.3.
* IF one additionally accepts the rational Hodge theorem for CM abelian
varieties (a separate, external result), then every such specialized class
is represented by a single rational algebraic cycle on A_0, in all the
realizations simultaneously. See Corollary 5.4. The main theorem does NOT
need this extra input.
What the paper does NOT prove: the Hodge conjecture, the Tate conjecture, or any
lift of the special-fiber cycle back to the original variety A. It is a statement
about one fixed reduction place.
0.7 How the proof is organized (so you know where things are going)
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The reduction is genuinely hard because the obstruction -- the non-lifting of
divisors -- must be dodged. The strategy has four moves, spread across Sections
2 through 5:
1. TRANSPORT TO A CM VARIETY. After taking a power, replace the (possibly
non-CM) situation on the special fiber by one coming from a CM abelian
variety C, via a single rational homomorphism on reductions. The Hodge
class gamma_0 becomes the pullback of a Hodge class kappa_0 on C_0, with
the same equality in every realization. (Corollary 2.7.)
2. REDUCE CM CLASSES TO BALANCED WEIL CLASSES. On the CM variety, express any
Hodge class as a rational sum of pullbacks of "Weil classes" carried by
balanced factors of powers of C. (Lemma 5.1.)
3. DIVIDE AND CONQUER ON ONE WEIL CLASS. For a single balanced Weil class, a
dichotomy holds. Either every complementary intersection number vanishes
(no invertible endomorphism of the right kind exists downstream), or the
class is already a rational polynomial in divisors in every realization
at once, and the desired number is a rational intersection number. This is
the heart of the argument. (Proposition 4.2, plus the filtration theorem
of Section 3 and the divisor-lifting lemma of Section 2.)
4. PULL EVERYTHING BACK. The Weil-class information transports back down the
chains of rational homomorphisms to gamma_0. (Section 5.)
With this map in hand, turn to Section 1.
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1. INTRODUCTION
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Suppose an abelian variety has good reduction at a prime. A rational algebraic
cycle on the variety specializes to a cycle on the reduction; its intersection
with any complementary product of divisor classes is a rational degree; and
every cycle-class realization of the cohomology class computes that same
number. Milne's rationality conjecture asks for exactly this conclusion when
the original class is only known to be a rational HODGE class rather than an
algebraic cycle. We prove it for every residue characteristic.
Why the distinction matters. Good reduction can acquire divisors that do not
lift to the original variety. Deligne's absolute-Hodge theorem gives compatible
realizations of the Hodge class in all cohomology theories at once, but it does
not by itself identify the pairing of that class with these NEW divisors as the
degree of a rational cycle. The conjecture isolates this rationality question
from the stronger, still-open problem of representing the Hodge class by an
algebraic cycle.
Notation. Fix an embedding Q-bar inside C, a prime p, and a p-adic place w of
Q-bar. Write
F = the algebraic closure of F_p,
K0 = Frac W(F) (W = Witt vectors),
Cw = the completion of Q-bar at w.
We view K0 and an algebraic closure of it as subfields of Cw. "Good reduction"
means good reduction after descending to a number field and a finite extension
at the chosen place.
THEOREM 1.1 (Main theorem). Let A/Q-bar be an abelian variety of dimension d
with good reduction A_0/F at w. Let 0 <= r <= d, and let gamma in H^{2r}(A(C),Q(r)) be a rational Hodge
class. For every list of Cartier divisors D_1,...,D_{d-r} on A_0 there is a
rational number q such that
Tr_ell( gamma_{0,ell} c_{1,ell}(D_1) ... c_{1,ell}(D_{d-r}) ) = q
in Q_ell, for every prime ell != p,
Tr_p( gamma_{0,p} c_{1,cris}(D_1) ... c_{1,cris}(D_{d-r}) ) = q
in Cw.
Here gamma_{0,ell} is the smooth proper specialization of the ell-adic
realization, and gamma_{0,p} is the de Rham-crystalline specialization lying in
H^{2r}_cris(A_0/W(F))[1/p](r) tensor_{K0} Cw.
The traces use the top Tate twist and send the class of a zero-cycle to its
degree. When r = d the divisor product is empty, i.e. equal to 1.
By linearity the same conclusion holds for every rational Lefschetz class of
complementary codimension -- every rational linear combination of products of
divisors. The number q is common to all realizations for the fixed data; no
comparison between different reduction places is needed.
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1.1 Origins and consequences
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Milne introduced a weak-rationality formulation while studying the reduction of
Shimura varieties and the Langlands-Rapoport conjecture (Milne 2000, Conjecture
6.1(A)). It asks that a specialized CM Hodge class whose components lie, in
every realization, in the local spans of divisor products should itself come
from one rational Lefschetz class. The pairing formulation used here appears in
his AIM notes and in his study of rational Tate classes (Milne 2007, Section 10;
Milne 2009, Conjecture 4.1). A previously known positive case is a CM abelian
variety with simple ordinary reduction, together with all its powers: there
every divisor class on the reduction lifts (Milne 2009, Example 4.2).
The broader purpose is to obtain rational cohomological correspondences for
abelian motives in characteristic p, compatible with reduction from
characteristic zero (Milne 2000, Sections 5-6; Milne 2025, Introduction and
Section 2). For ell != p, Tate classes are twisted ell-adic classes fixed by a
power of Frobenius, and there is a matching Q_p-algebra of crystalline Tate
classes. A GOOD THEORY OF RATIONAL TATE CLASSES chooses, on each variety, one
graded Q-algebra whose scalar extensions reproduce these Tate algebras. Such a
theory is stable under pullback and pushforward, contains divisor classes, and
contains specializations of CM Hodge classes (Milne 2009, Definitions 2.1 and
3.1). These are cohomological correspondences; their definition does not assert
that they are algebraic.
Milne proved that the rationality statement for all CM abelian varieties is
equivalent to the existence of such a good theory on varieties over F whose
connected components are products of abelian varieties and projective spaces
(Milne 2009, Theorem 4.5). Our proof additionally expresses each specialized
Hodge class as the pullback, along a rational homomorphism of special fibers,
of a specialized Hodge class from a CM abelian variety. This extends the
theory's specialization property beyond the CM case. In Corollary 5.3 every
rational Hodge class on a good-reduction abelian variety specializes to a
unique class in one fixed good theory, and its crystalline component gives the
stated specialization after extension to Cw.
A separate additional input gives algebraicity on the special fiber. Combining
the same transport with the Hodge theorem for CM abelian varieties (OpenAI 2026,
Theorem 1.1), Corollary 5.4 represents each specialized Hodge class by one
rational algebraic cycle, simultaneously in all realizations. Its pairing with
any complementary algebraic cycle is then a rational intersection number. This
consequence USES the CM Hodge theorem; the proof of the main theorem does not.
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1.2 The method and its antecedents
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Three foundational results make the question accessible.
* Deligne proved that every Hodge class on an abelian variety is absolute
Hodge (Deligne 1982, Main Theorem 2.11).
* Blasius established compatibility with p-adic comparison (Blasius 1994,
Theorems 0.3 and 5.3); Moonen presents the Blasius-Wintenberger comparison
in the perfect-residue-field form needed for the local auxiliary lifts
below (Moonen 1998, Theorem 5.6.3).
* Milne's theory of Lefschetz classes supplies a common rational divisor
algebra with a perfect complementary pairing, independent of the choice of
Weil cohomology (Milne 1999, Section 5).
The determinant classes used in the proof belong to the theory of Weil classes
developed by Weil, Deligne and Andre; see (Deligne 1982, Section 4 and Endnotes
M.10-M.12). Andre's refinement of the CM reduction expresses Hodge classes by
pullbacks from varieties of split Weil type (Andre 1992); a detailed account is
(Milne 2020, Theorem 1). We prove directly the BALANCED form of that reduction
needed here (Lemma 5.1). Balanced signatures mean equal Hodge multiplicities at
conjugate embeddings; split Weil type additionally requires the associated
Hermitian form to have a half-dimensional isotropic subspace. Andre's later
theory of motivated cycles provides a broader unconditional motivic home for
abelian Hodge classes (Andre 1996, Theorem 0.6.2), without identifying them with
algebraic cycles.
For an imaginary quadratic field, Markman has proved algebraicity of Weil
classes on abelian fourfolds of Weil type and on abelian sixfolds of split Weil
type (Markman 2026, Theorem 1.2). Those are characteristic-zero algebraicity
results. The pairing argument here works in arbitrary dimension and begins only
with the Hodge class and its good reduction.
There is also a direct antecedent for the lifting strategy. Milne asked whether
a specialized CM Hodge class and a divisor can be lifted together, up to
isogeny, to a CM variety, and explained the resulting rationality implication
(Milne 2007, Question 10.6 and Proposition 10.7). His current account gives a
conditional criterion through weak lifting of Lefschetz classes on CM Weil
triples, and describes the reduction from general Hodge classes through CM
varieties to Weil classes (Milne 2025, Introduction, Question 2.17 and Theorem
2.18). Our proof follows this general route using two auxiliary varieties with
different jobs. The first is CM. The second realizes a prescribed admissible
filtration over a local field and need not be CM. The realization machinery
comes from Grothendieck-Serre-Tate deformation theory, Tate's extension theorem,
and the integral-model and local-Shimura results of Kisin-Zhou and
Pappas-Rapoport (Grothendieck 1970; Tate 1967; Kisin-Zhou 2025; Pappas-Rapoport
2024). For the prescribed filtration we choose a framed local Shimura point
over its admissible period and use integral uniformization; the required
uniformization condition (U_x) is supplied by (Gleason-Lim-Xu 2026, Corollary
6.3). We state each precise input where it enters and prove the connecting
arguments for the prescribed flag.
The contribution along this route is to build a filtration that preserves the
extra divisor, then obtain a rational divisor identity on its lift. A separate
normalization by rational intersection numbers makes the tensor transports
SIMULTANEOUS in all realizations. Kisin-Zhou's compatibility theorem concerns
Weil-Deligne representations across coefficient fields (Kisin-Zhou 2025,
Theorem 1.2); the extra issue here is the rational VALUE of a Hodge-divisor
intersection.
The key step is a dichotomy for balanced Weil classes: either all the
complementary Lefschetz pairings vanish, or the specialized class is already a
rational polynomial in divisors, simultaneously in every realization.
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1.3 Proof overview
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Four stages. The order below is the deduction. Sections 2-4 build the reusable
constructions first; Section 5 assembles them.
STAGE 1. TRANSPORT TO A CM VARIETY.
Corollary 2.7 constructs a CM abelian variety C/Q-bar, a quasi-isogeny
a_CM : A_0^n -> C_0, and a rational Hodge class kappa such that
gamma_0 = (a_CM o j)* kappa_0,
where j includes the first factor. This is ONE rational algebraic homomorphism
on the special fibers and ONE equality in every realization. Kisin-Zhou's
special-point theorem supplies the CM lift (Kisin-Zhou 2025, Theorem 2.2.7). The
local tensors initially determine similitude scalars; a ratio of rational
divisor intersections shows those scalars are all the same rational number.
That normalization is what transports the ORIGINAL rational class, rather than a
separate local multiple in each cohomology theory.
STAGE 2. REDUCE CM CLASSES TO BALANCED WEIL CLASSES.
Lemma 5.1 constructs rational-idempotent factors of actual powers of C. Their
homology has rank 2s over a CM field E, with signatures (s,s) at every
embedding, and their polarizations induce conjugation on E. Their Weil classes
are the rational forms that, after splitting E, are top alternating forms on
individual E-eigenspaces. Pullbacks of these Hodge classes span the given CM
Hodge class over Q. Good reduction is preserved. Having one rational value for
each complementary Lefschetz pairing is preserved under rational pullback
(Lemma 2.2). So it remains to prove the statement for balanced Weil classes.
STAGE 3. FIND A DIVISOR AND A COMPATIBLE FILTRATION.
For a balanced Weil variety Z, let dagger denote Rosati on its reduction, and
consider
U = { u in End^0(Z_0) : u^dagger = u, u e = conj(e) u for all e in E }.
Proposition 4.2 proves a dichotomy. If U has no invertible element, every
complementary Lefschetz pairing of a specialized Weil class is zero. If it has
an invertible element, the specialized Weil class is a simultaneous rational
Lefschetz class: one rational polynomial in divisors gives its class in every
realization. In this second branch, such a u yields a divisor D^u whose
alternating form is nondegenerate on each E-eigenspace. Theorem 3.1 constructs a
weakly admissible balanced filtration on the fixed crystalline module that
preserves E, u, and the polarization. Generic transversality plays its classical
role here (Fontaine-Rapoport 2005, Theorem 1 and Lemmas 1.1-1.2). Bounded-degree
avoidance makes all the constrained choices over a single finite extension, and
a maximality argument reduces the weak-admissibility test to subobjects stable
under the endomorphisms.
STAGE 4. LIFT THE DIVISOR AND EXTRACT A RATIONAL IDENTITY.
The second case of Proposition 2.4 realizes this flag on a local auxiliary
abelian variety B, with a marked quasi-isogeny a_fl : Z_0^m -> B_0. The
prescribed flag makes the first-summand projector, its algebraic E-action, and
the transported divisor FILTERED. These specified algebraic maps lift to B.
After choosing an abstract complex realization of B, polynomial coefficient
extraction expresses the relevant rational Betti Weil class as a rational
linear combination of divisor products (Lemma 4.4). The same rational identity
specializes in every realization. Pullback returns it to Z_0, finishing the
invertible-element branch and, through Stages 1-2, the main theorem.
Figure 1 (end of this part) separates the two lifts and their special-fiber
markings.
SECTIONS:
Section 2 fixes comparisons, traces and rational pullback, and proves both
transport alternatives.
Section 3 constructs the prescribed endomorphism-stable filtration.
Section 4 proves the balanced-Weil dichotomy.
Section 5 proves the CM reduction and finishes the argument.
FIGURE 1. The two uses of transport.
Each lower horizontal arrow is one actual quasi-isogeny preserving the
specified tensors in all realizations. Vertical arrows denote good reduction.
No map between the generic fibers is claimed.
CM transport:
A^n/Q-bar --(reduction)--> A_0^n
|
| a_CM
v
C/Q-bar (CM) --(reduction)--> C_0
and gamma_0 = (a_CM o j)* kappa_0.
Prescribed-filtration transport:
Z^m/Q-bar --(reduction)--> Z_0^m
(balanced Weil) |
| a_fl
v
B/K (K/K0 finite) --(reduction)--> B_0
and a_fl* Fil^1 H^1_dR(B) = (F_Z^1)^{+m}.
The upper construction starts from the original variety. The lower
construction is applied separately to a balanced Weil variety Z arising in
the CM reduction; its lift B realizes the chosen filtration F_Z^{direct sum m}
under a_fl*. The CM and prescribed-filtration auxiliaries need not coincide.
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2. COMPARISONS, RATIONAL LEFSCHETZ CLASSES, AND TRANSPORT
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This carries the two preparatory engines of the proof. Section 2.1 sets the
cohomological conventions and records that the target property is preserved
under rational pullback. Section 2.2 proves the transport proposition: one
rational homomorphism of special fibers moving the Hodge class onto an
auxiliary variety, simultaneously in every realization, with the auxiliary
variety optionally CM or optionally carrying a prescribed admissible
filtration.
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2.1 Comparisons and rational Lefschetz classes
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We first put the realizations appearing in Theorem 1.1 into a common notation
and express its conclusion as a property of cohomology classes. The functorial
fact that drives everything is that this property is preserved by rational
pullback. We also record the criterion (Lemma 2.3) that will lift an extra
divisor once a suitable Hodge filtration has been realized.
We work throughout in the category of abelian varieties up to isogeny. Thus
Hom^0(X,Y) = Hom(X,Y) tensor Q, and divisors may have rational coefficients.
Pullback by a rational homomorphism is defined on H^1 and extended to its
exterior algebra: if N f is an actual homomorphism, then on H^{2r},
f* = N^{-2r} (N f)*. (2.1)
This convention is the same in every realization.
For an abelian variety X over F, set
k_ell = Q_ell for ell != p,
k_p = Cw,
and write H_v^{2r}(X)(r) for its ell-adic cohomology if v = ell != p, and for
its crystalline cohomology extended to Cw if v = p. Cup products are usually
written without a cup symbol. All trace maps and divisor classes carry their
cycle-class normalizations. The Betti Tate line Q(r) supplies the twists in the
other realizations through comparison. Thus Tr_v is the trace on top cohomology
with its top Tate twist, and it sends the class of a zero-cycle to its degree.
For a good-reduction lift, gamma_{0,ell} denotes smooth proper specialization of
the absolute-Hodge ell-adic realization, and gamma_{0,p} denotes de
Rham-crystalline specialization, extended from K0 to Cw. We keep first
crystalline cohomology over K0 when building filtrations, and specify each
extension of the filtration field separately.
We use the exterior-algebra description of abelian cohomology and the
faithfulness of rational homomorphisms on H^1; see (Milne 1986, Theorems 12.5
and 15.1) for the prime-to-characteristic realizations. The uniform
Weil-cohomology formalism, including crystalline cohomology, exterior powers,
cycle-normalized traces and the projection formula, is set out in (Milne 1999,
Section 1 and Appendix, Proposition A.2). Smooth proper specialization away from
p is (Milne 2013, Theorem 20.4).
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2.1.1 Comparison of Hodge tensors
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LEMMA 2.1 (Comparison). For an abelian variety with good reduction, the
realizations of an absolute Hodge class are compatible with the de Rham and
crystalline comparison isomorphisms. In particular, the p-adic crystalline
tensor obtained from its p-adic ell-adic realization agrees, after extension of
scalars, with the de Rham-crystalline realization in Theorem 1.1. These
compatibilities respect tensor operations, duals, Tate twists, divisor classes,
and rational homomorphisms. The same assertion applies to auxiliary abelian
varieties with good reduction over a finite extension of K0, using a complex
embedding of an algebraic closure of their field of definition.
Proof. For abelian varieties over Q-bar, Deligne's absolute-Hodge theorem
(Deligne 1982, Main Theorem 2.11) and Blasius's comparison theorem (Blasius
1994, Theorems 0.3 and 5.3) give the assertion. Compatibility of the
crystalline comparison with the de Rham comparison is part of (Blasius 1994,
Section 5.1). The formulation for a complete mixed-characteristic discrete
valuation ring with perfect residue field is given in (Moonen 1998, Theorem
5.6.3); the paragraph following that theorem explains how to remove the
number-field hypothesis. A finite extension of K0 is within that scope. The
stated functorialities are those of the comparison isomorphisms. []
An abstract complex embedding used here need not be continuous. To see that it
can be chosen compatibly with Q-bar inside C, take a transcendence basis of the
algebraic closure of K0 over Q-bar, embed it in a transcendence basis of C over
Q-bar, and extend algebraically. Cardinalities permit this construction. No
descent of a general auxiliary lift to Q-bar is inferred.
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2.1.2 The common rational Lefschetz space
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Let Lef^r(X) be the rational vector space of polynomial expressions in divisors
of codimension r, modulo numerical equivalence. We use the following
established properties:
(i) its cycle map is injective, and Lef^r(X) tensor k_v is the
divisor-generated subspace of H_v^{2r}(X)(r);
(ii) intersection and degree give a perfect pairing
Lef^r(X) x Lef^{dim X - r}(X) -> Q;
(iii) pushforward by a homomorphism of abelian varieties preserves the
divisor-generated subspaces in each realization;
(iv) hard Lefschetz and its inverse restrict to these subspaces.
These are (Milne 1999, Proposition 5.2, Corollaries 5.3 and 5.5, Theorem 5.9),
with the author's correction to the proof of Theorem 5.9. Crystalline
cohomology is included. Extending its coefficient field to Cw preserves these
statements. We call the tuple of realizations of a fixed element of Lef^r(X) a
SIMULTANEOUS RATIONAL LEFSCHETZ CLASS.
DEFINITION. A tuple alpha = (alpha_v)_v, with alpha_v in H_v^{2r}(X)(r), has
the LEFSCHETZ PAIRING PROPERTY if, for every delta in Lef^{dim X - r}(X), the
numbers Tr_v(alpha_v delta_v) are the images of one number in Q.
Simultaneous rational algebraic classes have this property, by the intersection
formula. The property is closed under rational linear combinations. We shall
need one more closure property.
LEMMA 2.2 (Pairing functoriality). If alpha has the Lefschetz pairing property
on Y, and f is in Hom^0(X,Y), then f* alpha has that property on X.
Proof. If r > dim Y the input class is zero; if r > dim X its pullback is
zero. We may therefore suppose f is an actual homomorphism and r <= min(dim X,
dim Y). For delta in Lef^{dim X - r}(X), the class f_* delta_v lies in the
Lefschetz subspace of codimension dim Y - r in every realization. Choose
rational bases a_i of Lef^{dim Y - r}(Y) and b_j of Lef^r(Y). The matrix
M_{ij} = deg(a_i b_j) is an invertible rational matrix. If
f_* delta_v = sum_i c_{i,v} (a_i)_v, the projection formula gives
sum_i c_{i,v} M_{ij} = deg(delta f* b_j).
The right side is a rational number independent of v. Solving this one rational
system shows c_{i,v} = c_i in Q for every v. Thus f_* delta is simultaneous
rational Lefschetz. Applying the projection formula once more proves the
assertion. For rational f, use (2.1). []
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2.1.3 Lifting a prescribed rational homomorphism
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The second transport construction will produce an abelian lift with a chosen
Hodge filtration. The following criterion then lifts the algebraic projector,
field action, and divisor already specified on its special fiber. Its input is
a rational algebraic homomorphism, not an arbitrary map between cohomology
spaces.
In the lemma, R is a complete mixed-characteristic discrete valuation ring
finite over W(F), with fraction field K. For an abelian scheme X/R, the Hodge
filtration is regarded as a filtration of
H^1_cris(X_0/W(F))[1/p] tensor_{K0} K.
LEMMA 2.3 (Lifting a filtered rational homomorphism). Let X, Y be abelian
schemes over R, with special fibers X_0, Y_0, and let f_0 be in Hom^0(X_0,Y_0).
If its contravariant crystalline map preserves the Hodge filtrations, then f_0
lifts uniquely to a rational homomorphism X -> Y.
In particular, a symmetric rational homomorphism X_0 -> X_0^dual with this
property lifts to a rational divisor class on the generic fiber. Algebraic
identities among finitely many such lifted maps remain valid.
Proof. The crystalline map already commutes with Frobenius. Full faithfulness
for rational filtered Dieudonne modules gives a rational morphism of the
associated p-divisible groups; see (Grothendieck 1970) and the
perfect-residue-field formulation in (Illusie 2015, Section 4.2(v)).
Equivalently, crystalline comparison first gives the rational map on Tate
modules. Multiply by an integer N clearing both its Tate-lattice denominator
and the denominator of f_0. Tate's extension theorem (Tate 1967, Theorem 4)
extends the resulting integral generic-fiber map of p-divisible groups over R.
Faithfulness of the special-fiber Dieudonne functor identifies its reduction
with the map induced by N f_0. Here equality after inverting p is already
equality of integral maps, because the integral Dieudonne modules are torsion
free. The integer N is fixed for the rest of the construction.
The specified-map form of Serre-Tate deformation theory now gives a lift over
each R / m_R^n (Grothendieck 1970, p. 432). One can also deduce it from the
fixed-object formulation (Illusie 2015, Theorem 2.3.3) as follows. On
X_0 x Y_0 form the shear s_0(x,y) = (x, y + N f_0(x)). The lifted
p-divisible-group map gives the same shear on the product p-divisible group.
View the existing product deformation with two special-fiber markings differing
by s_0. Full faithfulness lifts the shear to an algebraic isomorphism between
these marked deformations; its second component restricted to X lifts N f_0.
Uniqueness makes these maps compatible at every finite order.
Formal existence algebraizes the resulting formal map (Stacks Project, Tag
0A42): its source is proper over the complete noetherian ring R, and its target
is separated and of finite type. Dividing by the fixed N gives the desired
rational lift. Rigidity gives uniqueness and injectivity of specialization on
rational homomorphisms (Milne 1986, Proposition 20.1 and Corollary 20.2).
If f_0 is symmetric, injectivity shows its lift is symmetric. The usual
identification
NS(X_K) tensor Q = Hom(X_K, X_K^dual)^sym tensor Q
gives the rational divisor class, after a finite extension if necessary (Milne
1986, Proposition 17.2). Finally an identity between the maps specializes to
the given identity, so injectivity proves it upstairs as well. []
Only prescribed rational algebraic maps on the special fiber are being lifted
in Lemma 2.3. An arbitrary Q_p-linear map of p-divisible groups is NOT asserted
to be a rational algebraic map. We also use the extension of generic-fiber
homomorphisms between abelian schemes over a discrete valuation ring: their
Neron mapping property extends such a homomorphism uniquely over the base
(Conrad 2003, Definition 5.1 and Example 5.3). Clearing one rational
denominator therefore makes specialization of these maps well defined.
================================================================================
2.2 Transport through Hodge-type integral models
================================================================================
After taking a power of the original abelian variety, we transport a rational
Hodge class to an auxiliary variety by ONE quasi-isogeny on special fibers, with
the same equality in every realization. The auxiliary variety can be chosen
either of CM type, or with a prescribed admissible Hodge filtration. These two
choices serve different parts of the proof but share the tensor construction and
the rational normalization of Proposition 2.4.
Throughout this section, a polarization on a rational homology space is written
as a rational alternating form, using the fixed Betti trivialization of the Tate
line. If its similitude character is c, an alternating form xi on the 2r-fold
product V^{2r}, of MULTIPLIER c^r, means
xi(g v_1, ..., g v_{2r}) = c(g)^r xi(v_1, ..., v_{2r}). (2.2)
Such a form is regarded as a tensor with its codimension-r Tate twist. All
homomorphisms and polarizations may be rational: an isomorphism in the isogeny
category induces the usual isomorphisms on cohomology, including its tensor
constructions.
Filtered isocrystal data have FINITE LOCAL-FIELD DESCENT if they are obtained by
scalar extension from an isocrystal, with its tensors, over the maximal
unramified subfield H_0 of a finite extension H/Q_p, together with a filtration
over H. The scalar extension is to K0 H. This condition lets us apply weak
admissibility over a finite local field. For a group in a faithful
representation, a TENSOR-PRESERVING IDENTIFICATION means one respecting a finite
family of rational tensors whose stabilizer is that group, together with their
comparison realizations.
PROPOSITION 2.4 (Transport). Let A/Q-bar be an abelian variety of positive
dimension with good reduction at w, and put V = H_1(A(C),Q). Suppose
G subset GSp(V,psi)
is connected reductive, its Hodge homomorphism h gives the Hodge structure of A,
and its conjugacy class X makes this inclusion a Hodge embedding. Let gamma be a
rational Hodge class of codimension r, represented on V by an alternating form
satisfying (2.2).
Then there exist a positive integer n, an auxiliary good-reduction abelian
variety B, a quasi-isogeny a: A_0^n -> B_0, and a rational Hodge class kappa of
codimension r on B such that, for the first-factor inclusion
j: A_0 -> A_0^n,
(a j)* kappa_0 = gamma_0. (2.3)
The equality holds simultaneously in every ell-adic realization with ell != p,
and in the crystalline realization after extension to Cw. Either of the
following two further specifications can be imposed.
(1) The variety B is of CM type and is defined over Q-bar, with its indicated
good reduction at w.
(2) Put D_A = H^1_cris(A_0/W(F))[1/p]. Suppose F_A is a Hodge filtration in
degrees 0,1 on D_A, over a finite extension of K0, which is weakly
admissible and has finite local-field descent together with the
specialized tensors specifying the G-structure. Suppose also that, after
a geometric tensor-preserving identification, it lies in the G-orbit of
the original Hodge filtration. Then B can be defined over a finite
extension K/K0 inside Cw, and a can be chosen so that
a* Fil^1 H^1_dR(B/K) = (F_A^1)^{direct sum n}. (2.4)
The equality is made after a common finite extension and a
crystalline-de Rham identification. To discuss Hodge classes on this B,
choose a complex embedding of an algebraic closure of K extending the
fixed embedding of Q-bar.
In the second case, the realizations of kappa are supplied by the canonical tensor
sections of the Hodge-type model and the polarization. In the first case, these
agree with the usual realizations and specialization of its absolute Hodge class.
Two ingredients prepare the proof. First, we enlarge the given Hodge datum so
that integral-model theorems apply at every prime, while its rational
representation remains a sum of copies of V. Second, we realize a prescribed
admissible filtration by choosing a framed local Shimura point above its
crystalline period and applying integral uniformization; the resulting abelian
lift retains that point's integral tensors and special-fiber marking. After
these preparations, the proof builds the transported tensor and normalizes it
using rational divisor intersections.
--------------------------------------------------------------------------------
2.2.1 An auxiliary strongly admissible datum
--------------------------------------------------------------------------------
We recall the hypotheses on integral models that we will use. For a reductive
group H_p over Q_p, a FULL BRUHAT-TITS STABILIZER is the smooth group scheme
fixing a point of its extended building; its identity component is the
associated parahoric group scheme. Now let (H, X_H) be a Shimura datum over Q.
In the terminology of (Kisin-Zhou 2025, Definition 2.1.6), the triple
(H, X_H, H-script) is STRONGLY ADMISSIBLE if (H, X_H) is of Hodge type, H-script
is a full stabilizer of H_{Q_p} with connected special fiber, the centralizer of
a maximal K0-split torus of H_{K0} is R-smooth, and the relative root system over
K0 is reduced when p = 2. Here R-smoothness is the condition on tori used in
(Kisin-Zhou 2025a, Section 2.4); we will use it only for products and extensions
of restrictions of scalars of split tori, for which it holds by (Kisin-Zhou
2025a, Proposition 2.4.6).
LEMMA 2.5 (Auxiliary Hodge datum). Let
(G, X) -> (GSp(V,psi), X_Siegel)
be a Hodge embedding, with G connected and similitude character c. For every
prime p there exist a totally real field P, a connected reductive group J, and a
Hodge embedding
(J, X_J) -> (GSp(W, psi_W), X'_Siegel)
with the following properties.
(1) There are inclusions of Shimura data induced by
G -> J -> Res_{P/Q} G_P,
where the first map is diagonal and J is the subgroup on which the
similitude multipliers in the restriction of scalars agree.
(2) The group J_{Q_p} is quasi-split and has a full connected stabilizer
J-model making (J, X_J, J-model) strongly admissible.
(3) As a rational J-representation,
W = (V tensor_Q P)^{direct sum m} (2.5)
for some positive integer m. The form psi_W has multiplier the common
character c, and J-model -> GL(W_{Q_p}) extends to a closed immersion
into GL(Lambda_p) for a self-dual Z_p-lattice Lambda_p.
Proof. Choose a finite extension of Q_p splitting G_{Q_p}. Polynomial
approximation, together with Krasner's lemma (Milne 2020b, Propositions
7.60-7.61 and Corollary 7.62), gives a totally real number field P whose
completions at p split G_{Q_p}. For example, approximate the minimal polynomial
of a primitive element of a splitting extension at p, and a polynomial with
distinct real roots at the real place. The resulting number field can have just
one place above p; this extra condition is not needed.
Define the multiplier fiber product
J = (Res_{P/Q} G_P) x_{Res_{P/Q} G_m} G_m, (2.6)
where the last map is diagonal. The Hodge cocharacter, or its inverse according
to convention, pairs to 1 with c. Since c factors through G^ab, its character on
this torus is primitive. Thus ker(G^ab -> G_m) is a torus, and ker(c: G -> G_m)
is connected reductive. The exact sequence
1 -> Res_{P/Q}(ker c)_P -> J --c--> G_m -> 1
proves connectedness and reductivity of J. Moreover
J^der = Res_{P/Q} G_P^der. (2.7)
The action on U = V tensor_Q P, with alternating form Tr_{P/Q} psi_P, gives a
symplectic embedding of J with similitude character c. The diagonal Hodge
homomorphism gives X_J. The adjoint Hodge-type and Cartan-involution axioms
follow componentwise from those for X. We also check that extending the totally
real field introduces no rational adjoint factor on which the Hodge homomorphism
is trivial. Write a rational simple adjoint factor as Res_{L/Q} H, with L
totally real. Every field factor L' of P tensor_Q L is totally real, and every
real embedding of L extends to a real embedding of L'. Hence the existence of a
nontrivial Hodge component for H persists in each resulting rational factor.
Finally the diagonal weight is rational and the trace form is a polarization.
This proves that (J, X_J) is a Shimura datum of Hodge type.
The local conditions are explicit. Put P tensor Q_p = prod_{v | p} P_v and
choose a split maximal torus T_v of G_{P_v}. The character c on T_v has a
cocharacter section: a conjugate of the Hodge cocharacter pairs to 1 with it,
and all cocharacters of a split torus are defined over P_v. The same argument
applies to G_{P_v}^ab. Consequently the maximal torus of J_{Q_p} obtained from
the T_v, and the abelianization of J_{Q_p}, have the forms
T_J = G_m x prod_{v|p} Res_{P_v/Q_p} G_m^{rk G - 1}, (2.8)
J_{Q_p}^ab = G_m x prod_{v|p} Res_{P_v/Q_p} G_m^{dim G^ab - 1}. (2.9)
The split Borels over the P_v give a Borel of J_{Q_p}, so this group is
quasi-split. Its derived group is a product of restrictions of split groups by
(2.7). The vertices from the hyperspecial vertices in those split groups give a
very special parahoric of J_{Q_p}. The inertia coinvariants of the cocharacter
module in (2.9) are torsion-free, since that module is a direct sum of
permutation modules. The full stabilizer is therefore connected by (Kisin-Zhou
2025a, Lemma 4.2.4).
The centralizer of a maximal K0-split torus is obtained from (2.8) by unramified
base extension; its torus factors are still induced split tori and G_m. It is
R-smooth by (Kisin-Zhou 2025a, Proposition 2.4.6). Finally, restrictions of
split root systems give reduced relative root systems, also at p = 2. These are
precisely the strong-admissibility checks in the construction of (Kisin-Zhou
2025, Proposition 5.3.8). They use the stipulated Hodge embedding, not the
assertion that G is the full Mumford-Tate group of a particular point.
Apply (Kisin-Zhou 2025, Proposition 2.1.9), starting with the representation U
just constructed. Its proof first tensors U with an auxiliary totally real
Q-vector space, then takes direct sums, and finally applies the symplectic
self-duality construction. The group acts trivially on that extra vector-space
factor. In the last construction the dual representation twisted by c is
identified with U by its symplectic form. Thus every resulting rational summand
is a copy of U, with the same multiplier. This gives (2.5), together with psi_W
and Lambda_p. The cited proposition includes p = 2; its proof uses the
reduced-relative-root condition through (Kisin-Zhou 2025, Lemma 2.1.8). []
Choose a sufficiently small neat prime-to-p level K^p and put K_p = J-model(Z_p).
The adapted embedding gives the integral model S_K and its universal abelian
scheme. A rational Z_(p)-lattice can be chosen with completion Lambda_p. We use a
finite family of tensors defining its stabilizer J-model, as in (Kisin-Zhou 2025,
Section 2.1.4). Any additional rational J-invariant tensors may be included in
this family after clearing denominators. Indeed, their integral stabilizer
condition already holds on the generic fiber; flatness of J-model makes it hold
on J-model itself. Adding them therefore changes neither the group scheme nor
its tensor trivializations.
The added tensors have compatible sections in every realization, as follows. Use
the homological degree-one realizations in this paragraph, that is the duals of
the corresponding H^1, so that their reference representation is W. For such a
realization H_nu over its coefficient field k_nu, let
P_nu = Isom_{s_alpha}(W tensor k_nu, H_nu)
be its torsor of frames preserving the original defining tensors. For a rational
J-invariant tensor u and a frame e, set u_nu = e(u). Two frames differ by an
element of J, so this definition is independent of the frame and descends from a
trivialization of the torsor. It also works integrally after clearing the fixed
denominator of u.
The Betti and etale frame torsors are supplied by the canonical tensor sections
of (Kisin-Zhou 2025, Section 2.1.4). The construction of the universal
Breuil-Kisin-Fargues module and its structure tensors gives the crystalline
torsor and its compatibility with those sections (Kisin-Zhou 2025, Sections
2.1.13-2.1.14). Indeed, a comparison map carrying every s_alpha to its specified
realization carries a frame e to the comparison of e; it therefore carries e(u)
to the same tensor evaluated in the new frame. Thus it preserves u_nu for every
added invariant u, without requiring u to be an expression in the chosen defining
tensors. At a lift over Q-bar, these are the realizations of its rational Hodge
tensor, by Lemma 2.1. Their crystalline specializations are independent of the
lift because they are evaluations in the fixed crystalline frame torsor.
The same observation applies to isogenies. One quasi-isogeny preserving the
defining tensors identifies their frame torsors in every realization in which
those tensors are preserved. It therefore preserves all the added invariant
tensors as well. This proves the tensor comparison and the simultaneous
preservation used below; flatness alone is used only to keep the integral
stabilizer unchanged.
Specification (1) of Proposition 2.4 uses the special-point theorem (Kisin-Zhou
2025, Theorem 2.2.7) directly. The next lemma is needed only for specification
(2), where the filtration has been prescribed in advance.
--------------------------------------------------------------------------------
2.2.2 Realizing a prescribed admissible filtration
--------------------------------------------------------------------------------
For a point x in S_K(F), let C_x be the corresponding abelian variety and put
D_x = H^1_cris(C_x/W(F)), D_x = D_x[1/p].
The crystalline structure tensors are denoted s_{alpha,0,x}; the tensors away
from p are denoted s_{alpha,ell,x}. They are the specializations of the
canonical tensor sections and are independent of the chosen lift of x.
The TENSOR ISOGENY CLASS I_x consists of the points y in S_K(F) for which a
single quasi-isogeny C_x -> C_y preserves s_{alpha,0} and every s_{alpha,ell}
for ell != p. This definition concerns actual quasi-isogenies of abelian
varieties. It is the class defined in (Kisin-Zhou 2025, Section 2.2.4).
LEMMA 2.6 (Realization of an admissible filtration). Let (J, X_J, J-model) be
strongly admissible, with J_{Q_p} quasi-split, and choose the adapted embedding
and neat level above. Let x in S_K(F) have a lift to characteristic zero.
Suppose a filtration F on D_x, after a finite extension of K0, has the following
properties:
(1) the resulting filtered isocrystal is admissible;
(2) geometrically its filtration is induced by a J-valued cocharacter in the
conjugacy class prescribed by X_J.
Then, after extending that field finitely, there are y in I_x, a
tensor-preserving quasi-isogeny b: C_x -> C_y, and a lift y~ over the extended
valuation ring such that
b* Fil^1 H^1_dR(C_{y~}) = F^1,
under the crystalline-de Rham identification.
In particular, condition (1) follows if the filtered isocrystal and its tensor
data have finite local-field descent and F is weakly admissible.
Proof of Lemma 2.6. Write k = F and let E^breve be the completed local reflex
field. We work over a common finite extension K of K0 containing E^breve and a
field of definition of F. Every further field extension below is finite. Put
K_p = J-model(Z_p).
First fix the relation between the group and cohomological conventions. Let rho:
J_{Q_p} -> GL(W_{Q_p}) be the adapted homological representation and rho^vee
its dual. Choose the tensor-preserving identification D_x = Lambda_p^vee
tensor_{Z_p} W(k) of (Kisin-Zhou 2025, Section 2.1.14). The local shtuka
parameter is b_x in J(K0), whereas crystalline Frobenius on this cohomological
Dieudonne module is
F_x = rho^vee(sigma(b_x)) o sigma.
The additional sigma on b_x comes from the comparison with the vector shtuka:
its special-fiber module is (sigma^{-1})*(D_x^vee) (Pappas-Rapoport 2024,
Example 2.3.4 and Remark 2.3.10). Write {mu} for the class prescribed by X_J;
then [b_x] lies in B(J, mu^{-1}).
For precision, identify the general linear crystalline period flag with the
appropriate filtration variety on D_x by its marked Rapoport-Zink
interpretation. If a deformation H has special-fiber marking q: H_k -> C_x[p^inf],
this identification sends its period to (q*)^{-1} Fil^1 H^1_dR(H). Pushout
through the faithful Hodge representation gives the commutative period square
M_{J,b_x,mu,K_p} --> M_{GL,rho(b_x),rho(mu)}
| pi_GM | pi_GM
v v
F_{J,mu} --> Flag(D_x).
The lower map is a closed immersion: the parabolic stabilizing the faithful
filtration intersects J in the parabolic for mu. Define f_F to be the unique
J-flag with image F. It exists by condition (2). The square uses crystalline
periods and the dual Frobenius descent above, so no extension of sigma to the
ramified filtration field is chosen. When q = b^{-1} its lower right period is
exactly b* Fil^1.
A FRAMED POINT OVER THE PRESCRIBED PERIOD. Admissibility of the faithful
filtered isocrystal implies that the image of f_F in the general linear period
flag variety is in its admissible locus. To see this in the cohomological
coordinates of the square, let U be a crystalline representation realizing
(D_x, F). Its crystalline comparison identifies the associated vector bundle
away from the untilt with U tensor O. The filtered de Rham comparison identifies
the modification at the untilt with the standard lattice U tensor B_dR^+. They
glue to the trivial bundle U tensor O on the Fargues-Fontaine curve, which is
the general linear admissibility condition. Since rho is faithful,
rho^{-1}(Z_GL(W)) is contained in Z_J: an element acting as a scalar commutes
with the faithful image of J. The admissible-locus identity in
(Pappas-Rapoport 2024, Proposition 3.1.1(i), 3.1.9) therefore puts f_F in
F_{J,mu}^adm.
This locus is the image of the crystalline period map
pi_GM: M_{J,b_x,mu,K_p} -> F_{J,mu}^adm.
The map is etale (Pappas-Rapoport 2024, Section 3.1.1, 3.1.4). Its fiber at the
classical point f_F is nonempty; an etale rigid space over K has a point over a
finite extension of K. After making that extension, choose
z in M_{J,b_x,mu,K_p}(K), pi_GM(z) = f_F.
In particular z supplies the integral level as well as the rational isocrystal
marking. If C is the completed algebraic closure of K, its moduli description
gives an actual J-torsor P_z, a Frobenius isomorphism bounded by mu, and, for
sufficiently large a, an isomorphism
i_a: J_{Y_{[a,inf)}(C^flat)} ~ P_z|_{Y_{[a,inf)}(C^flat)},
varphi_{P_z} o varphi* i_a = i_a o varphi_{b_x}.
These are precisely the torsor and annular framing in (Pappas-Rapoport 2024,
Section 3.1.1). Its associated pro-etale J(Z_p)-torsor gives integral tensor
frames. Thus the integral structure and Frobenius-compatible framing are part of
the chosen point, with all their tensor compatibilities.
INTEGRAL REALIZATION AND ITS SPECIAL FIBER. Our adapted datum is of global
Hodge type: the group scheme is a full connected Bruhat-Tits stabilizer, and the
chosen Hodge embedding is integral on Lambda_p. The center also satisfies the
split-rank hypothesis of (Pappas-Rapoport 2024, Section 4.1). Indeed, its real
subgroup of multiplier one preserves the positive definite form
psi_W(v, h(i) w) and is compact. Its real split rank is therefore at most one,
while the rational weight torus supplies one split central direction over Q;
both split ranks are one. The model here is the one in (Kisin-Zhou 2025,
Section 2.1.11 and Theorem 2.1.12), obtained from this adapted Hodge embedding
and satisfying the Pappas-Rapoport characterization. Condition (U_x) holds by
(Gleason-Lim-Xu 2026, Corollary 6.3); see also (Pappas-Rapoport 2024, Remark
4.10.4). Consequently (Pappas-Rapoport 2024, Theorem 4.10.6) represents the
integral local Shimura variety by a normal formal scheme M_{J,b_x,mu} and gives
the morphism
Theta_x: M_{J,b_x,mu} -> S_K-hat.
More precisely, (Pappas-Rapoport 2024, Section 4.10.2, 4.10.2-4.10.6) identifies
this formal scheme with the tensor-compatible open and closed part of the
pullback of the Siegel Rapoport-Zink space. This construction identifies the
pullback of the universal J-shtuka with its tautological framed shtuka, and the
faithful vector realization with the universal p-divisible group.
Its rigid generic fiber is the local Shimura variety (Pappas-Rapoport 2024,
Proposition 3.5.1(i)). The classical point z of this rigid generic fiber gives
Spf O_K -> M_{J,b_x,mu}. Indeed it lies in the generic fiber of a formal affine
chart; for a special formal chart one may take an affinoid member of its standard
generic-fiber covering containing z. Evaluation of the formal chart's ring is
integral, and its ideal of definition maps to topologically nilpotent elements of
O_K. Let z_0 be the special point and put y = Theta_x(z_0) and y~ = Theta_x(z).
The formal map y~ gives a point of the integral model over O_K by restricting to
an affine neighborhood of y. Pullback of the universal abelian scheme gives
C_{y~} and its p-divisible group H_z.
We record the integral comparison at this point. For a uniformizer pi of K, put
S = W(k)[[u]], with u |-> pi, and let E_pi(u) generate its kernel. The
crystalline representation and integral level of z give the J-Breuil-Kisin torsor
of (Pappas-Rapoport 2024, Section 3.5), with
Phi: varphi* P_BK[1/E_pi] ~ P_BK[1/E_pi].
Its pushout to GL(Lambda_p) is the Breuil-Kisin module of H_z. Reduction at u = 0,
followed by the dual Frobenius descent fixed above, therefore gives the actual
Dieudonne lattice of H_{z,k}. (Pappas-Rapoport 2024, Proposition 3.5.1) compares
the two specializations of this already constructed framed point. In the
coordinates of (Kisin-Zhou 2025, Theorem 2.2.5) it gives
g in X_J(mu^{-1}, b_x), D(H_{z,k}) = rho^vee(sigma(g)) D_x subset D_x.
Indeed, the Frobenius matrix on this lattice, written in its induced basis, is
rho^vee(sigma(g^{-1} b_x sigma(g))) o sigma,
which explains the same Frobenius descent in the Deligne-Lusztig condition and in
the lattice formula.
THE ABELIAN MARKING AND THE FILTRATION. Keep the identity prime-to-p coordinate
in the uniformization. The description of its special fiber in
(Pappas-Rapoport 2024, Section 4.10.3), or equivalently the isogeny-class map of
(Kisin-Zhou 2025, Theorem 2.2.5), gives y = i_x(g) and a single abelian
quasi-isogeny b: C_x -> C_y preserving the crystalline tensors and every
prime-to-p tensor. Its crystalline marking is the one just used for H_z, because
the uniformization is induced by the marked Siegel Rapoport-Zink space.
The crystalline period map of that space is the Hodge filtration of its universal
deformation pulled back by its marking. Compatibility of Theta_x with the
tautological shtuka, and the equation pi_GM(z) = f_F, consequently give
b* Fil^1 H^1_dR(C_{y~}/K) = F^1.
The deformation criterion (Kisin-Zhou 2025, Proposition 2.2.3) is satisfied
concretely: the filtration is of type mu, and the crystalline structure tensors
compare with the integral etale tensors in the J(Z_p)-frames of z. The bounded
integral shtuka and its annular framing were obtained before using this criterion
or comparing specializations. All cited global Hodge-type and strongly admissible
results apply at p = 2 as well.
Finally, under finite local-field descent, the weak-admissibility inequalities
restrict to the descended Frobenius subobjects. Colmez-Fontaine's theorem
(Colmez-Fontaine 2000, Theorem A) makes those descended data admissible, and
scalar extension supplies condition (1). []
--------------------------------------------------------------------------------
2.2.3 Proof of the transport proposition
--------------------------------------------------------------------------------
Proof of Proposition 2.4.
THE ORIGINAL POINT AND ITS REDUCTION. Apply Lemma 2.5. Choose a rational
J-equivariant identification W = (V tensor_Q P)^{direct sum m} and set
n = m [P:Q]. Choose a basis of P whose first vector is 1. The resulting
identification of the Hodge structure at the diagonal point h with V^{direct sum n}
makes the first-factor map equal to
v |-> (v tensor 1, 0, ..., 0). (2.10)
The rational homology/isogeny correspondence (Milne 2005, Corollary 6.9)
therefore makes the abelian variety C at that point isogenous to A^n. Fix a
rational isogeny f: A^n -> C realizing this identification.
We explain the field and reduction assertions for this starting point. The Hodge
embedding maps it to the Siegel point of the polarized variety C with finite
level. The abelian variety C is defined over Q-bar: clearing a denominator in f
realizes it as a quotient of A^n by a finite subgroup of its torsion, and every
such subgroup is defined over Q-bar. Its rational polarization and its finite
level also descend to Q-bar, since homomorphisms between abelian varieties over
Q-bar do not enlarge after extending algebraically closed fields (Milne 1986,
Corollary 20.4). The Hodge embedding induces a finite morphism onto its image at
sufficiently small finite level (Milne 2005, Theorem 5.16). Its fiber over a
Q-bar-point therefore consists of Q-bar-points. This places the diagonal point on
the canonical model over Q-bar.
After a finite number-field extension the variety C has good reduction, since it
is isogenous to A^n (Milne 2008, Chapter IV, Theorem 3.5). Its polarization and
prime-to-p level extend after a further finite extension. The integral Hodge-type
model is the normalization of the closure of the generic fiber in the adapted
Siegel model. The valuative criterion for this finite normalization therefore
extends our point to its integral model. Denote the reduction by x. The rational
isogeny f specializes to a quasi-isogeny f_0: A_0^n -> C_x.
THE TRACE TENSOR AND ITS DESCENT. Extend the alternating form defining gamma
P-multilinearly and take its trace. Composing with projection onto the first copy
in (2.5) gives
eta = Tr_{P/Q}(gamma_P) o (pr_1)^{tensor 2r}
in wedge^{2r} W^*. (2.11)
For r = 0 this is interpreted as the scalar trace. The equal-multiplier condition
implies eta has multiplier c^r under J. On (2.10), its restriction is
j* f* eta_C = [P:Q] gamma. (2.12)
The class eta_C is understood with its Tate twist.
We turn this semi-invariant into the untwisted tensors to which the integral-model
theorems directly apply. Write psi_W^{-1} in wedge^2 W for the inverse symplectic
bivector, and include
u_psi = psi_W tensor psi_W^{-1},
u_eta = eta tensor (psi_W^{-1})^{tensor r} (2.13)
among the structure tensors. Both are J-invariant tensors in ordinary tensor
constructions of W and W^*, requiring no separately trivialized Tate object. They
can be made integral and added to the defining family as explained above. Their
crystalline and prime-to-p realizations at x are the specializations from the
original point, with the comparison conventions fixed earlier.
The entire finite family of structure tensors, including u_psi and u_eta, has
finite local-field descent. To see this, choose a finite extension L/Q_p over
whose valuation ring the original number-field point is defined, and let L_0 be
its maximal unramified subfield. The canonical etale tensor sections of
(Kisin-Zhou 2025, Section 2.1.4), pulled back to that point, are
Gal(L-bar/L)-invariant. Applying the tensor functor D_{cris,L} to these morphisms
from the unit object places them in the crystalline tensor spaces over L_0. Their
extension to K0 is exactly the family s_{alpha,0,x}, by good-reduction comparison.
The map f_0 also descends after a finite residue-field extension, so this descent
is retained under its marking. Taking a compositum with the finite field of
definition of a prescribed filtration therefore gives one finite local field for
all the data. No tensor-preserving trivialization over L_0 is being asserted or
needed.
THE TWO CHOICES OF LIFT. For specification (1), apply (Kisin-Zhou 2025, Theorem
2.2.7) to x. It gives a point y in I_x with a special-point lift and a
tensor-preserving quasi-isogeny b: C_x -> C_y. The special-point lift supplies a
CM abelian variety B. Special points on the canonical Shimura variety are
algebraic over its reflex field by the canonical-model theory (Milne 2005,
Definition 12.8 and Proposition 14.14), so B and its level data can be taken over
Q-bar. The given local lift specifies their embedding at w and their good
reduction B_0 = C_y.
For specification (2), identify D_x with D_A^{direct sum n} using f_0^*, and
prescribe the direct-sum filtration F_A^{direct sum n}. It is weakly admissible,
with the required finite local-field descent. It is also in the required
J-cocharacter orbit. Indeed a geometric element of the transported G taking the
original filtration on D_A to F_A acts diagonally on all the copies of V in W;
its image in J takes the original filtration on D_x to the prescribed one. The
tensor-preserving identifications used here come from the initial point and
comparison, so they include the J-structure tensors as well. Geometric conjugacy
can be realized over a finite extension, since the corresponding isomorphism
scheme is of finite type. Lemma 2.6 now gives y, its lift B, and a
tensor-preserving quasi-isogeny b: C_x -> B_0 with the prescribed marked
filtration. This proves (2.4) for a = b f_0.
A COMMON RATIONAL NORMALIZATION. It remains, in both specifications, to recover
the twisted class and normalize it by a single rational scalar. Choose a complex
description of the lifted Shimura point. Its rational marking by W supplies a
rational Hodge class eta_B corresponding to eta, and a rational divisor class
theta_B corresponding to psi_W. Write theta_C for the analogous class on C and
put
theta_* = f_0^* theta_{C,0} in NS(A_0^n) tensor_Z Q.
The class theta_* is rational ample. Tensor preservation of u_psi says that, in
each realization,
a* theta_{B,0} = lambda_v theta_*, (2.14)
with a nonzero local scalar lambda_v. Let D = n dim A. Pairing with
theta_*^{D-1} gives
lambda_v = deg(a* theta_{B,0} . theta_*^{D-1}) / deg(theta_*^D)
=: lambda in Q^*. (2.15)
Both intersection numbers refer to rational divisor classes on the same abelian
variety; their cycle-class traces are identical in every realization. The
denominator is positive because theta_* is ample, and the numerator is nonzero by
(2.14). Thus all the scalars in (2.14), including the crystalline one, are the
single rational number lambda.
Preservation of u_eta, together with (2.15), then yields
a* eta_{B,0} = lambda^r f_0^* eta_{C,0} (2.16)
simultaneously. In fact the inverse bivector scales by lambda^{-1}, so that the
equality for the untwisted tensor in (2.13) gives exactly (2.16). Set
kappa = (lambda^{-r} / [P:Q]) eta_B.
Specializing (2.12) and using (2.16) proves (2.3).
For the arbitrary local lift, the crystalline realization just used is
unambiguous: the canonical model supplies the untwisted tensor u_eta, and the
algebraic polarization supplies its Tate line and inverse bivector. The comparison
isomorphism and these operations define eta_B in the crystalline realization and
agree with its canonical etale tensors. Consequently any identity expressing this
class by algebraic divisor operations specializes with the same rational
coefficients in all realizations. For the Q-bar-valued CM lift, these are also the
usual absolute-Hodge and de Rham realizations. This completes the two
constructions. []
COROLLARY 2.7 (CM transport of a Hodge class). Let A/Q-bar be an abelian variety
of positive dimension with good reduction at w, and let gamma in
H^{2r}(A(C),Q(r)) be a rational Hodge class, where 0 <= r <= dim A. There are a
positive integer n, a CM abelian variety B/Q-bar with good reduction at w, a
rational Hodge class kappa of codimension r on B, and a quasi-isogeny
a: A_0^n -> B_0 such that
gamma_0 = (a j)* kappa_0,
where j: A_0 -> A_0^n is the first-factor inclusion. This is an equality of the
usual absolute-Hodge specializations in every prime-to-p realization and in
crystalline cohomology extended to Cw.
Proof. Choose a polarization psi on V = H_1(A(C),Q), and let G = MT(V). The
Mumford-Tate group is connected reductive and is characterized by its Hodge
tensors (Deligne 1982, Section 3, Propositions 3.4 and 3.6). The Hodge
homomorphism and its G(R)-conjugacy class define the Mumford-Tate Shimura datum
of Hodge type, with faithful symplectic representation
G -> GSp(V, psi).
The polarization supplies the rational weight and the Cartan involution
condition. The definition of the Mumford-Tate group excludes an adjoint factor on
which the Hodge homomorphism is trivial. The class gamma, with its Tate twist, is
fixed by G. Equivalently, its alternating form on V has multiplier c^r, where c is
the similitude character. Thus the first alternative of Proposition 2.4 applies
and gives the asserted data and simultaneous equality. []
================================================================================
3. AN ENDOMORPHISM-STABLE WEAKLY ADMISSIBLE FILTRATION
================================================================================
We construct a filtration on first crystalline cohomology that preserves an
additional divisor. The construction takes place entirely on the given
isocrystal; its realization by an abelian lift is a separate step (Lemma 2.3). The
classical ingredient of admissible-filtration existence theorems is generic
transversality (Fontaine-Rapoport 2005, Theorem 1 and Lemmas 1.1-1.2). We impose
the additional endomorphism and polarization constraints explicitly. The
bounded-degree argument below chooses the required coordinates over a finite
algebraic extension, while testing every subisocrystal over the fixed coefficient
field.
Throughout this section K0 = Frac W(F-bar), and valuations are normalized by
v(p) = 1. If D is an isocrystal over K0 and K/K0 is finite, a subspace L subset
D_K defines the one-step filtration
Fil^i D_K = D_K for i <= 0,
L for i = 1,
0 for i >= 2.
For an isocrystal subobject P subset D, its Hodge number for this filtration is
t_H(P) = dim_K(P_K cap L). Its Newton number t_N(P) is the sum of its slopes with
multiplicities. Weak admissibility means t_H(D) = t_N(D) and t_H(P) <= t_N(P) for
every isocrystal subobject P over K0.
Finite local-field descent is understood as in Section 2.2: the isocrystal and its
indicated tensors descend to the maximal unramified subfield of a finite extension
of Q_p, and the filtration descends to that finite extension.
THEOREM 3.1 (The filtration construction). Let A_0/F-bar be an abelian variety
equipped with a rational action of a CM field E and a polarization whose Rosati
involution restricts to conjugation on E. Suppose that
D = H^1_cris(A_0/W(F-bar))[1/p]
has rank 2s over E tensor_Q K0, where s >= 1. Suppose also that there is an
invertible u in End^0(A_0) such that
u^dagger = u, u e = conj(e) u for all e in E.
Let beta be the dual polarization pairing on D, with values in the isocrystal line
of slope 1.
Then there are a finite extension K/K0 and a subspace L subset D_K with the
following properties:
(1) The one-step filtration with Fil^1 D_K = L is weakly admissible.
(2) L is stable under the cohomological actions of E and u, and is Lagrangian
for beta.
(3) After extending to an algebraically closed field Omega/K, one has
dim_Omega L_tau = s for every embedding tau: E -> Omega.
(4) The isocrystal, its indicated tensors, and this filtration have finite
local-field descent.
Moreover, geometrically L belongs to the balanced polarized cocharacter orbit.
More explicitly, define
G_D = { g in GSp(D,beta) cap GL_{E tensor K0}(D) :
det_{E tensor K0}(g) = c(g)^s },
where the scalar on the right is viewed in (E tensor K0)^x. The balanced
E-stable beta-Lagrangians form a single G_D(Omega)-orbit. Each is the weight-1
space of a cocharacter of G_{D,Omega} with weights 0,1. In particular L belongs to
the orbit of any original filtration of this type under a tensor-preserving
comparison.
The proof constructs L from Lagrangians in finitely many symplectic spaces. The
two lemmas below supply the required simultaneous choice: minimum intersection
with a subspace is ensured by a nonzero minor of bounded degree, and all such
minors over a fixed coefficient field can be made nonzero over one finite
extension.
LEMMA 3.2 (Bounded-degree avoidance over a finite extension). Let K_1/K0 be a
finite extension, and let r, b >= 1 be integers. There exist an element theta
algebraic over Q_p and a point
(x_0, ..., x_{r-1}) in K_1(theta)^r
at which every nonzero polynomial in K_1[X_0, ..., X_{r-1}] of total degree at
most b is nonzero. The coordinates x_i can all be chosen in Q_p(theta).
Proof. Choose an integer N > b and put
B = b sum_{i=0}^{r-1} N^i.
Let e be the ramification index of K_1/K0, so that v(K_1^x) = (1/e) Z. Choose a
prime q > max(B, e), and choose theta with theta^q = p in an algebraic closure of
K_1. The value v(theta) = 1/q has order q modulo (1/e) Z. Consequently the
ramification index, and hence the degree, of K_1(theta)/K_1 is at least q. The
polynomial X^q - p gives the reverse inequality, so
[K_1(theta) : K_1] = q > B.
Set x_i = theta^{N^i}. Under the substitution X_i = T^{N^i}, distinct monomials
of total degree at most b have distinct exponents: each exponent of an individual
variable is smaller than N, and uniqueness follows from base-N expansion. Thus a
nonzero polynomial of total degree at most b becomes a nonzero polynomial in T of
degree at most B. It cannot vanish at theta, whose degree over K_1 exceeds B. []
REMARK. The coefficient field in Lemma 3.2 is the fixed field K_1. The conclusion
concerns all bounded-degree polynomials over that field, with no restriction on
their cardinality. It would be false with K_1(theta) in place of K_1 as the
coefficient field.
LEMMA 3.3 (Intersections with Lagrangians). Let M be a symplectic space of
dimension 2m over a field k of characteristic zero, and let W subset M have
dimension a. A nonempty open subset of the Lagrangian Grassmannian consists of
subspaces L such that
dim(L cap W) = max(0, a - m).
On any fixed symplectic affine big cell, at least one nonzero polynomial of degree
at most m has nonvanishing locus contained in this open subset. This degree bound
is independent of W.
Proof. We first work over an algebraic closure. The Lagrangian Grassmannian is
geometrically integral of dimension m(m+1)/2; its affine big cells are parametrized
by symmetric m x m matrices. If 1 <= a <= m, consider the projective incidence
variety of pairs (ell, L) with ell subset W a line and ell subset L a Lagrangian.
The fiber over ell is the Lagrangian Grassmannian of the symplectic space
ell^perp/ell, and therefore the incidence variety has dimension
a - 1 + m(m-1)/2 < m(m+1)/2.
Its image is a proper closed subset. Its complement consists precisely of the
Lagrangians missing W. The case a = 0 is immediate. For a > m, use
dim(L cap W) = a - m + dim(L cap W^perp)
and apply the preceding argument to W^perp.
For the assertion on equations, choose coordinates on the fixed big cell in which
L is the column space of
( I_m ; T ), T = T^transpose.
The minimum-intersection condition says that the composite of this matrix with the
quotient map M -> M/W has rank min(m, 2m - a). Its entries are affine linear
polynomials in the independent entries of T. The nonempty open subset already
proved meets this big cell, so some minor of the required size is a nonzero
polynomial over k. Its degree is at most m, and its nonvanishing implies the
required rank. If the required rank is zero, use the constant polynomial 1. []
--------------------------------------------------------------------------------
3.1 Proof of the filtration theorem
--------------------------------------------------------------------------------
Proof of Theorem 3.1. We write e and u also for their cohomological actions on
D. These actions commute with Frobenius and satisfy
beta(u x, y) = beta(x, u y),
beta(e x, y) = beta(x, conj(e) y),
u e = conj(e) u. (3.1)
There is no change to the last relation from contravariance. Indeed, pulling back
u e = conj(e) u gives e* u* = u* conj(e)*; replacing e by conj(e) gives
u* e* = conj(e)* u*. Likewise, if J is the homological polarization matrix,
U^transpose J = J U implies U J^{-1} = J^{-1} U^transpose, which is the
self-adjointness of the cohomological action U^transpose for the dual pairing. The
Rosati-conjugate adjoint identity for E follows the same way.
The pullback U^transpose of an individual endomorphism is distinct from the
contragredient group action g |-> g^{-transpose}. If a homological similitude g has
multiplier c_V and satisfies det_E(g) = c_V^s, its dual action has multiplier
c_D = c_V^{-1} and satisfies det_E(g^{-transpose}) = c_D^s. Thus G_D is the dual
form of the homological group defined by this determinant condition, with its
cohomological multiplier.
REDUCTION TO THE STABLE SUBOBJECTS. First consider any E,u-stable candidate
L subset D_K, where K/K0 is finite. We show that the weak-admissibility
inequalities need only be tested on E,u-stable subobjects. This resembles the use
of canonical destabilizing objects in (Fontaine-Rapoport 2005, Section 2), but we
use total defect, not defect divided by dimension, and give the required
elementary argument. For any isocrystal subobject P subset D over K0, set
f(P) = t_H(P) - t_N(P).
These values form a finite set: the Hodge number is a bounded integer, and the
Newton number is determined by bounded integer multiplicities among the finitely
many slopes of D. Let a be their maximum. Newton numbers are additive in short
exact sequences, while
(P_K cap L) + (Q_K cap L) subset (P + Q)_K cap L.
Consequently
f(P) + f(Q) <= f(P cap Q) + f(P + Q). (3.2)
If P and Q both attain a, the right-hand terms are each at most a, so both attain
a. Thus maximizers are closed under intersection and sum. Choose a maximizer P_max
of largest dimension. Adding any other maximizer to it shows that the other one is
contained in it. Hence P_max is the unique largest maximizer.
The operators u and e in E^x are filtration-preserving isocrystal automorphisms.
They preserve f and hence fix P_max, by its uniqueness. Thus P_max is E,u-stable.
Consequently, if t_H(D') <= t_N(D') for every E,u-stable subobject D', then
a = f(P_max) <= 0. All the weak-admissibility inequalities follow. We must
therefore construct an E,u-stable Lagrangian with these restricted inequalities and
t_H(D) = t_N(D).
SEMISIMPLICITY AND SYMPLECTIC BLOCKS. The Rosati trace form
<a,b> = Tr(a b^dagger)
on End^0(A_0) tensor_Q R is positive definite (Milne 1986, Theorem 17.3). Left
multiplication by u is self-adjoint for this form, since trace cyclicity gives
<u a, b> = Tr(u a b^dagger) = Tr(a b^dagger u) = <a, u b>.
It is therefore diagonalizable. Its minimal polynomial equals the minimal
polynomial of u in the endomorphism algebra, as is seen by applying a polynomial in
the multiplication operator to the identity. Thus u, and hence u^2, acts
semisimply on D. The relation in (3.1) also shows that u^2 commutes with E.
Put
B(x, y) = beta(x, u y).
This is a nondegenerate alternating pairing. Indeed, invertibility is immediate,
and B(y, x) = -beta(u x, y) = -B(x, y). In addition,
B(e x, y) = beta(x, conj(e) u y) = beta(x, u e y) = B(x, e y),
B(u^2 x, y) = B(x, u^2 y).
After a finite splitting extension K_1/K0, write
D_{K_1} = direct sum_{tau, z} M_{tau,z},
M_{tau,z} = { x : e x = tau(e) x (e in E), u^2 x = z x }.
We omit zero summands. The self-adjoint identities for B show that distinct
summands are orthogonal; its restriction to each summand is therefore
nondegenerate. Write dim M_{tau,z} = 2 m_{tau,z}.
For conj(tau) = tau o conj, the operator u maps M_{tau,z} isomorphically to
M_{conj(tau),z}. On these spaces,
B(u x, u y) = z B(x, y). (3.3)
Here z != 0 and tau != conj(tau). Choose a B-Lagrangian L_{tau,z} in one member of
each opposite pair, and prescribe
L_{conj(tau),z} = u L_{tau,z}. (3.4)
Equation (3.3) shows that the latter is Lagrangian as well. Their direct sum L is
E-stable and u-stable; on applying u twice, one obtains z L_{tau,z} = L_{tau,z}.
Since
beta(x, y) = B(x, u^{-1} y),
L is also beta-Lagrangian. Its dimension at each E-embedding is s. It remains to
make the independent choices appropriately.
THE INHERITED NEWTON GRADING. Use the canonical slope decomposition
D = direct sum_lambda D_lambda.
The isocrystal category over F-bar is semisimple, and this decomposition is
functorial (Colmez-Fontaine 2000, Section 3.2, Proposition 3.3). The slopes of D
belong to [0,1]: its integral Dieudonne operators F, V satisfy F V = p, so both
the slopes of F and those of V are nonnegative. All the endomorphisms above
preserve every D_lambda, and the projectors onto the simultaneous eigenspaces
preserve its scalar extension. Thus each block has a grading
M_{tau,z} = direct sum_lambda M_{tau,z,lambda},
M_{tau,z,lambda} = M_{tau,z} cap (D_lambda)_{K_1}.
We use this grading even if Frobenius permutes the labels (tau, z). No extension of
Frobenius to K_1, and no isocrystal structure on an individual block, is required.
Before scalar extension, B is a pairing of isocrystals into the slope-1 line.
Hence B(D_lambda, D_mu) = 0 unless lambda + mu = 1. Nondegeneracy on an individual
block M then gives perfect pairings between M_lambda and M_{1-lambda}. In
particular, if dim M = 2m,
sum_lambda lambda dim M_lambda = m. (3.5)
Let D' subset D be an isocrystal subobject over K0 stable under E and u. Its
intersections M' = D'_{K_1} cap M decompose D'_{K_1} and inherit the same slope
grading. Put a = dim M'. The slope sum of the graded quotient M / M' is at most
2m - a, because all its slopes belong to [0,1]. By (3.5),
sum_lambda lambda dim M'_lambda >= max(0, a - m). (3.6)
The additional bound by zero follows from the nonnegative slopes. The right side is
also the smallest possible dimension of the intersection of a Lagrangian in M with
M'. Thus attaining this minimum in every block will give the weak-admissibility
inequalities for all the stable subobjects.
SIMULTANEOUS CHOICES AND FINITE LOCAL FIELD DESCENT. We may arrange the splitting
field and coordinates used above as follows. Choose a finite field k_0 subset F-bar
over which A_0, its polarization, u, and a rational basis of E are defined. The
underlying isocrystal and the indicated tensors then descend to Frac W(k_0). Choose
a finite local field H/Q_p containing this field and splitting the actions of E and
u^2. The eigenspaces and their symplectic bases can be constructed over H by linear
algebra. We take K_1 = K0 H. In particular, the independent Lagrangian
Grassmannians, fixed symplectic big cells in them, and the maps induced by u all
have coordinates over H.
Take the product of these big cells, one for each independent choice in (3.4), and
let r be its number of affine coordinates. For every E,u-stable isocrystal
subobject D' over K0 and each independent block M, the space M' = D'_{K_1} cap M is
defined over the FIXED field K_1. Lemma 3.3 supplies a nonzero polynomial over K_1,
of degree at most m, whose nonvanishing ensures
dim(L_{tau,z} cap M') = max(0, dim M' - m). (3.7)
We regard this as a polynomial on the whole product by ignoring the other
coordinates. A single bound b = max m_{tau,z} works for all these polynomials,
independently of D'. On the opposite block, the same intersection equality follows
from u-stability of D' and (3.4).
Apply Lemma 3.2 to K_1, r, b. Its one point satisfies (3.7) for every one of the
subobjects just specified, simultaneously. This use of the lemma does not assume
that the collection of subobjects is countable or that its members descend to H.
Their definition over K_1 and the uniform polynomial degree bound are the only
requirements. There are nonzero blocks with m >= 1, so r, b >= 1 as required.
Put K = K_1(theta) = K0 H(theta). The resulting Lagrangian is defined over H(theta)
in the chosen coordinates, and H(theta) is finite over Q_p. Extending the original
isocrystal to the maximal unramified subfield of H(theta) therefore gives finite
local field descent of the complete filtered data. In particular, the construction
has not obtained its generic point by introducing transcendental parameters.
Summing (3.7) and (3.6) over all blocks gives
t_H(D') <= t_N(D') for every E,u-stable isocrystal subobject D' subset D.
(3.8)
For D itself, both numbers equal (1/2) dim D by (3.5). The reduction to stable
subobjects proved at the start now gives all the weak-admissibility inequalities
for subobjects of D over K0. Each subobject over the smaller unramified field of
descent extends to one of these subobjects. Hence the descended filtered isocrystal
is weakly admissible as well.
THE GEOMETRIC COCHARACTER ORBIT. Over Omega, the polarization pairs D_tau only
with D_{conj(tau)}. A balanced E-stable Lagrangian is determined by an
s-dimensional subspace of D_tau for one member of each pair; the subspace in the
opposite member is its annihilator. The subgroup of G_D with multiplier 1 contains
the product of SL(D_tau) acting on opposite spaces by the inverse dual
representations. These special linear groups are transitive on the s-dimensional
subspaces. Thus all the indicated Lagrangians form one G_D(Omega)-orbit.
Finally choose an E-stable beta-Lagrangian complement to one such L. Such a
complement is obtained separately on each pair of opposite embedding spaces. Acting
by t on L and by 1 on the complement gives a cocharacter with multiplier t and
E-determinant t^s at each embedding. It lies in G_D and has the stated filtration.
This proves the final assertions. []
COROLLARY 3.4 (The filtration is admissible). The filtration of Theorem 3.1,
together with its finite local-field descent, is admissible: the descended filtered
isocrystal is the filtered crystalline module of a crystalline representation.
Proof. Apply the weakly-admissible-implies-admissible theorem of Colmez-Fontaine
with monodromy operator zero (Colmez-Fontaine 2000, Theorem A). []
COROLLARY 3.5 (The divisor filtration condition). Let D, L be as in Theorem 3.1,
and let psi and U denote the homological polarization and the homological action of
u on D_K^dual. The untwisted divisor form psi(., U .) belongs to
Fil^1( wedge^2 D_K ) = L wedge D_K,
and hence to filtration degree zero after its codimension-one Tate twist. If a
marked power is realized by an abelian lift, the corresponding pulled-back divisor
satisfies the filtration condition for its symmetric homomorphism in Lemma 2.3.
Proof. Let H subset D_K^dual be the annihilator of L. It is Lagrangian for psi and
stable under U: for a in L and h in H,
a(U h) = (u a)(h) = 0,
using u-stability of L. Therefore psi(h, U h') = 0 for h, h' in H. The divisor form
thus lies in
ker( wedge^2 D_K -> wedge^2 H^dual ) = L wedge D_K.
This proves the assertion on the fixed filtered vector space. After realization of a
marked power, pullback respects this filtration condition, which is precisely the
condition on the associated symmetric homomorphism in Lemma 2.3. []
================================================================================
4. BALANCED WEIL CLASSES
================================================================================
Let A/Q-bar have good reduction at the fixed place w, and let a CM field E act on A
up to isogeny. We write e |-> conj(e) for its complex conjugation. Suppose a
polarization lambda induces this involution on E. Put
V = H_1(A(C),Q), dim_E V = 2s, g = dim A = s [E:Q],
where s >= 1. We assume that the Hodge multiplicities at every embedding of E are
(s,s). The polarization is an alternating form psi: V x V -> Q(1) satisfying
psi(e x, y) = psi(x, conj(e) y).
DEFINITION 4.1 (Weil space). The rational space of WEIL CLASSES associated with E
is the subspace
W_E^s(A) subset wedge^{2s} V^dual tensor Q(s) = H^{2s}(A(C),Q(s))
whose scalar extension to a splitting field Omega of E is
W_E^s(A) tensor_Q Omega
= direct sum_{tau: E -> Omega} wedge^{2s} V_tau^dual tensor Q(s)_Omega. (4.1)
Here each summand is extended by zero on the other eigenspaces. The right side is
Galois stable, so it defines the indicated rational subspace.
The balanced multiplicities imply that every element of W_E^s(A) is a Hodge class:
each determinant in (4.1) has untwisted type (s,s). We use its compatible
realizations as an absolute Hodge class; see (Deligne 1982). The same eigenspace
description of its support holds after specialization, since the E-action
specializes compatibly.
Our goal is the Lefschetz pairing property for these specialized classes. In fact
the reduction satisfies a dichotomy determined by its rational endomorphisms. Let
dagger be Rosati for the reduced polarization and put
U-set = { u in End^0(A_0) : u^dagger = u, u e = conj(e) u for every e in E }.
(4.2)
PROPOSITION 4.2 (Weil dichotomy). Every gamma in W_E^s(A) satisfies the Lefschetz
pairing property at w. More precisely, if U-set in (4.2) has no invertible element,
all complementary pairings are zero. If it has an invertible element, gamma_0 is a
simultaneous rational Lefschetz class.
We first prove the vanishing alternative. For the other alternative, an invertible
u supplies a divisor whose form is nondegenerate on each E-eigenspace separately. We
will lift this divisor and use it to obtain a rational identity for the determinant
classes.
--------------------------------------------------------------------------------
4.1 The orthogonality alternative
--------------------------------------------------------------------------------
LEMMA 4.3 (Weil orthogonality). If U-set contains no invertible element, every
specialization of a class in W_E^s(A) is orthogonal to all complementary Lefschetz
classes, in each cohomological realization.
Proof. We prove the contrapositive. A nonzero pairing will produce a divisor form
nondegenerate on one E-eigenspace. We will then show that a rational endomorphism in
U-set with this property is invertible.
Fix one realization and extend its coefficient field to an algebraically closed
field Omega. At the crystalline component we also include the coefficient extension
used for the specialized class. In this proof V_tau also denotes the corresponding
eigenspace in the dual of first cohomology of A_0, identified by specialization and
comparison. Write L^j_Omega(A_0) for the Omega-span of products of j divisor
classes. Since [E:Q] >= 2, we have g - 2s >= 0. Hard Lefschetz on the Lefschetz
algebra gives
L^{g-s}_Omega(A_0) = psi_0^{g-2s} L^s_Omega(A_0). (4.3)
The restriction of hard Lefschetz to this algebra, including for crystalline
cohomology, follows from the Lefschetz correspondences of (Milne 1999, Theorem 5.9).
Suppose a Weil class pairs nontrivially with D_1 ... D_s psi_0^{g-2s}, and choose a
pure term, supported on V_tau, whose contribution is nonzero. This term already has
degree 2s on V_tau, so no further factor in a nonzero top-degree product can use
that space. A factor from psi_0 which uses V_{conj(tau)} also uses V_tau; hence the
polarization factors contribute to neither space in this pair. All 2s degrees on
V_{conj(tau)} must therefore come from the s divisor factors. Each has degree two, so
each must contribute its restriction to V_{conj(tau)}, and those restrictions have
nonzero product. The coefficient of t_1 ... t_s in
( t_1 D_1 + ... + t_s D_s )^s |_{V_{conj(tau)}}
is s! times that product. Some Omega-linear combination of the divisors therefore
restricts to a nondegenerate alternating form on V_{conj(tau)}.
We identify which endomorphisms give these restrictions. Let p_xi in
E tensor_Q Omega be the eigenspace projectors. Rosati satisfies p_xi^dagger =
p_{conj(xi)}. A divisor corresponds under psi_0 to a Rosati-symmetric endomorphism t.
Define
Pi(t) = sum_xi p_{conj(xi)} t p_xi. (4.4)
This takes place inside End^0(A_0) tensor_Q Omega. Moreover
( p_{conj(xi)} t p_xi )^dagger = p_{conj(xi)} t^dagger p_xi,
Pi(t) e = conj(e) Pi(t).
Thus Pi(t) lies in U-set tensor_Q Omega. Conversely, these are exactly the blocks
allowed by the second equation in (4.2). For x, y in V_xi, the form psi_0(x, t y)
depends only on p_{conj(xi)} t p_xi. It follows that projecting by (4.4) preserves
all the same-embedding restrictions in question.
Nondegeneracy on V_{conj(tau)} is a polynomial open condition on
U-set tensor_Q Omega, and we have shown this open is nonempty. The rational points of
the rational vector space U-set are Zariski dense after extension to Omega. Hence
there is a rational u in U-set for which psi_0(., u .) is nondegenerate on
V_{conj(tau)}.
To prove invertibility, we use the fact that a nonzero E-stable abelian subvariety
has a nonzero homology space at every embedding of E. Suppose u is not invertible,
and choose an integer N > 0 such that N u is an endomorphism. The reduced identity
component C = ((ker(N u))^0)_red is a positive-dimensional abelian subvariety. The
relation u e = conj(e) u makes C stable under the rational action of E. This action
is unital and therefore faithful. Choose a primitive element e of E and an integer
M > 0 such that M e acts by an integral endomorphism of C. The exterior algebra
description of cohomology and normalized top-degree pullback give, in every
realization (Milne 1999, Appendix, Proposition A.2(a),(b)),
det(n - e | H^1(C)) = M^{-2 dim C} deg([M n] - M e)
for every integer n for which the map on the right is an isogeny. All but finitely
many integers have this property. The right side is rational and independent of the
realization. Polynomial interpolation therefore gives one characteristic polynomial
in Q[T], identical also in crystalline cohomology. Its roots lie among the
conjugates of e, and rationality forces their multiplicities to be equal. Since C is
nonzero, every embedding of E occurs with positive multiplicity. The same statement
holds on first homology, in the chosen realization. Rational Poincare reducibility
(Milne 1986, Proposition 12.1) makes the inclusion C -> A_0 injective on first
homology in every realization. This inclusion is E-equivariant, and its image is
killed by u. Thus the nonzero conj(tau)-eigenspace of H_1(C) shows that the kernel
of u meets V_{conj(tau)} nontrivially. Such a vector lies in the radical of
psi_0(., u .)|_{V_{conj(tau)}}, a contradiction.
A nonzero pairing would therefore give an invertible element of U-set. Taking the
contrapositive proves the lemma. The argument applies separately to every
realization, so all the pairings in this case have the common rational value zero.
[]
--------------------------------------------------------------------------------
4.2 A divisor identity on a rational summand
--------------------------------------------------------------------------------
For the rest of the section, suppose u in U-set is invertible. It gives a rational
divisor D^u on A_0 with alternating form
D^u(x, y) = psi_0(x, u y). (4.5)
This form is nondegenerate and pairs each embedding of E only with itself: u
exchanges opposite eigenspaces, whereas psi_0 pairs opposite eigenspaces.
Prescribed-filtration transport will replace A_0 by a power and then by an isogenous
special fiber B_0. We will retain the first factor through a rational projector on
B_0, and lift that projector, the E-action on its image, and the transported divisor
to B. The next lemma explains the resulting divisor identity: a divisor
nondegenerate separately on those eigenspaces generates their determinant classes. A
RATIONAL PROJECTOR here means an idempotent in the rational endomorphism algebra.
LEMMA 4.4 (Divisor identity on a rational summand). Let B be a complex abelian
variety, let R in End^0(B) be a rational projector, and suppose there is an algebra
action E -> R End^0(B) R whose unit maps to R. Let T = im(R) in rational first
homology, with dim_E T = 2s; the indicated endomorphisms act by zero on ker(R).
Suppose there is a rational divisor class D on B such that R^* D = D and, after
splitting E, its alternating form is
D = sum_tau D_tau, D_tau in wedge^2 T_tau^dual tensor Q(1),
with each D_tau nondegenerate. Then the rational subspace whose scalar extension is
direct sum_tau wedge^{2s} T_tau^dual tensor Q(s),
extended by zero on the complementary summand, consists of rational Lefschetz classes
on B.
Proof. For b in E, let b~ in End^0(B) act by b on T and by zero on the complement.
Its pullback of a rational divisor is a rational divisor: clear a denominator in b~,
and use that multiplication by N acts as N^2 on H^2. Over a splitting field Omega,
choosing Tate-line bases, we have
(b~* D)^s = ( sum_tau tau(b)^2 D_tau )^s.
The coordinates x_tau = tau(b) are independent coordinates on the scalar extension of
the rational vector group underlying E. In the corresponding vector-valued polynomial
P((x_tau)_tau) = ( sum_tau x_tau^2 D_tau )^s, (4.6)
the coefficient of x_tau^{2s} is D_tau^s. Nondegeneracy makes this coefficient
nonzero, so it spans wedge^{2s} T_tau^dual tensor Q(s)_Omega.
Let M be the rational span of the Lefschetz classes (b~* D)^s for b in E. The set E
of rational points is Zariski dense in its vector group, also after extension to
Omega. Any linear functional annihilating M tensor_Q Omega annihilates all
evaluations of (4.6) on those points. It therefore annihilates the polynomial, and
hence each coefficient. Every D_tau^s belongs to M tensor_Q Omega.
This proves the required inclusion after scalar extension. Both subspaces are
rational; applying faithful scalar extension to their quotient proves the inclusion
over Q. In particular it gives rational coefficients for every rational class in the
asserted subspace, without requiring an individual embedding line to be defined over
Q. []
--------------------------------------------------------------------------------
4.3 A lift preserving the divisor
--------------------------------------------------------------------------------
We now construct the auxiliary variety to which Lemma 4.4 applies. To transport the
Weil class we use the E-linear polarization similitudes whose determinant has the
same transformation law as a codimension-s Tate twist. Choose a rational basis for
the Betti Tate line, let c be the multiplier of GSp(V, psi), and set
G = { a in GSp(V, psi) cap GL_E(V) : det_E(a) = c(a)^s }. (4.7)
On points over a Q-algebra R, the determinant equality is in (E tensor_Q R)^x, with
the right side embedded diagonally.
LEMMA 4.5 (The Weil similitudes). The Hodge homomorphism of A factors through G,
and its conjugacy class defines a Shimura datum of Hodge type through the displayed
embedding. Over an algebraically closed coefficient field, the orbit of the
corresponding filtration on H^1 consists exactly of the E-stable Lagrangians for the
dual polarization pairing whose dimension at each embedding of E is s. Using a basis
of its Tate line, every alternating form in (4.1) transforms by c^s.
Proof. The polarization pairs V_tau with V_{conj(tau)}, and no other pair of
eigenspaces. Over an algebraically closed field, choose one embedding in each
conjugate pair. An element of G is specified by c and matrices a_tau in GL_{2s} with
det(a_tau) = c^s; the matrices on the opposite spaces are then determined by the
polarization. In particular,
ker(c: G -> G_m) = prod_{ {tau, conj(tau)} } SL_{2s}.
The multiplier is surjective geometrically. Thus G is connected reductive, with the
same adjoint group as the group of E-linear polarization similitudes.
The two Hodge characters occur s times at every embedding. Their product is the
polarization multiplier, so the Hodge homomorphism satisfies the determinant condition
in (4.7). Its weight is rational and central. On the adjoint Lie algebra the Hodge
types are among (-1,1), (0,0), (1,-1), as follows from the weight-one representation.
Conjugation by the image of i is a Cartan involution, by the polarization. At every
real embedding of the maximal totally real subfield of E, the adjoint group has
unitary signature (s,s). Its Hodge homomorphism is nontrivial there, so no rational
simple adjoint factor has trivial Hodge homomorphism. These are the Shimura-datum
conditions for the given Hodge embedding.
On H^1 the polarization still pairs opposite eigenspaces. An E-stable Lagrangian is
specified by an s-dimensional subspace at one embedding of each pair; the subspace at
the opposite embedding is its annihilator. The factor SL_{2s} acts transitively on
these subspaces, as in the cohomological orbit calculation of Theorem 3.1. Finally,
an alternating determinant form evaluated on a_tau v_1, ..., a_tau v_{2s} is
multiplied by det(a_tau) = c^s. This proves the asserted transformation law. []
Proof of Proposition 4.2. The vanishing alternative is Lemma 4.3. Suppose that u is
invertible, and use the divisor D^u from (4.5).
A FILTRATION PRESERVING THE DIVISOR. Let D = H^1_cris(A_0/W(F))[1/p], with its
contravariant convention, and let beta be the dual polarization pairing, taking values
in the isocrystal line of slope one. The actions on D satisfy
beta(u x, y) = beta(x, u y),
beta(e x, y) = beta(x, conj(e) y),
u e = conj(e) u.
All these endomorphisms and the polarization have finite field descent. Theorem 3.1
supplies, over a finite extension with finite local field descent, a weakly admissible
filtration Fil^1 D = L which is E- and u-stable, is Lagrangian for beta, and has
dimension s at every embedding of E. By Lemma 4.5 it lies geometrically in the
required G-orbit. The finite-descent argument in Proposition 2.4 shows that its full
finite family of Hodge-type structure tensors descends from the original number-field
point by crystalline comparison. Taking a compositum with the local field of
definition of L gives the common finite local-field descent required by its
prescribed-filtration clause.
TRANSPORT AND THE FIRST RATIONAL SUMMAND. Apply Proposition 2.4 to G and gamma with
this filtration. It gives a lift B, a quasi-isogeny a: A_0^n -> B_0, and a rational
Hodge class kappa on B, with
f_0 = a j : A_0 -> B_0, f_0^* kappa_0 = gamma_0,
simultaneously in the realizations under consideration. The filtration on B, marked by
a, is the direct sum of n copies of the chosen filtration. Here j includes the first
factor. The map f_0 identifies A_0 with a rational direct summand of B_0. Its
retraction q_0 and the corresponding projector R_0 are
q_0 = pr_1 a^{-1}, R_0 = f_0 q_0. (4.8)
Thus q_0 f_0 = id and R_0^2 = R_0. Pulling back f_0^* kappa_0 = gamma_0 by q_0 gives
R_0^* kappa_0 = q_0^* gamma_0.
Act by E on im(R_0) using e |-> f_0 e q_0, and by zero on the complement. Also put
D_0 = q_0^* D^u.
LIFTING THE PROJECTOR, ACTION, AND DIVISOR. The projector and this E-action preserve
the repeated filtration. By Corollary 3.5, D^u satisfies the divisor lifting
condition for the chosen filtration on the first summand. The crystalline map induced
by q_0 = pr_1 a^{-1} respects that filtration and the marked repeated filtration on
B_0. Hence D_0 = q_0^* D^u satisfies the same condition on B. Apply Lemma 2.3 to R_0,
to a finite Q-basis of the action of E on im(R_0), and to the symmetric homomorphism
h_{D_0}: B_0 -> B_0^dual defined by D_0. Denote their lifts by R, the corresponding
basis of maps, and h_D. Injectivity of specialization preserves R^2 = R, the rational
multiplication table of that basis, and its unit R. It therefore gives an algebra
action E -> R End^0(B) R. The symmetric map h_D defines a rational divisor class D.
Moreover, q_0 R_0 = q_0, so
R_0^dual h_{D_0} R_0 = h_{D_0}.
Injectivity lifts this identity to R^dual h_D R = h_D, which is exactly R^* D = D.
Thus the required projector, action, and divisor all exist on the one auxiliary
variety B.
THE RATIONAL IDENTITY AND ITS SPECIALIZATION. Consider now rational Betti cohomology
at a complex realization of B. The summand T = im(R) has dimension 2s at each
embedding of E. The form of D is supported on this summand and pairs each embedding
with itself nondegenerately. Also R^* kappa is supported on the pure determinant
spaces of this summand. These assertions are checked after extension to any one
prime-to-p realization: for the divisor they follow from its specified
specialization, and for the Hodge class they follow from the equality
R_0^* kappa_0 = q_0^* gamma_0 and the pure support of gamma_0. Compatibility of the
transported realizations then gives these support and nondegeneracy assertions in
Betti cohomology.
Lemma 4.4 now expresses the rational Betti class R^* kappa as a finite rational linear
combination of products of rational divisor classes on B. The realization and
specialization compatibilities in Proposition 2.4, together with those for divisors
and homomorphisms, specialize this identity with the same rational coefficients in
every component. Hence R_0^* kappa_0 is a simultaneous rational Lefschetz class.
Finally,
gamma_0 = f_0^* R_0^* kappa_0,
because R_0 f_0 = f_0. Pullback by a rational homomorphism preserves simultaneous
rational Lefschetz classes, so the same is true of gamma_0. Its complementary traces
are therefore one common rational intersection number, proving the second alternative
and the proposition. []
================================================================================
5. CM REDUCTION AND CONSEQUENCES
================================================================================
The remaining step in the main proof is to express CM Hodge classes as pullbacks of
balanced Weil classes. Together with CM transport (Corollary 2.7) and the Weil
dichotomy (Proposition 4.2), this proves the main theorem. We then derive the two
specialization consequences.
================================================================================
5.1 CM classes as pullbacks of Weil classes
================================================================================
We give the CM reduction in the form needed here. The stronger reduction to classes
on abelian varieties of split Weil type is due to Deligne and Andre (Andre 1992); see
the reconstruction in (Milne 2020, Theorem 1 and Section 4). Only balanced signatures
and a compatible polarization are needed for our application, and we prove this form
directly using rational idempotents on powers.
LEMMA 5.1 (CM reduction). Let A/Q-bar be an abelian variety of CM type with good
reduction at w, and let r > 0. For every rational Hodge class
gamma in H^{2r}(A(C),Q(r)), there are finitely many abelian varieties B_i/Q-bar with
good reduction at w, rational homomorphisms f_i: A -> B_i, and rational Weil classes
eta_i in H^{2r}(B_i(C),Q(r)) such that
gamma = sum_i f_i^* eta_i. (5.1)
Each B_i has an action of a CM field K for which H_1(B_i(C),Q) has rank 2r over K,
the Hodge signatures are (r,r) at every embedding of K, and a polarization induces
complex conjugation on K. The equality specializes simultaneously in all the
realizations used in Theorem 1.1.
Proof. We first identify the eigenspace components that can support a rational Hodge
class. For each such component we construct a balanced abelian factor of a power of A,
whose Weil classes pull back onto that component. Taking the sum of these rational
pullback spaces will then give the decomposition over Q.
BALANCED EIGENSPACE COMPONENTS. Set V = H_1(A(C),Q). We work throughout up to
isogeny. There is a CM algebra E', a product of CM fields, acting on A such that V is
free of rank one over E'. To obtain it, decompose A up to isogeny into simple CM
factors and take the product of their CM fields, with a separate label for each
repeated factor. Transporting this action back along an isogeny gives the asserted
action on A itself.
Choose a finite Galois CM field K subset Q-bar containing the images of all the
factors of E', and put S = Hom_{Q-alg}(E', K). Then
V_K := V tensor_Q K = direct sum_{rho in S} V_rho, dim_K V_rho = 1.
The Hodge decomposition is diagonal for this decomposition after extension to C. We
suppress the common Tate twist in the following linear-algebra calculations.
Alternating forms of degree 2r decompose as
wedge^{2r} V_K^dual
= direct sum_{ Delta subset S, |Delta| = 2r } L_Delta,
L_Delta = wedge^{2r}( direct sum_{rho in Delta} V_rho )^dual,
where each form on the right is extended by zero on the other summands. Every L_Delta
is one-dimensional and has a definite Hodge type.
Suppose the component of gamma in L_Delta is nonzero. Since gamma is rational, its
component in L_{sigma Delta} is the semilinear conjugate of that component and is
nonzero for every sigma in Gal(K/Q). The Hodge condition therefore implies that every
sigma Delta contains r lines of each of the two Hodge types. We call such a subset
BALANCED under all conjugates.
AN ABELIAN FACTOR WITH BALANCED SIGNATURES. Fix one of these subsets Delta. Form the
abelian variety A tensor_Q K in the category up to isogeny: it is a power of A whose
index space is the underlying rational vector space of K. The regular action of K on
this index space is an action by rational endomorphisms of the power. Its rational
homology is
T = V tensor_Q K, dim_Q T = [K:Q] dim_Q V.
In contrast with the coefficient space V_K, we regard T here as a Q-vector space,
with K acting on its second factor. The abelian variety remains over Q-bar. The
actions on the two factors give an action of the rational algebra
E' tensor_Q K = prod_{rho in S} K, e tensor k |-> (rho(e) k)_rho.
Let e_Delta be the idempotent with coordinates 1 on Delta and 0 elsewhere. It is a
rational endomorphism of A tensor_Q K, so its image is an abelian variety B_Delta up
to isogeny. In particular no Galois invariance of Delta is required for this
construction. Write
U = H_1(B_Delta(C),Q) = e_Delta T.
The index-field action of K preserves U.
Now extend the coefficients of U to K, and use the fixed inclusion K subset C to
identify the embeddings of K with Gal(K/Q). Evaluation of the index factor at tau
identifies its eigenspace in T tensor_Q K with V_K. In that eigenspace the idempotent
retains exactly the summands indexed by tau Delta. Thus
U_tau = direct sum_{rho in Delta} V_{tau rho}, (5.2)
and the same description holds after extension to C. Explicitly, evaluation of the
idempotent at the pair of embeddings (xi, tau) is 1 exactly when xi = tau rho for some
rho in Delta. Thus dim_K U_tau = 2r for every tau. Since K is Galois and Delta is
balanced under all conjugates, each U_tau has r lines of each Hodge type.
Consequently U has rank 2r over K, dim B_Delta = r[K:Q], and the required Hodge
signatures are (r,r).
A COMPATIBLE POLARIZATION. Let psi be a rational polarization form on V, and equip
the index space K with the positive definite rational symmetric form
t(k, l) = Tr_{K/Q}(k conj(l)).
The form Psi = psi tensor t is a rational polarization form on the power
A tensor_Q K: its alternating and Hodge properties follow from those of psi, and its
positivity follows by tensoring the positive symmetric form associated to psi with t.
A positive integral multiple gives an actual polarization. For a in K,
t(a k, l) = t(k, conj(a) l),
so the Rosati involution of Psi induces conjugation on the index-field action.
Restrict Psi to B_Delta. This is again a polarization, and because U is K-stable its
adjoint identity still induces conjugation on K. This restriction does not require
e_Delta to be self-adjoint.
PULLBACK AND DESCENT OF THE RATIONAL SPAN. The factor B_Delta now has the field
action, balanced signatures, and polarization needed to define its Weil classes. The
rational homomorphism f_Delta: A -> B_Delta defined on homology by
v |-> e_Delta (v tensor 1)
has the needed effect on Weil classes. After splitting coefficients, its component at
the identity embedding of K is the projection
direct sum_{rho in S} V_rho -> direct sum_{rho in Delta} V_rho.
Thus pullback maps wedge^{2r} U_id^dual isomorphically onto L_Delta. More generally the
component indexed by tau pulls back onto L_{tau Delta}. The rational Weil-class space
W_K^r(B_Delta) is characterized, with the twist restored, by
W_K^r(B_Delta) tensor_Q K
= direct sum_{tau: K -> K} wedge^{2r} U_tau^dual (r).
In particular f_Delta^* W_K^r(B_Delta) is a rational subspace whose extension to K
contains L_Delta(r).
Take the sum of these rational pullback spaces over the finitely many subsets
supporting gamma. Its scalar extension contains every component of gamma, and
therefore contains gamma itself. Since extension of scalars is injective on the
quotient by a rational subspace, the sum already contains gamma over Q. This proves
(5.1) with rational Weil classes. Notice that an individual embedding component need
not descend to Q; only the entire Weil-class space and its pullback are used in this
descent argument.
GOOD REDUCTION AND SPECIALIZATION. All the constructions use finitely many rational
endomorphisms of powers of A and their images. They therefore exist over Q-bar and
descend, together with the maps, to a common number field after a finite extension.
Choose that extension so that A has good reduction at the place specified by w. For
any ell != p, the rational Tate module of B_Delta is a Galois-stable direct summand of
the rational Tate module of the power of A. It is unramified at this place. The
Neron-Ogg-Shafarevich criterion gives good reduction of B_Delta there; it also shows
that the isogenies used in the construction do not affect this conclusion (Milne 2008,
Chapter IV, Theorem 3.5). Rational homomorphisms between the resulting good-reduction
varieties specialize after clearing denominators. The equality of rational Hodge
classes (5.1) is respected by their compatible realizations and by specialization,
with the same rational coefficients in every component. This proves the last
assertion. []
COROLLARY 5.2 (CM pairing property). The Lefschetz pairing property holds for the
specialization of every rational Hodge class on a CM abelian variety over Q-bar with
good reduction at w.
Proof. In codimension zero the class is a rational multiple of the unit, so the
assertion follows from the compatibility of divisor intersection numbers with the
normalized traces. In positive codimension use Lemma 5.1. Proposition 4.2 gives the
Lefschetz pairing property for the specialization of each eta_i. Lemma 2.2 preserves
this property under rational pullback and rational linear combinations. Applying it to
(5.1) proves the assertion. []
================================================================================
5.2 Proof of the main theorem
================================================================================
Proof of Theorem 1.1. The case r = 0 follows from divisor intersection numbers as in
the preceding corollary. This includes dim A = 0, where the abelian variety is a
point. If r = dim A, a rational top class is a rational multiple of the point class;
its normalized trace is preserved by specialization, and the complementary product is
1. We may therefore assume 0 < r < dim A.
Corollary 2.7 gives a CM abelian variety B/Q-bar with good reduction at w, a rational
Hodge class kappa on B, a quasi-isogeny a: A_0^n -> B_0, and the first-factor
inclusion j: A_0 -> A_0^n, such that
gamma_0 = (a j)* kappa_0
simultaneously in all the realizations under consideration. Corollary 5.2 gives the
Lefschetz pairing property for kappa_0, and Lemma 2.2 gives it for gamma_0. Hence, for
each prescribed product of complementary divisor classes on A_0, all the normalized
traces in Theorem 1.1 equal one rational number. This proves the theorem. []
================================================================================
5.3 Specialization in a fixed good theory
================================================================================
We now use CM transport to place the specialization of every rational Hodge class in
one fixed theory of rational Tate classes.
Continue to fix Q-bar inside C, p, and w, with F, K0, and Cw as in the introduction.
Let S_0 be Milne's class of F-varieties whose connected components are products of
abelian varieties and projective spaces. The CM case of Theorem 1.1, by linearity for
complementary Lefschetz classes, satisfies the hypothesis of (Milne 2009, Theorem 4.5).
Fix ONE good theory R-theory of rational Tate classes on S_0 supplied by that theorem,
before varying any abelian variety or Hodge class.
For rho in R-theory^r(X), write (rho)_ell for its ell-adic component when ell != p, and
(rho)_p for its native crystalline Tate component in
H^{2r}_cris(X/W(F))[1/p](r).
We use three properties of the theory. Its spaces are rational vector spaces preserved
by pullback, its realization maps are injective, and specializations of CM Hodge classes
belong to it. These are the properties in axioms (R1), (R4), and the goodness axiom
(R3), respectively (Milne 2009, Definitions 2.1 and 3.1). The following consequence
extends the last property to all rational Hodge classes, while keeping this one theory
fixed.
COROLLARY 5.3 (Specialization of all rational Hodge classes). For every abelian
variety A/Q-bar of dimension d with good reduction A_0/F at w, every 0 <= r <= d, and
every rational Hodge class gamma in H^{2r}(A(C),Q(r)), there is a unique class
rho_gamma in R-theory^r(A_0) such that
(rho_gamma)_ell = gamma_{0,ell} (ell != p),
(rho_gamma)_p tensor 1 = gamma_{0,p}.
The prime-to-p equalities are in H^{2r}_et(A_0, Q_ell(r)); the crystalline equality is
in
H^{2r}_cris(A_0/W(F))[1/p](r) tensor_{K0} Cw.
The specialization components are those of Theorem 1.1. The assertion holds for every
prime p, including p = 2, with the prime, place, and theory fixed before A and gamma
vary.
Proof. In codimension zero, take the corresponding rational multiple of the unit,
which belongs to the theory by pullback from the point. This also covers d = 0. In top
codimension r = d > 0, take q times the point class, where q in Q is the normalized
trace of gamma. The point class belongs to R-theory^d(A_0) (Milne 2009, Section 2.3),
and its normalized trace is one. Compatibility of traces with specialization gives the
required components in these cases.
Suppose now that 0 < r < d. Choose the CM variety B, Hodge class kappa, and rational
homomorphism
b = a j in Hom^0(A_0, B_0), gamma_0 = b* kappa_0
from Corollary 2.7. The equality holds in all the stated realizations, with the usual
absolute-Hodge specializations of kappa. Axiom (R3) gives one class rho_kappa in
R-theory^r(B_0) with
(rho_kappa)_ell = kappa_{0,ell} (ell != p),
(rho_kappa)_p tensor 1 = kappa_{0,p}.
Choose an integer N > 0 such that h = N b is an actual homomorphism A_0 -> B_0.
Pullback by h preserves the theory by axiom (R1) of (Milne 2009, Definition 2.1). Since
R-theory^r(A_0) is a Q-vector space, we may set
rho_gamma = N^{-2r} h* rho_kappa in R-theory^r(A_0).
Functoriality of the component maps and (2.1) give
(rho_gamma)_ell = N^{-2r} h* kappa_{0,ell} = b* kappa_{0,ell} = gamma_{0,ell}
(ell != p),
(rho_gamma)_p tensor 1 = N^{-2r} h* kappa_{0,p} = b* kappa_{0,p} = gamma_{0,p}.
The second line is taken after extension from K0 to Cw. Pullback in (R1) applies to any
homomorphism h, and rational scalar multiplication allows any denominator N. Thus the
argument requires neither an isogeny nor separability, and remains valid when p | N,
including at p = 2.
Finally, two classes with the required components have the same ell-adic component for
any one ell != p. Injectivity of that component map by (R4) makes them equal. Thus the
class is unique in the fixed theory, independently of the auxiliary choices. []
This cohomological membership statement alone gives no algebraic cycle and proves
neither the Hodge nor the Tate conjecture. The construction takes place at the fixed
reduction place: its CM auxiliary may have dimension n dim A, and the map b is between
special fibers. It supplies no same-dimensional CM lift or map between the generic
fibers.
================================================================================
5.4 Algebraicity after good reduction
================================================================================
The CM Hodge theorem (OpenAI 2026, Theorem 1.1), combined with our CM transport,
strengthens the preceding specialization statement. It is an additional input to the
following consequence, not to the proof of Theorem 1.1.
COROLLARY 5.4 (Algebraic specialization of rational Hodge classes). Fix the prime and
place as in Theorem 1.1. Let A/Q-bar be an abelian variety of dimension d with good
reduction A_0/F at w, and let gamma in H^{2r}(A(C),Q(r)) be a rational Hodge class,
where 0 <= r <= d. There is one rational algebraic cycle z_gamma in CH^r(A_0)_Q such
that
cl_ell(z_gamma) = gamma_{0,ell} (ell != p),
cl_cris(z_gamma) tensor 1 = gamma_{0,p}.
The crystalline equality is in
H^{2r}_cris(A_0/W(F))[1/p](r) tensor_{K0} Cw.
This holds for every prime p, including p = 2. Consequently, for every
y in CH^{d-r}(A_0)_Q, the traces of gamma_{0,ell} cl_ell(y) for ell != p and of
gamma_{0,p} cl_cris(y) are all the same rational number deg(z_gamma . y).
Proof. In codimension zero or top codimension, use the corresponding rational multiple
of the unit or the point class, as in the preceding proof. This includes dimension zero.
For 0 < r < d, use the CM variety B, rational Hodge class kappa, and map
b = a j in Hom^0(A_0, B_0), gamma_0 = b* kappa_0
given by Corollary 2.7, with the equality in every stated realization.
By (OpenAI 2026, Theorem 1.1), kappa is the Betti class of a rational algebraic cycle
on B_C. A representative can be chosen over Q-bar: spread each of its finitely many
components in the Hilbert scheme of the fixed projective variety B, and choose a
Q-bar-point on the same geometrically connected component. The universal flat family
preserves the Betti cycle class. Thus there is Z in CH^r(B)_Q with class kappa. Descend
B and Z to a number field and extend the chosen place so that B has an abelian scheme
model. Specialization gives Z_0 in CH^r(B_0)_Q. Compatibility of cycle classes with
smooth proper specialization and the de Rham-crystalline comparison identifies its
classes with kappa_{0,ell} and kappa_{0,p}, respectively. In particular the same
rational cycle Z_0 gives every realization.
Choose N > 0 with h = N b an actual homomorphism A_0 -> B_0, and put
z_gamma = N^{-2r} h* Z_0 in CH^r(A_0)_Q.
Here h* is the Chow pullback between smooth varieties, defined by the graph and refined
intersection; it requires neither flatness nor separability of h. Functoriality of
cycle classes and (2.1) give the required equalities. Rational coefficients permit this
denominator even if p | N; the crystalline realization has already inverted p. Finally,
compatibility of cycle classes with products and normalized traces gives the asserted
intersection number for every y. []
This conclusion concerns only the special fiber at the fixed place. It neither asserts
the Hodge conjecture for arbitrary abelian varieties in characteristic zero nor lifts
z_gamma to the original generic fiber. The representing cycle is not asserted to be
unique modulo rational equivalence, and no comparison between different reduction places
is made.
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REFERENCES
================================================================================
[Andre 1992] Y. Andre. Une remarque a propos des cycles de Hodge de type CM. In
S. David (ed.), Seminaire de Theorie des Nombres, Paris, 1989-90, Progress in Math.
102, Birkhauser Boston, 1992, 1-7.
[Andre 1996] Y. Andre. Pour une theorie inconditionnelle des motifs. Publ. Math.
Inst. Hautes Etudes Sci. 83 (1996), 5-49. doi:10.1007/BF02698643.
[Blasius 1994] D. Blasius. A p-adic property of Hodge classes on abelian varieties.
In Motives (Seattle, WA, 1991), Part 2, Proc. Sympos. Pure Math. 55, Part 2, AMS,
Providence, RI, 1994, 293-308. doi:10.1090/pspum/055.2/1265557.
[Colmez-Fontaine 2000] P. Colmez and J.-M. Fontaine. Construction des
representations p-adiques semi-stables. Invent. Math. 140 (2000), no. 1, 1-43.
doi:10.1007/s002220000042.
[Conrad 2003] B. Conrad. Minimal models for elliptic curves, notes, 20 October 2003.
math.stanford.edu/~conrad/vigregroup/vigre03/minimalmodel.pdf.
[Deligne 1982] P. Deligne. Hodge cycles on abelian varieties (notes by J. S. Milne).
In P. Deligne, J. S. Milne, A. Ogus, and K.-y. Shih, Hodge Cycles, Motives, and
Shimura Varieties, Lecture Notes in Math. 900, Springer-Verlag, 1982, 9-100.
www.jmilne.org/math/Documents/Deligne82.pdf (corrected online version).
[Fontaine-Rapoport 2005] J.-M. Fontaine and M. Rapoport. Existence de filtrations
admissibles sur des isocristaux. Bull. Soc. Math. France 133 (2005), no. 1, 73-86.
doi:10.24033/bsmf.2479.
[Gleason-Lim-Xu 2026] I. Gleason, D. G. Lim, and Y. Xu. The connected components of
affine Deligne-Lusztig varieties. Invent. Math. 243 (2026), 805-861.
doi:10.1007/s00222-025-01386-1; arXiv:2208.07195v3.
[Grothendieck 1970] A. Grothendieck. Groupes de Barsotti-Tate et cristaux. In Actes
du Congres International des Mathematiciens (Nice, 1970), Tome 1, Gauthier-Villars,
Paris, 1971, 431-436.
[Hartl 2008] U. Hartl. On period spaces for p-divisible groups. C. R. Math. Acad.
Sci. Paris 346 (2008), nos. 21-22, 1123-1128. doi:10.1016/j.crma.2008.07.003.
[Illusie 2015] L. Illusie. Grothendieck at Pisa: crystals and Barsotti-Tate groups.
In U. Zannier (ed.), Colloquium De Giorgi 2013 and 2014, Publications of the Scuola
Normale Superiore 5, Edizioni della Normale, Pisa, 2015, 79-107.
doi:10.1007/978-88-7642-515-8_5.
[Kim 2012] W. Kim. The classification of p-divisible groups over 2-adic discrete
valuation rings. Math. Res. Lett. 19 (2012), no. 1, 121-141.
doi:10.4310/MRL.2012.v19.n1.a10; arXiv:1007.1904v3.
[Kisin 2006] M. Kisin. Crystalline representations and F-crystals. In V. Ginzburg
(ed.), Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld's 50th
Birthday, Progress in Math. 253, Birkhauser Boston, 2006, 459-496.
doi:10.1007/978-0-8176-4532-8_7.
[Kisin 2017] M. Kisin. Mod p points on Shimura varieties of abelian type. J. Amer.
Math. Soc. 30 (2017), no. 3, 819-914.
[Kisin-Zhou 2025a] M. Kisin and R. Zhou. Independence of ell for Frobenius conjugacy
classes attached to abelian varieties. Ann. of Math. (2) 202 (2025), no. 3, 1077-1156.
doi:10.4007/annals.2025.202.3.3; arXiv:2103.09945v2.
[Kisin-Zhou 2025] M. Kisin and R. Zhou. Strongly compatible systems associated to
semistable abelian varieties. Preprint, 2025, arXiv:2505.02165v1 (dated 10 June 2025).
[Markman 2026] E. Markman. Secant sheaves and Weil classes on abelian varieties.
Preprint, contribution to the ICM 2026, arXiv:2509.23403v2 (13 February 2026).
[Milne 1986] J. S. Milne. Abelian varieties. In G. Cornell and J. H. Silverman
(eds.), Arithmetic Geometry (Storrs, CT, 1984), Springer-Verlag, 1986, 103-150.
doi:10.1007/978-1-4613-8655-1_5; corrected author's version (AVs.pdf), 2 January 2022.
[Milne 1999] J. S. Milne. Lefschetz classes on abelian varieties. Duke Math. J. 96
(1999), no. 3, 639-675; see the author's online correction to the proof of Theorem 5.9.
[Milne 2000] J. S. Milne. Towards a proof of the conjecture of Langlands and Rapoport.
Notes for a talk at the Institut Henri Poincare, 28 April 2000, version dated 30 April
2000.
[Milne 2005] J. S. Milne. Introduction to Shimura varieties. In Harmonic Analysis,
the Trace Formula, and Shimura Varieties, Clay Math. Proc. 4, AMS, 2005, 265-378;
author's revision of 16 September 2017.
[Milne 2007] J. S. Milne. The Tate conjecture over finite fields (AIM talk). Notes for
the AIM workshop, July 2007, version 2.2, 7 May 2008.
[Milne 2008] J. S. Milne. Abelian Varieties, course notes, version 2.00, 2008,
vi+166 pp.
[Milne 2009] J. S. Milne. Rational Tate classes. Mosc. Math. J. 9 (2009), no. 1,
111-141; page numbers refer to the author's version.
[Milne 2013] J. S. Milne. Lectures on Etale Cohomology, course notes, version 2.21,
22 March 2013.
[Milne 2017] J. S. Milne. Algebraic Groups: The Theory of Group Schemes of Finite Type
over a Field, Cambridge Studies in Advanced Mathematics 170, CUP, 2017; corrected
reprint 2022. doi:10.1017/9781316711736.
[Milne 2020] J. S. Milne. Hodge classes on abelian varieties. Preprint, 2020,
arXiv:2010.08857v2.
[Milne 2020b] J. S. Milne. Algebraic Number Theory, course notes, version 3.08,
19 July 2020. (Cited for Krasner's lemma, Propositions 7.60-7.61 and Corollary 7.62.)
[Milne 2025] J. S. Milne. Abelian motives and Shimura varieties in nonzero
characteristic. Preprint, version 2.2, 2 November 2025.
[Moonen 1998] B. Moonen. Models of Shimura varieties in mixed characteristics. In
A. J. Scholl and R. L. Taylor (eds.), Galois Representations in Arithmetic Algebraic
Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, CUP, 1998, 267-350.
doi:10.1017/CBO9780511662010.008.
[OpenAI 2026] OpenAI. The rational Hodge conjecture for CM abelian varieties. OpenAI
Math Release preprint, 2026. (Cited for Theorem 1.1 in Corollary 5.4.)
[Pappas-Rapoport 2024] G. Pappas and M. Rapoport. p-adic shtukas and the theory of
global and local Shimura varieties. Camb. J. Math. 12 (2024), no. 1, 1-164.
doi:10.4310/CJM.2024.v12.n1.a1; arXiv:2106.08270v6.
[Stacks Project] The Stacks Project Authors. The Stacks Project, Lemma 30.28.3,
Tag 0A42. stacks.math.columbia.edu/tag/0A42.
[Tate 1967] J. T. Tate. p-Divisible Groups. In T. A. Springer (ed.), Proceedings of a
Conference on Local Fields (Driebergen, 1966), Springer-Verlag, 1967, 158-183.
doi:10.1007/978-3-642-87942-5_12.
[Wintenberger 1997] J.-P. Wintenberger. Proprietes du groupe tannakien des structures
de Hodge p-adiques et torseur entre cohomologies cristalline et etale. Ann. Inst.
Fourier (Grenoble) 47 (1997), no. 5, 1289-1334. doi:10.5802/aif.1600.
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