Lattices in Tate modules

Refining a theorem of Zarhin, we prove that, given a g-dimensional abelian variety X and an endomorphism u of X, there exists a matrix A∈M2g(ℤ) such that each Tate module TℓX has a ℤℓ-basis on which the action of u is given by A, and similarly for the covariant Dieudonné module if over a perfect field of characteristic p.


Introduction
Let X be an abelian variety of dimension g over a field k of characteristic p ≥ 0. Let End X be its endomorphism ring. Let End these are free rank 2g modules over the ring of Witt vectors W := W (k ), its fraction field the productẐ (p) × W , and respectively. Definition 1.1: Given rings R ⊆ R and corresponding modules L ⊆ L , say that L is an R-lattice in L if L has an R-basis that is an R -basis for L .
Zarhin (ref. (1), Theorem 1.1) proved that, given u ∈ End • X , there exists a matrix A ∈ M2g (Q) such that, for every = p, there is a Q -basis of V on which the action of u is given by A; equivalently, there exists a u-stable Q-lattice in the ( =p Q )module =p V . Our main theorem refines this, as follows.  The following restatement of (b) answers a question implicit in ref.
(1), Remark 1.2. Corollary 1.3. Let u ∈ End X . Then there exists a matrix A ∈ M2g (Z) such that, for every = p, there is a Z -basis of T X on which the action of u is given by A, and such that, if p > 0 and k is perfect, there is a W-basis of M * (X ) on which the action of u is given by A.
The characteristic 0 case of Theorem 1.2 can be proved by reducing to the case k = C and taking rational or integral homology (ref. (1), Remark 1.2). But pairs (X , u) in characteristic p > 0 cannot always be lifted to characteristic 0 (ref. (2), Example 14.5), so the general case does not seem to follow easily from this.

Proof
Lemma 2.1. Suppose that p > 0 and k is perfect. Let L be a finite extension of Qp. Let N be an (L ⊗ Qp K )-module with an automorphism F that is L-linear and K-semilinear with respect to the Frobenius automorphism φ of K. Then N is free over L ⊗ Qp K .
Proof: The residue field of L is finite, so it has a largest subextension embeddable in k. Let L ⊂ L be the corresponding unramified extension of Qp. Let I = Hom Qp -algebras (L , K ). Then where each Li is a field since K is absolutely unramified and any tensor product ⊗ k is a field. Now N i∈I Ni , where each Ni is a Li -vector space.
The action of φ on K induces a permutation π of I that is transitive since L /Qp is Galois with group generated by the Frobenius automorphism. If i ∈ I and j = π(i ), then the compatible actions of φ of K and F on N induce compatible isomorphisms Li Author contributions: B.P. and S.R. performed research and wrote the paper.

The authors declare no competing interest.
This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY). 1 To whom correspondence may be addressed. Email: poonen@math.mit.edu or (v) If p > 0 and k is perfect, then the (E ⊗ Q AW )-module VW is free of rank h.

Proof:
(i) This is ref.

Proof of Theorem 1.2:
(a) We work in the category of abelian varieties over k up to isogeny. By ref.
(1), Theorem 2.4, u is contained in a subring of End • X isomorphic to i Mr i (Ei ) for some number fields Ei . Then X is isogenous to Y r i i for some abelian varieties Yi with Ei ⊆ End • Yi . If we can find an Ei -stable Q-lattice Vi ⊂ VYi for each i, then we may take V = V r i i . In other words, we have reduced to the case that u ∈ E ⊆ End • X for some number field E. By Lemma 2.2 (iv), for some Q-vector space P. Then V := P ⊗ Q E is a u-stable Q-lattice in V. (b) Given u ∈ End X , choose V as in (a). We have which we interpret as Z if p = 0. Then V ∩ T is a Z[1/p]lattice in T. Since Z[u] ⊂ End X is a finite Z module, the Z[u]-submodule generated by any Z[1/p]-basis of V ∩ T is a u-stable Z-lattice. (c) As in the proof of (a), we reduce to the case in which u ∈ E ⊆ End • X for some number field E. By Lemma 2.2 (v), for some Q-vector space P.

Generalizations and Counterexamples
In Theorem 1.2, suppose that, instead of fixing one endomorphism u, we consider a Q-subalgebra R ⊂ End • X (or subring R ⊂ End X ) and ask for an R-stable Q-lattice (respectively, Zlattice), that is, one that is r stable for every r ∈ R.
(1) If R is contained in a subring of End • X isomorphic to i Mr i (Ei ) for some number fields Ei , then the proof of Theorem 1.2 shows that an R-stable lattice exists.
(2) Serre observed that if X is an elliptic curve such that End • X is a quaternion algebra, then for R = End • X , there is no R-stable Q-lattice in any V , since R cannot act on a twodimensional Q-vector space. (3) If R is assumed to be commutative, then the conclusions of Theorem 1.2 can still fail. For example, suppose that Y is an elliptic curve such that End • Y is a quaternion algebra B, and X = Y 2 , and The ideal ( 0 B 0 0 ) has square zero, so R is commutative. For each nonzero b ∈ B , we have Suppose that there is an R-stable Q-lattice V in VX . Let U := V ∩ (0 × VY ), which is a Q-vector space of dimension at most 2. Then, for every nonzero b ∈ B , the image ( 0 b 0 0 ) V is a 2-dimensional Q-lattice in 0 × VY , contained in U, and hence equal to U. Thus we obtain a Q-linear injection ACKNOWLEDGMENTS. This article arose out of a discussion initiated at the virtual conference "Arithmetic, Geometry, Cryptography and Coding Theory" hosted by the Centre International de Rencontres Mathématiques in Luminy in 2021. B.P. was supported, in part, by NSF Grants DMS-1601946 and DMS-2101040 and Simons Foundation Grants 402472 and 550033. We are grateful to a reviewer for suggesting that we prove Theorem 1.2 (c) and (d).