ON THE ESSENTIAL TORSION FINITENESS OF ABELIAN VARIETIES OVER TORSION FIELDS

Abstract The classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension 
$K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$
 by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension 
$K_B$
 obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of 
$A(K_B)_{\mathrm tors}$
 in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.


INTRODUCTION
Suppose A is an abelian variety defined over a number field K.The celebrated Mordell-Weil theorem states that for any number field L containing K, the subgroup A(L) tors of torsion points of A defined over L is finite (e.g., [Mum08,Appendix II]).At the opposite extreme, over the algebraic closure K of K, using the geometry of A one easily sees that the geometric torsion group A(K) tors is infinite.Then it is natural to ask whether the finiteness property of the torsion subgroup A(L) tors is still preserved for various infinite algebraic extensions L/K.This kind of question can be traced back at least to [Maz72], in which Mazur asked whether the group A(K cyc,p ), where K cyc,p = K(∪ n ζ p n ) is the field obtained by adjoining all p-power roots of unity to K, is still finitely generated.The torsion part A(K cyc,p ) tors of this group is then proved to be finite by Imai [Ima75] and Serre [Ser68] independently.Their results are then generalized by Ribet in his article [KL81,Appendix].Let K cyc := ∪ p K cyc,p = K(∪ n ζ n ) be the infinite extension of K obtained by adjoining all roots of unity.Then Ribet showed that for every abelian variety A defined over the number field K, one has Zarhin then further generalized this result [Zar87] by showing that if A is a simple abelian variety over its ground field K, then over the maximal abelian extension K ab of K, the torsion group A(K ab ) tors is finite if and only if A is not of CM-type over K, i.e., if and only if the K-endomorphism algebra End(A) Q := End(A) ⊗ Z Q is not a number field of degree 2 dim A. As a cohomological generalization of Ribet's result, R össler and Szamuely [RS19] proved that for any projective, smooth, and geometrically connected variety X over a number field K, the groups H i ét (X, Q/Z(j)) Gal(K/K cyc ) are finite for all odd positive integers i and all integers j.In contrast, when K is a p-adic field, then the analog of Imai and Serre's result is generalized by Ozeki in [Oze09].In addition, an analogue of Zarhin's result is proved for Drinfeld modules by Li [Li02].Quite recently, Lombardo studied a problem which, while perhaps superficially different, turns out to be closely related [Lom23]; we discuss his work at the end of this introduction.
In this paper, we focus on the generalization of (1.1) in another direction.Notice that by the Kronecker-Weber theorem, the cyclotomic extension K cyc = K(G m,tors ) is exactly the extension of K obtained by adjoining all the geometric torsion points of the algebraic torus G m .Due to the fact that there is no nontrivial isogeny, or even nonconstant geometric morphism, between G m and an abelian variety, one is naturally led to ask the following question: Question 1.1.Suppose two abelian varieties A and B are defined over a number field K; assume that over K they share no common nontrivial isogeny factor.Let K B denote the infinite extension of K obtained by adjoining all the geometric torsion points of B. Is the torsion group of A over K B finite, i.e., is We answer this question in the present paper, up to a finite extension of the base field and under the Mumford-Tate conjecture.We state our results after introducing a few definitions.

Definition 1.2. Given two abelian varieties A and B defined over a number field K, we say that A is torsion finite for B over K if A(K B
) tors is finite.Otherwise, we say that A is torsion infinite for B over K.
Moreover, if there is a finite extension L/K such that A(L B ) tors is infinite, we say A is potentially torsion infinite for B. If such L does not exist, we will say that A is essentially torsion finite for B.
Although not stated in this language, Serre gave a positive answer to Question 1.1 in [Ser72, Théoremè 6 and 7] when A and B are both elliptic curves which are not geometrically isogenous.In fact, Serre proved that for such A and B the image of the adelic representation induced by A × B equals the product of the images induced by A and B, up to a finite index; the claim readily follows.Our strategy is inspired by Serre's work.However, one of the advantages of working with elliptic curves, as opposed to higher dimensional abelian varieties, is the open image theorem [Ser68,Ser72].This theorem, together with its analogue for CM elliptic curves [ST68], classifies the ℓ-adic representation images of elliptic curves in terms of Q-algebraic groups.With the help of these algebraic groups, the answer to Question 1.1 is essentially (but nontrivially) a consequence of Goursat's lemma.
The open image theorem for higher-dimensional abelian varieties is not known in general -indeed, it cannot hold for an abelian variety which is not Hodge maximal.Nonetheless, the Mumford-Tate conjecture claims that, for an abelian variety A over a sufficiently large number field, its ℓ-adic Galois representation images are still classified by a Q-algebraic group, the Mumford-Tate group MT(A), which is defined in terms of the Hodge structure H 1 (A(C), Q). (For a quick review of the Mumford-Tate group and the related conjecture, see Section 3.2.For an abelian variety A defined over a subfield K ⊂ C, we will often abuse notation and write H 1 (A, Q) for the homology group H 1 ((A × Spec K Spec C)(C) an , Q), endowed with its Hodge structure, and define H 1 (A, Q) in an analogous fashion.) In this article, assuming the Mumford-Tate conjecture, using Galois theory, and generalizing Serre's idea to algebraic groups beyond GL 2 , we are able to prove a criterion for the essential torsion finiteness of pairs of abelian varieties.
Theorem 1.3 (see Theorem 4.10 for a more detailed version).Suppose A and B are two absolutely simple abelian varieties defined over a number field K, and suppose that the Mumford-Tate conjecture holds for A and B.
Then A is potentially torsion infinite for B if and only if When this holds, for each prime ℓ there exists a finite extension L ℓ /K such that Insofar as (the image of) the action of Galois on torsion points is constrained by the Mumford-Tate group, it is not surprising that a relation on Mumford-Tate groups can force a resonance among torsion fields.What is perhaps more interesting is that just the presence of infinite torsion -for example, if A[ℓ](K B ) is nontrivial for an infinite, but sparse, set of primes -is enough to constrain the relation between MT(A × B) and MT(B).In particular, we will see that the existence of ℓ-torsion for infinitely many ℓ forces the presence of ℓtorsion for ℓ in a set of positive density.
The Mumford-Tate group MT(A) is canonically an extension of G m by the Hodge group, or special Mumford-Tate group, sMT(A).Ichikawa [Ich91] and Lombardo [Lom16] have investigated conditions under which (1.3) sMT(A × B) = sMT(A) × sMT(B).
(For example, this holds if A and B satisfy a certain "odd relative dimension" condition and at least one is not of Type IV in the Albert classification.)When A and B satisfy (1.3), we have dim MT(A × B) = dim(MT(A)) + dim(MT(B)) − 1 > max {dim MT(A), dim MT(B)} .
Theorem 1.3 then immediately implies that A and B are mutually essentially torsion finite.
(See the last part of Section 4.4 for more details.)Taken together with our main theorem, Ichikawa and Lombardo's results often imply a positive answer to our main question 1.1, except when both A and B are of Type IV.Thus Question 1.1 is particularly interesting when both A and B are of Type IV, such as when both A and B have complex multiplication (CM) over K.
In fact, there do exist examples where Question 1.1 has a negative answer.For instance, the Jacobian of a certain genus 4 curve [Shi82,Example 6.1] decomposes into a product of a potentially CM elliptic curve and a simple potentially CM abelian threefold.However, one can check that the elliptic curve is torsion infinite for the threefold (see [Lom23, Theorem 1.2]).In addition, Lombardo [Lom23] constructed infinitely many pairs of nonisogenous CM abelian varieties for which the answer to Question 1.1 is again negative.As a complement to Lombardo's work, we give a sufficient and necessary condition for answering our main question for CM pairs, as follows.
Let A be an isotypic abelian variety over a number field K with CM by a CM field E, and suppose that K contains E. (Recall that an abelian variety is said to be isotypic if it is isogenous to some power of a simple abelian variety.)We will see in Section 5.1 that there is a surjection of algebraic tori which induces an inclusion of character groups In fact, MT(A) only depends on the CM type of A. With this reminder, we can state a version of our main theorem for abelian varieties with complex multiplication.For a torus Theorem 1.4 (see Theorem 5.4 for a more detailed version).Let A 1 and A 2 be isotypic potentially CM abelian varieties over a sufficiently large number field K, with respective Mumford-Tate groups T 1 and T 2 .Using the inclusions X * (T i ) ֒→ X * (T K ), either: Theorem 1.4 is unconditional because the Mumford-Tate conjecture is known for CM abelian varieties (see Lemma 5.1).
We briefly compare this result to Zarhin's work [Zar87].Suppose B has complex multiplication over K but A does not even have potential complex multiplication.Note that K B is an abelian extension of K. Zarhin's result implies that A(K ab ) tors is finite; a fortiori, A is essentially torsion finite for B. However, if A is also of CM type, then Theorem 1.4 gives finer information on whether A is essentially torsion finite for B.
A morphism A → B induces a map of homology groups H 1 (A, Q) → H 1 (B, Q), and thus a morphism of Tannakian categories H 1 (A, Q) → H 1 (B, Q) and ultimately of Mumford-Tate groups MT(B) → MT(A).More generally, a correspondence between A m and B n induces a relation between MT(A) and MT(B); and the class of such a correspondence is a Hodge class on A m × B n .
In Section 5.2, we will see that if the CM abelian variety A is torsion infinite for the CM abelian variety B, then there is a nonempty Q-vector space of interesting Hodge classes on some product A m × B n ; perhaps not surprisingly, we call such a class a torsion infinite class.These Hodge classes are "extra", in the sense that they are not in the span of classes pulled back from A m and B n .Conversely, we show that the presence of such a class implies that A is torsion infinite for B.
Of course, the Hodge conjecture predicts that torsion infinite classes are actually the classes of cycles on A m × B n .It would be interesting to see, even in special cases, if one can geometrically realize torsion infinite classes.
In addition to the above applications, thanks to the work of Moonen and Zarhin on the Hodge groups of abelian varieties of low dimension [MZ99], one can compare the Mumford-Tate groups of every possible pair of absolutely simple abelian varieties up to dimension 3.As a consequence, we give a positive answer to Question 1.1 for most pairs of such abelian varieties.Precisely, following the classification in their paper, we prove: Theorem 1.5 (also Theorem 5.14).Suppose A and B are absolutely simple abelian varieties over a common number field and assume that they are non-isogenous over C. Suppose that dim A ≤ dim B ≤ 3. Then A and B are mutually essentially torsion finite except for the following cases: (a) A is a CM elliptic curve and B is a CM abelian threefold.Then B is essentially torsion finite for A; and A is potentially torsion infinite for B exactly when there is an embedding of Q-algebras End 0 (A) ֒→ End 0 (B).(b) A is a CM elliptic curve and B is an abelian threefold of type IV but not CM.Then B is essentially torsion finite for A; and A is potentially torsion infinite for B exactly when there is an isomorphism of Q-algebras End 0 (A) ∼ = End 0 (B).(c) A and B are both CM abelian threefolds.
(In (c), the essential torsion finiteness depends on the CM-types of A and B as in Theorem 5.4.) Again, this result is unconditional since the Mumford-Tate conjecture is known to hold for simple abelian varieties of dimensions less than 4 [MZ99].
This paper is structured as follows.In Section 2, we collect some basic results on representations of algebraic groups.In particular, we introduce the notion of a collection of subgroups of bounded index (of the F ℓ -points of a group scheme over Z[1/N]); this allows us to infer information about a representation of an algebraic group from data about the behavior of abstract subgroups of its finite-field-valued points.In Section 3, we establish notation and review facts (and conjectures) concerning the Galois representations attached to abelian varieties.We finally turn to the torsion-finiteness question itself in Section 4.1, establishing our main result (Theorem 1.3) in Section 4.4.The paper concludes with a detailed analysis of CM ( §5.1) and low-dimensional ( §5.3) pairs of abelian varieties, and of certain extra Hodge classes which are the hallmark of torsion-infinite pairs of CM abelian varieties ( §5.2).
It turns out that while we were working out these results, Lombardo studied a similar problem with somewhat stronger restrictions [Lom23].Two abelian varieties A and B over a number field K are said to be strongly iso-Kummerian if for each positive integer d we have (1) Condition (1.4) is much stronger than our (potentially) torsion infinite condition.In fact, (1.4) forces ) is finite for every ℓ, but nontrivial for infinitely many ℓ.One of our main contributions in this paper is to rule out this possibility (under the Mumford-Tate conjecture, as usual).
(3) Lombardo shows that if A and B are strongly iso-Kummerian, then the natural projections MT(A × B) → MT(A) and MT(A × B) → MT(B) are isogenies [Lom23, Lemma 3.2].We are able to deduce this conclusion from the weaker hypothesis that A and B are mutually potentially torsion infinite (Corollary 4.14).

PRELIMINARIES
2.1.Reminders on algebraic groups.We collect some standard, useful facts on algebraic groups.
Lemma 2.1 (Goursat's Lemma).Let G 1 , G 2 , and G 12 be either abstract groups or algebraic groups over a field.Suppose G 12 is endowed with an inclusion ι : Let M 12 = ker(π 2 • ι), and let H 12 ∼ = M 12 be the image of M 12 under the isomorphism G 1 × {e} ∼ = G 1 ; define M 21 and H 21 analogously.Then under the composite map Proof.This is standard; see, e.g., [Rib76, Lemma 5.2.1] for the case of abstract groups.The constructions of H ij and M ij also make sense in the category of algebraic groups, and the asserted properties may be verified pointwise, as in [Rib76].
Remark 2.2.Under the hypotheses of Lemma 2. Finally, when studying CM abelian varieties in Section 5, we will need to work with algebraic tori.
Let K be a perfect field.An algebraic torus T/K is an algebraic group such that . Let X * (T) be the (absolute) character group X * (T) = Hom(T K , G m,K ), and let X * (T) Q = X * (T) ⊗ Q; then T → X * (T) gives a contravariant equivalence between the category of algebraic tori over K and the category of finite free Z-modules with a continuous action by the absolute Galois group Gal(K) := Gal(K/K).This extends to a contravariant equivalence between the category of K-groups of multiplicative type and the category of finitely generated Z-modules with continuous Gal(K) action.We have 2.2.Representations of algebraic groups.
2.2.1.Fixed spaces.Let G/K be an algebraic group over a field.Let V/K be a finite-dimensional representation of G, i.e., a finite-dimensional vector space V equipped with a morphism G → GL V of algebraic groups.The schematic fixed space of V under G is We define the naïve fixed space as More generally, if Γ ⊂ G(K) is an abstract subgroup, the subspace fixed by Γ is Lemma 2.5.We have V G ⊆ V G(K) , with equality if K is infinite and G/G • is a split étale group.
Proof.The first statement is trivial; for the second, use the fact that under the stated hypotheses, G(K) is Zariski dense in G.
Lemma 2.6.Let ρ : G → GL V be a morphism of algebraic groups over K, and let M ⊂ G be a normal algebraic subgroup.Then V M is stable under G, and thus is a sub-G-representation of G.
Proof.It suffices to verify this after passage to the algebraic closure of K, so we may and do assume that G(K) is dense in G, and that M(K) is dense in M. It now suffices to show that, for each g ∈ G(K), gW ⊂ W. Since W is fixed by the normal subgroup M, gW is fixed by gMg −1 = M, and so gW ⊂ V M = W.
Lemma 2.7.Let V/F ℓ be a finite-dimensional vector space, and let G be an abstract group equipped with a representation ρ : Proof.Choose some w ∈ V ρ(H ℓ ) {0}, and let W = F ℓ [G]w be the subspace spanned by its G-orbit.The representation Proof.Suppose #H ℓ > C H • #H(F ℓ ) and (using Lemma 2.9) #α(H(F ℓ )) We then have the easy estimates This proves (a).Let P ℓ = ker α| H ℓ .Then = C H #P(F ℓ ).
be the multiplicity of 1 as a root of the characteristic polynomial of ρ(g).Let G ρ,≥m be the locus of those g for which m(g, ρ) ≥ m. (Schematically, G ρ,≥1 may be constructed by pulling back the composite morphism / / G a by the zero section Spec Z[1/N] → G a , where the map eval 1 means evaluating the characteristic polynomial at 1; for other values of m, G ρ,≥m may be constructed by considering higher derivatives of the characteristic polynomial.)Lemma 2.11.Suppose that there is an infinite collection of primes L such that, if ℓ ∈ L, then r ℓ (G, ρ) = r.Then we have: Proof.We assume r > 0, since (by specialization) the statement is trivial if r = 0.
For (a) it suffices to apply, to the characteristic polynomial of ρ(g), the following elementary observation.Let f (T) ∈ Q[T] be any polynomial; since "clearing denominators" does not alter the roots of f , we may and do assume f (T) ∈ Z[T].Suppose λ ∈ Z.If ℓ is sufficiently large, relative to the coefficients of f and to λ, then f (λ) = 0 if and only if f (λ) ≡ 0 mod ℓ; and, by taking the first r − 1 derivatives of f , a similar result holds for roots of higher multiplicity.
We now prove (b).For each ℓ ∈ L, let Y ℓ ⊂ V ℓ be the subspace fixed by ss be the open and dense semisimple locus (e.g., [Hum75, Theorem 22.2]), and let be the m 0 -dimensional subspace fixed by g.After choosing an integral model of W g , for all but finitely many ℓ the reductions g ℓ ∈ G(F ℓ ) and W g,ℓ ⊂ V ℓ are well-defined; and for ℓ ∈ L, we have If ℓ is sufficiently large as to avoid the finitely many primes of bad reduction for the By the density of G * (Q), W is fixed by all of G Q , and so r Q (G, ρ) ≥ r; again, by specialization, we find that equality holds.This proves (b).Since W = V G Q Q , we may conclude (c), as well.Now let {G ℓ } be a collection of bounded subgroups of G, and let In the statement below, "sufficiently large" depends only on dim V and the constant in (2.2); however, in our applications, we don't have control over this constant.
Proof.For any ℓ, let W ℓ ⊂ V ⊗ F ℓ be a subspace of dimension r, and let Fix G,W ℓ ⊂ G F ℓ be the subgroup scheme which fixes For any connected group H of dimension d over a finite field F, we have Lemma 2.13.Let {G ℓ } be a collection of bounded subgroups of G. Suppose that there is an infinite collection of primes ≥ r for all but finitely many ℓ.
Proof.Under the hypothesis, r(G Q ℓ 0 , ρ) ≥ r; for (a), it then suffices to note that, since G Q is connected, the formation of the fixed points of the action of G is stable under the base change Q ֒→ Q ℓ 0 ( §2.2.1).Part (b) follows by specialization.

Interlude on étale group schemes.
If (S, s) is a geometrically pointed connected scheme, then a (not necessarily connected) finite étale group scheme G → S is tantamount to an action, by group automorphisms, of π 1 (S, s) on the abstract finite group G s .We will say that G is split if this action is trivial.
Lemma 2.15.Let G/Z[1/N] be an étale group scheme, and let M ⊂ G be a normal sub-group scheme.Suppose that there exists an ℓ Proof.Let Spec(R) → Spec(Z[1/N]) be a Galois étale cover which trivializes G and M; let K = Frac(R).Let ℓ ∤ N be a prime, and let λ be a prime of R lying over ℓ.The Artin symbol (λ, K/Q) determines G(F ℓ ) and M(F ℓ ) as abstract groups, and the equality Under the hypotheses, for any ℓ in the set of positive density for which (ℓ, The claim when G is split is trivially true, since then M is split, too, and 2.2.5.Representations of group schemes.We now turn to working with a smooth group scheme G/Z[1/N].We will often assume that G has reductive connected component of identity, i.e., that for each s (In fact, we will use the same nomenclature, and deduce the same conclusions, for group schemes over an arbitrary base.)(Without this assumption, it is known that G/G • is an étale algebraic space [AHPL16, Lemma 2.1], but one may need to enlarge be a smooth affine algebraic group scheme with reductive connected component of identity.Let V be a free Z[1/N]-module of finite rank, and let ρ : G → GL V be a homomorphism of algebraic groups.Let {G ℓ } be a collection of bounded subgroups of G. Suppose that there is an infinite collection of primes Proof.By Lemma 2.13, applied to the bounded subgroups this proves (a).Now suppose that there is some ℓ 0 ∈ L for which G ℓ 0 meets every geometrically irreducible component of Replace V with the eigenspace where G • acts with eigenvalue one; then V has rank at least r.The representation ρ : Let M be the group scheme M = ker(ρ W ) ⊂ G.We have G ℓ 0 ⊆ M(F ℓ 0 ).Parts (b) and (c) now follow from Lemma 2.15.
Lemma 2.17.Let G/Z ℓ be a smooth affine group scheme with reductive connected component of identity, and suppose that ℓ we may write V uniquely as a direct sum of irreducible G representations over Z ℓ .By hypothesis, there is a free Z ℓ -module W ⊂ V, stable under G and of rank at least r, such that G(F ℓ ) acts trivially on W ⊗ F ℓ .We have a commutative diagram Suppose g ∈ G(Z ℓ ), and let α be any eigenvalue of ρ W (g).
Proof.By Lemma 2.14, r ℓ (G • , ρ) ≥ r for all ℓ.Using the same technique as in the proof of Lemma 2.16 to move from G • to G, we find that r Q (G, ρ) ≥ r; for ℓ in a set of positive density, r ℓ (G, ρ) ≥ r; and that this holds for all sufficiently large

TORSION POINTS ON ABELIAN VARIETIES
3.1.Torsion points and Galois representations.For the purpose of establishing notation, let A/K be an abelian variety over a perfect field.For a natural number N, we let K A,N be the field of definition of the N-torsion of A. We further let K A,ℓ ∞ = n K A,ℓ n , and let K A = N K A,N be the field obtained by adjoining the coordinates of all torsion points of A.

Finally, we let A N = A[N](K) be the geometric N-torsion, and
For a fixed prime ℓ, we have the usual representations with respective images Γ A/K,ℓ and Γ A/K,ℓ ∞ .
3.1.1.Independence.Serre has shown that, while the ℓ-adic representations attached to an abelian variety are compatible, they are also independent.For an abelian variety A/K and a prime ℓ, briefly let K ′ A,ℓ := ℓ∤N K(A N ).Say that A/K has independent torsion fields if, for each prime ℓ, the Galois extensions K A,ℓ ∞ and K ′ A,ℓ are linearly disjoint over K. (Note that the compositum K A,ℓ ∞ K ′ A,ℓ is simply K A , the field generated by the torsion points of A.) Lemma 3.1.Let A/K be an abelian variety over a number field.
(a) There exists a finite extension K ind /K such that A/K ind has independent torsion fields.(b) If L/K ind is any algebraic extension, then A/L has independent torsion fields.
Proof.See [Ser13, Théorème 1 and §3] or [BGP18,§1].Lemma 3.2.Let A and B be abelian varieties over a number field K, and suppose that A × B has independent torsion fields.Then , we denote by K(P) the extension of K by adjoining the coordinates of P. Then K(P) ⊂ K A×B,ℓ ∞ .And we also have which tells that every inclusion here is actually an equality.Hence one has that In general, G A/K,ℓ does not have to be connected, but when K is a number field G A/K,ℓ will be connected after a finite extension of K which is independent of ℓ.More precisely, Lemma 3.3.Suppose K/Q is a finite extension.Then (a) The finite quotient group G A/K,ℓ /G • A/K,ℓ is independent of ℓ.(b) There exists a finite extension K conn of K such that, if L is any finite extension of K conn and ℓ is any prime number, the corresponding G A/L,ℓ is connected.
(In contrast, Example 5.7 will show that if K is algebraic but infinite, then such a finite connectedness extension need not exist.)3.2.Mumford-Tate conjecture.This section is devoted to recalling the Mumford-Tate conjecture.In particular, we will review a result of Cadoret and Moonen [CM20, § 1] and of Hindry and Ratazzi [HR16] which states that as ℓ varies, the ℓ-adic image of the Galois group is a bounded index subgroup of the Z ℓ -points of the Mumford-Tate group.
Let K be a number field, embedded in C. To ease notation slightly, we write MT(A) for the Mumford-Tate group of an abelian variety A over K, i.e., MT(A) := MT(H 1 (A C , Q)) (cf.Section 4.1).This is a connected Q-algebraic group.Let G A be the Zariski closure of MT(A) in GL H 1 (A C ,Z) ; it is a group scheme over Z. Then G A is smooth over Z[1/N A ] for some positive integer N A , and G A,Q = MT(A).
If K = K conn , then it is known that there is a natural inclusion

, and thus an inclusion
algebraic groups over Q ℓ .The Mumford-Tate conjecture asserts that this inclusion is actually an isomorphism.More precisely, for every The following conjecture claims the comparison result of the two group schemes.
Conjecture 3.4.[CM20, Mumford-Tate Conjecture] With the above notations, G A,ℓ = G • A/K,ℓ .Remark 3.5.Conjecture 3.4 is equivalent to the usual statement In this paper, the Mumford-Tate conjecture is a standing assumption we require in order to make any significant progress.The conjecture is known to be true for large classes of abelian varieties.For example, it is known that an absolutely simple abelian variety A of dimension g satisfies the Mumford-Tate conjecture in any of the following settings: (1) g is prime [Rib83, Tan82, Tan87]; (2) g ≤ 3 [MZ99]; (3) End K (A) = Z and g satisfies certain numerical conditions (for instance, g is odd) [Pin98]; (4) A has complex multiplication [Poh68,Yu15].
Our list is far from complete.See also [Vas08] and the discussion in [Moo17, Section 2.4.] for additional references and known results.Moreover, if the Mumford-Tate conjecture is true for abelian varieties A and B, then it is also true for their product A × B [Com19].
In the presence of the Mumford-Tate conjecture we have good control over Γ A/K,ℓ ∞ .
Theorem 3.6.[CM20, Theorem A] [HR16, Théorème 10.1] Let A be an abelian variety defined over K, assume K = K conn , and assume that the Mumford-Tate conjecture is true for A. Then the index [G A (Z ℓ ) : In particular, {Γ A/K,ℓ } is a collection of bounded subgroups of G A .

TORSION-FINITE PAIRS
Equivalently, it is the Tannakian fundamental group (really, the group which represents automorphisms of the fiber functor which sends a Hodge structure V to its underlying vector space |V|) of H 1 (C, Q) , the tensor category generated by the Hodge structure and H 1 (C, Q) coincide, and we may use either description to compute MT(C).We have Thus, the three algebraic groups G A , G B and G A×B satisfy the hypotheses of Goursat's lemma (Lemma 2.1), and fit in a diagram as follows. (4.1) y y y y r r r r r r r r r r r x x x x q q q q q q q q q q q π B % % % % ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ sG A sG B In particular, sG A×B is the inverse image in sG A × sG B of the graph of an isomorphism (4.2) .
For future use, we record the following observation.Proof.The category H 1 (A × B, Q) is equivalent to the category of representations of MT(A × B).So H 1 (A, Q) and H 1 (B, Q) are isomorphic in the category of Hodge structures, or equivalently in the full subcategory generated by H 1 (A × B, Q), if and only if they are isomorphic representations of MT(A × B).For weight reasons, it suffices to verify this for the Hodge group sMT(A × B).Riemann's theorem -that the isogeny class of an abelian variety is determined by its Hodge structure -proves the equivalence of (a) and (b).
If A and B are isogenous, it's well-known that MT(A × B) ∼ = MT(A) ∼ = MT(B) (e.g., [Moo99, Rem.1.8]).Conversely, under the hypothesis of (c), weight considerations show that the corresponding hypothesis holds for Mumford-Tate gorup, too.Now use the fact that MT(A) is canonically isomorphic to the image of MT(A × B) is GL H 1 (A,Q) and the analogous statment for B in order to deduce (b).Now suppose that A and B have complex multiplication (see §5.1 for a review of this concept).Then A × B does, too, and the Mumford-Tate groups G A , G B and G A×B are all tori.Taking character groups in (4.1) yields a diagram of Z-modules In particular, we may use this diagram to compute H A,B , a group of multiplicative type; it is the group whose character group is If we identify X * (G A ) and X * (G B ) with their images under, respectively, the inclusions (π A • ι) * and (π B • ι) * , we may rewrite this as . (4.5) 4.2.Galois representations for a pair of abelian varieties.Now further suppose that K is finitely generated over Q, and assume that A and B satisfy the Mumford-Tate conjecture.
Since our main results concern potentially infinite torsion, we will assume that A × B has connected, independent Galois representations.
For a positive integer N, we identify the Galois group Γ A/K,N with a subgroup G A,N of G A (Z/N), and make similar identifications of the image of Gal(K) acting on the N-torsion of B and of A × B. The N-torsion fields of A and B are then arranged in the following tower, where each extension is labeled with its corresponding Galois group.

Lemma 4.2. Let A and B be abelian varieties over a number field K. Suppose that A × B has independent Galois representations. Then for each prime ℓ, A[ℓ](K B ) is nontrivial if and only if
Lemma 4.3.Let A and B be abelian varieties over a number field K. Suppose that A and B satisfy the Mumford-Tate conjecture, and that A × B has connected Galois representations.Then Proof.Since A × B satisfies the Mumford-Tate conjecture, {G A×B,ℓ } is bounded in G A×B (Theorem 3.6).Now apply Lemma 2.10(b) to the exact sequence to deduce (a).For a fixed ℓ, there exists an 4.3.Preliminaries on torsion finiteness.If two abelian varieties are isomorphic, or more generally isogenous, then it is easy to see that each is torsion infinite for the other: Lemma 4.4.Let A and B be abelian varieties over a number field K.

(a) If A and A ′ are isogenous over K, and if B and B ′ are isogenous over K, then A is torsion finite for B over K if and only if A ′ is torsion finite for B ′ over K. (b) If L/K is a finite extension, and if A L is torsion finite for B L over L, then A is torsion finite for B over K. (c) If m and n are two positive integers, then A is torsion finite for B over K if and only if A m
is torsion finite for B n over K.
Proof.Let g : B → B ′ be an isogeny of exponent N; there is an isogeny Then for any (not necessarily finite) field extension F/K one has In particular, B(F) tors and B ′ (F) tors are either both finite or infinite.Moreover, one can also deduce that K B = K B ′ .We deduce (a) after applying the same argument to A and A ′ .Part (b) is obvious since L B contains K B .Part (c) follows from the observation that K C,N = K C r ,N for any abelian variety C/K and any natural numbers r and N.

Lemma 4.5. Suppose A and B are abelian varieties over a field K. There exist a finite extension L and an isogeny of L-abelian varieties ⊕ r i=1 A m i i → A L with each A i absolutely simple; and A is essentially torsion finite for B if and only if each A i is essentially torsion finite for B L .
Proof.The existence of such an L and a factorization of A L is standard; since we are only concerned with essential torsion finiteness, we may and do assume L = K.Then A is essentially torsion finite for B if and only if ⊕ i A m i i is (Lemma 4.4(a)), which obviously holds if and only if each summand A m i i is essentially torsion finite for B. By Lemma 4.4(c), this holds if and only if each A i is essentially torsion finite for B.

4.4.
Potentially torsion infinite pairs.If H A,B is connected and if A acquires infinite torsion over K B , then A acquires ℓ-power torsion for all ℓ: Lemma 4.6.Let A and B be abelian varieties over a number field K. Suppose that A and B satisfy the Mumford-Tate conjecture, and that A × B has connected independent Galois representations.
Suppose that H A,B is connected.Then the following are equivalent.
Proof.It suffices to show that (a) implies each of (b) and (c).Note that, since By Lemma 4.3, {H A,B,ℓ } is a collection of bounded subgroups of the connected group H A,B , and each Suppose that (a) holds; then ) is infinite for some ℓ 0 .In the former case, A[ℓ](K B,ℓ ) is nontrivial for infinitely many ℓ (Lemma 4.2), and thus r ℓ (H A,B , ρ A ) is positive for infinitely many ℓ; in the latter, r Q ℓ 0 (H A,B , ρ A ) is positive.Thus by Lemma 2.13 or 2.14, ρ Q (H A,B , ρ A ) is positive; therefore, so is ρ ℓ (H A,B , ρ A ) and ρ Q ℓ (H A,B , ρ A ) for each ℓ.Therefore, both (b) and (c) hold.
In the absence of a connectedness hypothesis on H A,B , our results are less balanced.Moreover, the Mumford-Tate conjecture doesn't immediately imply that H A,B,ℓ meets every geometrically irreducible component of H A,B,ℓ .In situations where this is known, however, we can deduce the following statement.

Lemma 4.7. Let A and B be abelian varieties over a number field K. Suppose that A and B satisfy the Mumford-Tate conjecture and that A × B has connected, independent Galois representations.
Suppose that A(K B ) tors is infinite, and that there exists some ℓ 0 such that A

Additionally, assume that H A,B,ℓ 0 meets every geometrically irreducible component of H A,B,ℓ 0 , and H A,B,Z ℓ 0 is smooth. Then for ℓ in a set of positive density, A[ℓ](K B ) is nontrivial and A
) is nontrivial for infinitely many ℓ, this follows from Lemma 2.16, applied to the collection of bounded subgroups {H A,B,ℓ } of H A,B .If instead there exists some ℓ 1 such that A[ℓ ∞ 1 ](K B ) is infinite, then as in the proof of Lemma 2.18 we find that r Q (H • A,B , ρ A ) is positive; and that for ℓ in a set of positive density (namely, the set of ℓ relatively prime to [H A,B : H • A,B ] and with the same Artin symbol as ℓ 0 in some finite splitting field for ) is also positive.The hypothesis on ℓ 0 in Lemma 4.7 seems difficult to work with abstractly, although in explicit examples one can compute H A,B /H • A,B (e.g., Example 5.7) and thereby make progress.However, we can still make a uniform statement purely in terms of torsion, at the cost of surrendering some control of the precise field over which A acquires infinite ℓ-torsion.

Lemma 4.8. Let A and B be abelian varieties over a number field K. Suppose that A and B satisfy the Mumford-Tate conjecture and that A × B has connected independent Galois representations.
Suppose that A(K B ) tors is infinite.
then we may take n ℓ = 1.
Proof.Since A(K B ) tors is infinite, there exist infinitely many ℓ such that A[ℓ](K B ) is nontrivial, or there is some ℓ 0 such that A[ℓ ∞ 0 ](K B ) is infinite .By Lemma 2.16 or 2.14 as appropri- ate, r : Remark 4.9.In the context of Lemma 4.8(b), one might hope that there is a finite extension L B of K B such that A[ℓ](L B ) is nontrivial for each ℓ.(Even more optimistically, one might hope that such an L B is the compositum of K B and a finite extension of K.) However, there is no independence-of-ℓ connectedness result for Galois representations of infinite algebraic extensions of Q to which one might appeal.In fact, we will see below (5.7) that in general, no such uniform-in-ℓ finite extension exists.In this sense, Lemma 4.8(b) is optimal without additional hypotheses.

Theorem 4.10. Let A and B be abelian varieties over a number field K. Suppose that A and B satisfy the Mumford-Tate conjecture, and that A is absolutely simple. Then the following are equivalent: (a) A is potentially torsion infinite for B; (b) dim H
and (e) there exists a finite extension K ′ of K such that for all sufficiently large ℓ, Proof.Suppose that A is potentially torsion infinite for B.Then, possibly after replacing K with a finite extension, we find that r := r Q (H • A,B , ρ A ) > 0 (Lemma 4.8(a)).Note that H A,B , and thus H • A,B , are normal subgroups of MT(A) (Lemma 2.4).Because A is absolutely simple and MT(A) is reductive, This implies that H • A,B is trivial, and thus dim H A,B = 0.The converse, that (b) implies (a), is easy.Indeed, suppose dim H A,B = 0, and fix a prime ℓ.Then Gal(K B ) acts on T ℓ A through a subgroup of H A,B (Z ℓ ), which is by hypothesis a finite group; after passage to a finite extension, the Galois group acts trivially on all ℓpower torsion points.
The equivalence of (b), (c), and (d) is a standard observation about reductive groups (Lemma 2.3).Now suppose (b) holds.After replacing K with a suitable finite extension, we may and do assume that A × B has connected independent Galois representations.Then Lemma 4.8(c) shows that there exists some n ℓ such that Gal(K B K A,ℓ n ℓ ) acts trivially on T ℓ A; and that if ℓ ∤ [H A,B : H • A,B ], then we may take n ℓ = 1.The proof is completed with the trivial observation that (e) implies (a).
Remark 4.11.In Theorem 4.10, by Lemma 4.5, it suffices to assume that A is absolutely isotypic.

Corollary 4.12. Let B/K be an abelian variety over a number field for which the Mumford-Tate conjecture holds, and let E/K be an elliptic curve with complex multiplication. Then either E is potentially torsion infinite for B or sG
Proof.The Mumford-Tate conjecture holds for E and thus for E × B, and E is visibly absolutely simple; it therefore suffices to show that if E is essentially torsion finite for B then the special Mumford-Tate group of the product is the product of the special Mumford-Tate groups.By Theorem 4.10, dim H E,B > 0. Since H E,B is a positive-dimensional subgroup of the one-dimensional torus sG E , it follows that H E,B = sG E , and thus sG E×B = sG E × sG B (Remark 2.2).

Corollary 4.13. Let A and B be abelian varieties over a number field, of respective dimensions d A and d B . Suppose that A and B satisfy the Mumford-Tate conjecture, that A is absolutely simple, and that
Then A is essentially torsion finite for B.
Proof.Let r A and r B denote the respective ranks of the Mumford-Tate groups of A and B.
On one hand, we have the trivial bound r B ≤ d + 1.On the other hand, a weak form of [Orr15, Thm.1.2] implies that r A ≥ 1 3 (log 2 d A + 2).Therefore, hypothesis (4.6) implies that H B,A ), we find that the rank of H A,B is positive, and thus dim H A,B > 0. Now apply Theorem 4.10.
The embedding i induces an E-action on the Lie algebra Lie(A C ).The character of this Erepresentation is given by ∑ ϕ∈Φ ϕ for some subset Φ ⊂ Hom Q (E, C).Let c be the complex conjugation on C. Then (5.1) where Φ c = {c • ϕ|ϕ ∈ Φ}.The pair (E, Φ) is called the CM-type of A (or (A, i)).On the other hand, for any pair (E, Φ) where E is a CM-algebra and the subset Φ ⊂ Hom Q (E, C) satisfies (5.1), there exists a CM abelian variety A 0 defined over a number field equipped with an action i : E ֒→ End 0 (A 0 ) with CM-type (E, Φ).The correspondence between CM-types and isogeny classes of abelian varieties with action by a CM algebra is wellunderstood; see, e.g., [Mil20, Proposition 3.12].Fix an embedding K ֒→ C, and let K be the algebraic closure of K in C. Then any embed- ding E ֒→ C factors through K, and we have a bijection Hom Q (E, K) The fixed field E * := E H of H is called the reflex field of the CM-type (E, Φ).
We now introduce the reflex norm associated with the CM abelian variety A. Recall ( §2.1) that for any finite extension F of Q, we let T F := Res F/Q G m be the Weil restriction of the multiplicative group.We define ) is a torsor under Gal( E/Q), and the choice of embedding E ֒→ K gives a bijection between these two sets.We use this to define, for φ ∈ Hom defines a map of algebraic tori N Φ −1 : T E → T E .This map factors through T E * and has image contained in T E .More precisely, we have a commutative diagram of Q-tori where N E/E * is the usual norm map and N Φ is called the reflex norm of (E, Φ).For any finite extension L/E * , we define N L,Φ = N Φ • N L/E * .We will call N L,Φ the L-reflex norm of (E, Φ).Note that N L/E * is a surjective map.The image of the map N L,Φ : T L → T E is independent of L, and we denote it by T Φ .See, e.g., [Yu15,Lem. 4.2] for an explicit calculation of X * (T Φ ).
Consider the special case where E is a Galois extension of Q.Then E contains the reflex field E * , and T Φ , as the image of T E under N E,Φ : Proof.This is due to Pohlmann [Poh68, Thm.5]; see also [Yu15,Lemma 4.2] for a modern proof.(In fact, while [Yu15] focuses on simple abelian varieties, the proof given there works verbatim in the non-simple case, too.)Remark 5.2.Note in particular that (the character group of) the torus MT(A) can be explicitly described using a CM-type (E, Φ) of A. It is usually more convenient to assume that E is Galois over Q; and in studying the essential torsion finiteness problem for CM abelian varieties, we can always do this.Indeed, if E/Q is not Galois, choose a CM Galois extension ).Thus, for our purposes, we may restrict our attention to Galois CM fields.

Galois representations. For use in later examples, we recall the calculation of the Galois representation of a CM abelian variety following [ST68, §7
] and [Yu15,§3].Let A/K be an abelian variety with complex multiplication by E. Let ℓ be a rational prime which does not divide the index [O E : End(A)].(This condition only rules out finitely many primes.Alternatively, for our applications we may replace K by a finite extension and adjust A in its isogeny class, in which case we may assume that End(A) = O E .)Then O E ℓ := O E ⊗ Z ℓ is a direct sum of discrete valuation rings, and the Tate module T ℓ A is free of rank one over O E ℓ .The Galois group of K acts E-linearly on T ℓ A, and so the Galois representation ρ A/K,ℓ ∞ of Gal(K) factors through Gal(K) ab .Composing the ℓ-adic representation with the Artin reciprocity map (in the idelic formulation of class field theory), one obtains a continuous group homomorphism which we still denote by ρ A/K,ℓ ∞ :

Gal(K) ab
After possibly replacing K with a finite extension, we now assume that K contains E * , the reflex field of the CM-type of A, so that the reflex norm N K,Φ is defined ( §5.1).Then by [ST68, Theorem 10, 11] we can concretely describe the representation ρ A/K,ℓ ∞ by (5.4) × is induced by the reflex norm map from T K to T E ; and is the unique homomorphism satisfying the following conditions: (b) The homomorphism ε is continuous, in the sense that its kernel is open in A × K .(c) There is a finite set S of places of K, including the infinite ones and those where A has bad reduction, such that where each π v is the Frobenius element attached to v [ST68, p.511].
Lemma 5.3.Let A/K be an abelian variety with nondegenerate CM type (E, Φ).Then for each ℓ, Proof.By Lemma 3.3, it suffices to prove the statement for a single ℓ, and so we assume that ℓ ∤ [O E : End(A)].As we have seen above, T ℓ A admits commuting actions by O E ℓ and Gal(K), and thus the ℓ- . A choice of K-rational polarization on A induces a symplectic form ψ on T ℓ A, which is also preserved by Gal(K) up to a scaling.Thus, the image of Gal Since these are the Z ℓ -points of a maximal torus in GSp 2 dim A -indeed, both rank GSp 2 dim A and dim MT(A) are 1 + dim A -the result follows.

Torsion finiteness.
With this preparation, we can now use Theorem 4.10 to characterize the essential torsion finiteness of CM abelian varieties in terms of CM types.
Recall that an abelian variety A over a field K is called isotypic if it is isogenous to a power of a simple abelian variety over the same field K, i.e., up to isogeny, A has a unique simple factor [CCO14, Defn.1.2.5.2].Any CM abelian variety is isogenous to a product of isotypic CM abelian varieties [CCO14, Prop.1.3.2.1], and an isotypic CM abelian variety is geometrically isotypic [CCO14, Cor.1.3.7.2].
To state our next result, it is more convenient to name our abelian varieties A 1 and A 2 .In this case we will change notation slightly and write, for example, G 1 and H 12 for G A 1 and H A 1 ,A 2 .

Theorem 5.4. Let A 1 and A 2 be two isotypic abelian varieties over a number field K, with A
Moreover, if X * (T 1 ) ⊆ X * (T 2 ), and if A 1 is simple and nondegenerate, then Proof.By Theorem 4.10 and Remark 4.11, A 1 is potentially torsion infinite for A 2 if and only if dim H 12 = 0. Since H 12 is of multiplicative type, this happens if and only if dim Q X * (H 12 ) ⊗ Q = 0.After tensoring both sides of (4.5) with Q, we find that this hap- pens if and only if If A 1 is potentially torsion infinite for A 2 , then the description of K ℓ , etc. is in Lemma 4.8.Finally, suppose we have an inclusion of integral lattices X * (T 1 ) ⊆ X * (T 2 ) and that A 1 is simple and nondegenerate.The calculation (4.5) shows that H 12 is trivial.Briefly suppose that A 1 /K has CM actually defined over K, and thus (Lemma 5.3) has connected Galois representations.Then for each natural number N we have a containment Γ A 1 ,N ⊂ T 1 (Z/N), and thus Gal(K A 2 ) acts trivially on A 1,N .Now suppose that A 1 /K merely has potential complex multiplication.The surjection T 2 → T 1 means that the splitting field of T 2 contains the splitting field of T 1 ; equivalently, we have an inclusion of reflex fields Suppose N ≥ 3 is an integer.Then all geometric endomorphisms of A 2 are defined over K A 2 ,N [Sil92].Therefore K A 2 ,N contains E * 2 , and thus E * 1 [Mil20,Prop. 7.11].Because E * 1 is simple, all geometric endomorphisms of A 1 are defined over K A 2 ,N .Therefore, the image of the action of Gal(K A 2 ,N ) on A 1,N , and a fortiori that of Gal(K A 2 ), is contained in T 1 (Z/N) (Lemma 5.3), and we conclude as before.
Remark 5.5.In the context of Theorem 5.4, let E/Q be a Galois CM field containing E 1 and E 2 , and assume that A 1 and A 2 share no common geometric isogeny factor.As in Remark 5.2, after replacing A 1 and A 2 by suitable powers, we may assume A 1 and A 2 have CM by the same field E, with respective CM types Φ E,1 and Φ E,2 ; then A 1 × A 2 has a CM type (E × E, Φ 12 ), where Φ 12 = Φ E,1 ⊔ Φ E,2 .Then T Φ i = T E,Φ i , and the compatibility of the various (reflex) norm maps is expressed in the commutativity of the following diagram of tori: In particular, in Theorem 5.4, we may compare 5.1.5.Examples.In concrete cases, Theorem 5.4 gives a way to explicitly analyze essential torsion finiteness for pairs of CM abelian varieties.
Example 5.6.Let E = Q(ζ 13 ).Then Gal(E/Q) ∼ = σ|σ 12 = 1 .There are exactly six isomorphism classes of CM-types for E, with representatives Let A i be an abelian sixfold with CM-type (E, Φ i ).Then any abelian variety with CM by E is geometrically isogenous to one of the A i ; and for 1 ≤ i ≤ 5, A i is geometrically simple.By an explicit computation, one can check that in X * (T E ) ⊗ Q we have Then for any i ≤ 5, A i is essentially torsion finite for A 6 ; and for any j ≤ 6, A j is potentially torsion infinite for A i .
Example 5.7.Let E be Q(ζ 11 ).Then Gal(E/Q) ∼ = σ|σ 10 = 1 .There are exactly four isomorphism classes of CM-type for E, with representatives Let K be a number field containing E (and, in particular, the reflex fields of each Φ i ), and let A i /K be an abelian fivefold with CM-type (E, Φ i ).Further assume that each A i has independent representations over K, and that each A i has everywhere good reduction.Let S be a sufficiently large finite set of primes of K so that the description of the Galois representations in §5.1.2holds.Then any abelian variety with CM by E is geometrically isogenous to one of the A i .For 2 ≤ i ≤ 4, A i is geometrically simple, while A 1 is geometrically isogenous to the cube of an elliptic curve with complex multiplication by Q( √ −11).By an explicit computation, one can check that in X * (T E ) ⊗ Q we have For any i ≥ 2, A i is essentially torsion finite for A 1 ; and for any j ≤ 4, A j is potentially torsion infinite for A i .Now we focus on A 1 and A 2 .
On the other hand, H 12 is not split over Q, although it does admit Q-points.Indeed, H 12 (Q) = {(0, 0), (1, 1), (2, 2)} (Z/3) ⊕ (Z/3).Consequently, Lemma 4.8 only implies that, for each ℓ, there exists some finite extension We will now use the explicit calculation of the action of Galois to show that for any finite extension L/K, A 1 [ℓ ∞ ](L 2,ℓ ∞ ) is finite for all but finitely many primes ℓ.This shows that Lemma 4.8 is essentially optimal.
Let ρ 1,ℓ and ρ 2,ℓ denote ρ A 1 /K,ℓ ∞ and ρ A 2 /K,ℓ ∞ , respectively; and let ρ i,ℓ be the pullback of ρ i,ℓ to A × K .Now suppose ℓ ∈ S, and embed Then the restriction of the ℓ-adic representations (5.4) to O × K ℓ reads as (5.6) , and i = 1, 2. Now further assume that ℓ is a prime integer that totally splits in K, and use the fact that the reflex norm N K,Φ i = N E,Φ i • N K/E , where N K/E is the usual norm map of fields.Then K ℓ is unramified over E ℓ , and thus §2, Corollary of Proposition 3].Hence (5.6) factors through (which will still be denoted by ρ i,ℓ to ease notation) , and i = 1, 2.
Since ℓ splits in K, it is also totally split in the sub-extension E. Recall that Gal(E/Q) = σ ≃ Z/(10), hence With respect to this isomorphism, every element a ℓ of O × E ℓ can be expressed by a vector of the form a ℓ = (a 1 , a 2 , a 3 , • • • , a 10 ), and σ ∈ Gal(E/Q) acts on a ℓ by cyclically permuting its coordinates.By definition, Direct computation then shows that N E,Φ 1 (a ℓ ) = (xy, x 2 y 2 , xy, x 2 y 2 , xy, x 2 y 2 , xy, x 2 y 2 , xy, x 2 y 2 ) Now suppose that xy is a primitive third root of unity; this is possible, of course, exactly if ℓ ≡ 1 mod 3. On one hand, N E,Φ 1 (a ℓ ) = 1; on the other hand, because xy ≡ 1 mod ℓ and (xy) 2 ≡ 1 mod ℓ, N E,Φ 2 (a ℓ ) acts without fixed points on O E ⊗ Z/ℓ.In particular, a ℓ ∈ ker ρ 2,ℓ ker ρ 1,ℓ .Let g ∈ Gal(K) be the image of a ℓ under the reciprocity map.By (5.6) and the formula of Serre-Tate, g ∈ ker ρ 2,ℓ ker ρ 1,ℓ .Moreover, since ρ 1,ℓ (g) = ρ 1,ℓ (a ℓ ) ∈ E × , we know that it does not have eigenvalue 1, even when working with Z/ℓ-coefficients.In particular, Finally, notice that we can produce such an a ℓ for each ℓ with ℓ ≡ 1 mod 3 which is totally split in K. Since A 1 has independent extensions, there is not a finite extension L/K on which each art(a ℓ ) acts trivially.In particular, there is no finite extension L/K such that, for each ℓ, Example 5.8.In [Lom23, Thm.5.1], Lombardo gives a construction of an infinite family of iso-Kummerian CM pairs of abelian varieties.We briefly interpret his work in the framework developed here.
Given an a CM field E which is the compositum of a cyclic totally real field of dimension g and a quadratic imaginary field, and the auxiliary choice of two integers r and h, Lombardo defines two different CM types Φ 1 and Φ 2 , and chooses corresponding abelian varieties A 1 and A 2 .After passage to a suitably large common field of definition K, one shows that the kernels of the ℓ-adic representations ρ A k ,K,ℓ ∞ coincide.
This calculation shows that the characters of T E which vanish on the image of T K under N K,Φ 1 are the same as those characters which vanish on the image of T K under N K,Φ 2 .Consequently, MT(A 1 ) and MT(A 2 ) are the same sub-torus of T E , and thus A 1 and A 2 are mutually torsion infinite.

Extra Hodge classes and torsion infiniteness.
Following the notation in the previous section, let A 1 and A 2 be two isotypic CM abelian varieties over K.In this section, we will see that if A 1 is potentially torsion infinite for A 2 , then this is explained by a certain sort of Hodge class in some degree 2w on some product A m 1 × A n 2 .The particular values of w, m and n are not unique; and even once these are specified, the class itself, or even its Q-span is not canonical.Consequently, we will call any such Hodge class a "torsion-infinite Hodge class from A 1 to A 2 ", even though it actually lives on some unspecified product A m 1 × A n 2 .Suppose that A i has a CM-type (E, Φ i ) where E is a CM Galois extension of Q.We also assume that the base field K is sufficiently large (e.g., it contains E).
We first describe the Hodge classes on A m 1 × A n 2 (see [Poh68] for more details).Let Recall that V i , the tensor category generated by V i , is equivalent to the category Rep Q (MT(A i )) of representations of MT(A i ).By Lemma 5.1 and our assump- tion for E, the reflex norm defines a quotient map N E,Φ i : T E ։ MT(A i ), which induces a fully faithful map on the categories of representations Rep Q (MT(A i )) → Rep Q (T E ).This allows us to describe the Hodge classes on A m 1 × A n 2 using the representation theory of the algebraic torus T E for any positive integers m and n.
We denote the representation T E ։ MT(A i ) ֒→ GL V i by ρ i .Note that X * (T E ) ∼ = σ∈Gal(E/Q) Z σ , and the Galois group Gal(E/Q) acts on it by left multiplication.Since E/Q is Galois, we can identify Φ i with a subset of Gal(E/Q).
For any representation ρ : T E → GL V , we let Ξ V (or Ξ ρ ) be the collection of weights of this representation.The set Ξ V is a finite sub-multiset of X * (T E ); the support supp(Ξ V ) of Ξ V -that is, those elements of X * (T E ) with nonzero multiplicity -is finite, and all multiplicities are finite.For future use, we note that if Ξ V = {α 1 , . . . ,α d } is a set of distinct characters, then supp(Ξ V ⊕m ) is the same set, and each weight now occurs with multiplicity m.Moreover, the support of Ξ ∧ r (V ⊕m ) is then (5.7) supp(Ξ ∧ r (V ⊕m ) ) = ∑ e i α i : ∑ e i = r and 0 ≤ e i ≤ m for each i .
By the definition of the reflex norm, Since Gal(E/Q) acts transitively on this set, we have as Galois modules.We also denote the 1-dimensional representation Nm : If V is any Hodge structure, the group of Hodge classes in V is Hom Q−HS (1, V), where 1 is the trivial Hodge structure.For an integer w ≥ 0 we have So the Q-span of a Hodge class can be identified with an element α ∈ . Using the polarization on V 1 , we rewrite these conditions as . Moreover, the existence of a Hodge class in degree 2w on some product A m 1 × A n 2 is equivalent to the existence of α ∈ X * (T E ) and r with 0 ≤ r ≤ 2w such that α is a Z ≥0 -linear combination of the weights in Ξ V 1 and α + (w − r)χ is a Z ≥0 -linear combination of the weights in Ξ V 2 .Let s = 2w − r.The choices (r, s) = (0, 2w) and (r, s) = (2w, 0) correspond to Hodge classes which come from A m 1 and A n 2 by pullback, while classes with r and s positive are conjecturally the classes of nontrivial correspondences between A m 1 and . we call the related Hodge classes torsion-infinite Hodge classes from A 1 to A 2 , regardless of the choice of m and n (and of r and s).As a consequence of our definition, these classes are in H r,0 (A In particular, these classes are not in the Q-span of those classes which are pulled back from A m 1 or A n 2 , and thus the torsion-infinite Hodge classes are extra Hodge classes.Proposition 5.9.Let A 1 and A 2 be two isotypic CM abelian varieties over a number field K. Suppose that Hom K (A 1 , A 2 ) = (0), i.e., that A 1,K and A 2,K have no common nontrivial isogeny factor.
Then A 1 is potentially torsion infinite for A 2 if and only if there is a torsion-infinite Hodge class from A 1 to A 2 .Proof.Assume that A 1 is potentially torsion infinite for A 2 over K.By Theorem 5.4, where c + β and c + χ are non-negative rational numbers.Choose a positive integer m such that mc + β and mc + χ are integers.Then (5.8) Let n = ∑ β∈Ξ ρ 2 mc + β .Consider the embedding G m → T E induced by Q ֒→ E, and let τ be the standard (positive) generator of X * (G m ).Then γ| G m = gτ for any γ ∈ Ξ V 1 ∪ Ξ V 2 , while χ| G m = 2gτ.Thus, restricting (5.8) to G m and computing coefficients of τ yields Conversely, if there is a torsion-infinite Hodge class from A 1 to A 2 , then there exists a weight α Remark 5.10.In the proof, we choose r = m and s = n = ∑ β∈Ξ ρ 2 mc + β for convenience.However, sometimes smaller m and n can be chosen.See Examples 5.12 and 5.7.
Before displaying some concrete examples, let us prove the following lemma.Recall our discussion of nondegenerate abelian varieties (5.1.3).
Lemma 5.11.Let A 1 and A 2 be two CM abelian varieties.Suppose that A i has a CM-type (E, Φ i ) Proof.First, we let E + be the totally real subfield of E and let U E + 1 be the norm one subtorus of T E + .For any CM type (E, Φ), where the surjection f is induced by the inclusion (ker(N E,Φ 2 )) Example 5.12.Let E be Q(ζ 7 ).Then Gal(E/Q) ∼ = σ|σ 6 = 1 .There are two isomorphism classes of CM types for E: Let A i be an abelian variety with CM type (E, Φ i ).Then A 1 is geometrically isogenous to the third power of an elliptic curve with CM by Q( √ −7), while A 2 is nondegenerate.By Lemma 5.11, A 1 is potentially torsion infinite for A 2 .In fact, we have

Low dimension abelian varieties.
In [MZ99, §2], Moonen and Zarhin list all the possible Hodge groups for absolute simple abelian varieties with dimension g ≤ 3. We will follow their classification and use the notation (g, Type) to denote an absolutely simple abelian variety with dimension g and the indicated endomorphism type in the Albert classification.For instance, (2, IV(2, 1)) refers to an absolutely simple CM abelian surface.
Theorem 5.14.Suppose A and B are absolutely simple abelian varieties over a common number field and assume that they are non-isogenous over C. Suppose that dim A ≤ dim B ≤ 3. Then A and B are mutually essentially torsion finite except for the following cases: (a) A is a CM elliptic curve and B is of type (3, IV(3, 1)), i.e., B is a simple CM abelian threefold.Then B is essentially torsion finite for A; and A is potentially torsion infinite for B exactly when there is an embedding of Q-algebras End 0 (A) ֒→ End 0 (B).(b) A is a CM elliptic curve and B is of type (3, IV(1, 1)).Then B is essentially torsion finite for A; and A is potentially torsion infinite for B exactly when there is an isomorphism of Q-algebras End 0 (A) ∼ = End 0 (B).Then the essential torsion finiteness depends on the CM-types of A and B as in Theorem 5.4.
Proof.Our proof contains two parts.In the first part, we will assume that the pair (A, B) is not one of the cases (a), (b), or (c).The analysis of the special situations is carried out in the second part.
Recall that, if (5.9) sMT(A × B) = sMT(A) × sMT(B), then A and B are mutually torsion finite (Corollary 4.15).Since dim A ≤ dim B, (5.9) holds in each of the following cases.
(1) Suppose both A and B are of odd relative dimension, and they are not both of type IV.Then [Ich91, Theorem IA] states that (5.9) holds.
Hence we are left with two situations to discuss.For expository ease, we will let A 1 = A and A 2 = B in the following discussion.
Case 1. Suppose the pair is of type [(3,IV(1,1)), (3,IV(1,1))], and that A 1 is potentially torsion infinite for A 2 ; we will show that A 1 and A 2 are geometrically isogenous.We start by describing the Mumford-Tate groups of each A i although ultimately we will analyze their special Mumford-Tate groups, in order to exploit the fact that isogenous one-dimensional algebraic tori are actually isomorphic.Recall that if G is a reductive group with derived group G ′ and connected center Z, then Z is a torus and G is canonically isomorphic to G ′ × Z/(G ′ ∩ Z).
For i = 1, 2, the endomorphism algebra F i := End 0 (A i ) is an imaginary quadratic field.The Mumford-Tate group G i of A i is a unitary similitude group in three variables attached to the quadratic extension F i /Q, which we denote GU F i (3), and the Hodge group sG i is the unitary group U F i (3).The center of G i is T F i = Res F i /Q G m ; the connected center Z i of sG i is the norm one torus T F i ,1 = Res (1) Note that dim G 1 = dim G 2 .Under the assumption that A 1 is potentially torsion infinite for A 2 , we have dim G 12 = dim G 2 (Theorem 4.10).Therefore dim G 12 = dim G 1 as well, and thus A 1 and A 2 are mutually potentially torsion infinite.
The isogenies π i : G 12 → G i induce isomorphisms of Lie algebras g 12 → g i .We thus have an isomorphism of Q-Lie algebras gu F 1 (3) ∼ = gu F 2 (3), and so F 1 ∼ = F 2 .We relabel this common quadratic field F and proceed.
For each i, the inclusion H 1 (A i , Q) ֒→ H 1 (A 1 × A 2 , Q) is F-linear.Therefore, we have commutative diagrams where the right-hand diagram is the restriction of the left-hand diagram to Hodge groups.
Of course, the isomorphism class of sG 12 is independent of the choice of i; and we have seen that each π i is determined, up to isomorphism, by the isomorphism class of sG 12 .Therefore, sG 12 → sG i → GL V i is independent of i, and so A 1 and A 2 are isogenous (Lemma 4.1).(After the fact, using Lemma 4.1(c), we recognize that the second case happens, i.e., that sG 12 ∼ = U F (3).) Case 2. If the pair is of type [(3, IV(1, 1)), (3, IV(3, 1))], then the Hodge group of A 1 has been explained in Case 1.Note in particular that the center of sMT(A 1 ) is U F 1 (1), a onedimensional torus.Now consider A 2 .It has complex multiplication by a CM field E 2 .Since dim A 2 = 3 is prime, the CM type is nondegenerate, i.e., dim sMT(A 2 ) = 3.
In particular, there is no isogeny from the center of sMT(A 1 ) to sMT(A 2 ).By [MZ99, Lemma 3.6], A 1 and A 2 satisfy (5.9), and thus are mutually essentially torsion finite.This finishes the first part of the proof.
It remains to discuss cases (a), (b), and (c).Of course, there is nothing to prove for case (c).As for (a) and (b), since dim MT(A 1 × A 2 ) ≥ dim MT(A 2 ) > 2 = dim MT(A 1 ), we immediately deduce that A 2 is essentially torsion finite for A 1 .
OF ABELIAN VARIETIES 4.1.Mumford-Tate groups for a pair of abelian varieties.Let A and B be abelian varieties over a subfield K of C. Let G A , G B and G A×B denote the (Z-models of) the Mumford-Tate groups of, respectively, A, B and A × B, and let sG A , sG B and sG A×B denote their respective Hodge groups.Recall that the Mumford-Tate group G C of a complex abelian variety C is the Q-algebraic hull of the morphism Res

Lemma 4. 1 .
Let A and B be complex abelian varieties.The following are equivalent.(a) A and B are isogenous; (b) H 1 (A, Q) and H 1 (B, Q) are isomorphic representations of sMT(A × B); (c) The canonical surjections sMT(A × B) ։ sMT(A) and sMT(A × B) ։ sMT(B) are isomorphisms, and H 1 (A, Q) and H 1 (B, Q) are isomorphic representations of this common group.

Corollary 4. 14 .
Suppose that A and B are two absolutely simple abelian varieties over K, and that the Mumford-Tate conjecture holds for A × B. Then the following are equivalent.(a) A and B are mutually potentially torsion infinite.(b) The natural surjections G A×B → G A and G A×B → G B are isogenies.(c) The natural surjections sG A×B → sG A and sG A×B → sG B are isogenies.
(5.3) Lemma 5.1.If A is a CM abelian variety over a number field, then the Mumford-Tate conjecture holds for A. Moreover, if (E, Φ) is a CM type for A, then MT(A) ∼ = T Φ .
if the d-torsion points of A and B generate the same extension of K. Using the theory of the (special) Mumford-Tate group and assuming the Mumford-Tate conjecture, Lombardo proves that condition (1.4) puts a strong restriction on the Hodge groups of A, B and A × B. This constraint forces A to have the same isogeny factors as B when either dim A ≤ 3 and dim B ≤ 3 [Lom23, Theorem 1.2]; or every simple factor of A or B has dimension ≤ 2, is of odd relative dimension and not of type IV [Lom23, Theorem 1.4].As a complement, by studying certain simple CM types on cyclic CM fields, Lombardo also constructs infinitely many non-isogenous iso-Kummerian pairs [Lom23, Theorem 1.1].In spite of the obvious similarities, our work is differs from Lombardo's in its emphasis and results.

dim M 12 = 0 and thus π 2 • ι is an isogeny. Lemma 2.4. Let G be a connected algebraic group, and let M ⊂ G be a normal algebraic subgroup. Then M • is normal in G. Proof
1, suppose H 12 = G 1 .Then clearly H 21 = G 2 , and thus G 12 ∼ = G 1 × G 2 .At the opposite extreme, if H 12 and H 21 are trivial, then, up to a choice of isomorphismG 1 ∼ = G 2 , ι : G 12 ֒→ G 1 × G 2 is the diagonal embedding.. Since G and M • are connected, the image of M • under conjugation by G is connected and contains the identity element of G. Since this image is a subgroup of M, which is normal in G, it is contained in M • , and thus M • is stable under conjugation by G.
and α has connected kernel if and only if X * (T)/α * X * (S) is torsion-free.If F/Q is a finite extension, we let T F denote Res F/Q G m,F , the Weil restriction of the multiplicative group, and let T F,1 denote the norm one torus Res H is an ℓ-group, by the Sylow theorem, ρ W (G) is contained in the F ℓ -points of a maximal unipotent subgroup U of GL W . (Differently put, after a suitable choice of basis, the image of G in GL W is contained in the F ℓ -points of the group U of unipotent upper-triangular matrices.)SinceUhas a nontrivial fixed vector, so does G.2.2.2.Bounded subgroups.Let H/Z[1/N] be a smooth affine algebraic group scheme with geometrically connected fibers.Suppose that for each ℓ ∤ N, an abstract group H ℓ ⊂ H(F ℓ ) is specified.Let P = ker(α).Its formation commutes with base change, and it is the extension of a finite group scheme D by a connected group P • .Taking F ℓ -points, we have It suffices to show that H 1 (F ℓ , P) is bounded independently of ℓ.By Lang's theorem H 1 (F ℓ , P • ) is trivial, and so it suffices to show that H 1 (F ℓ , D) is bounded.Now use the fact that D is finite and #H 1 (F ℓ , D) = #D(F ℓ ) ([PR94, p. 290]).{(ker α| H ℓ ) ∩ P • (F ℓ )} ℓ has bounded index in P • .

[1/N]-module of
Let P ′ be any irreducible component of P F ℓ .Then P ℓ ∩ P ′ (F ℓ ), if nonempty, is a torsor under P ℓ ∩ P • (F ℓ ), and so rank n, and let ρ : G → GL V be a representation.For a field k equipped with a ring map Z[1/N] → k, let r k (G, ρ) be the multiplicity of the trivial representation of G(k): On one hand, α is an m th root of unity for some m|[G : G • ].On the other hand, by the commutativity of the diagram, α ≡ 1 mod ℓ.Consequently, α = 1.Since ρ W (g) ∈ GL W (Z ℓ ) has finite order it is semisimple, and we conclude that ρ W (g) = id W .
Lemma 2.18.Let G/Z[1/N] be a smooth affine group scheme with reductive connected component of identity.Let V be a free Z[1/N]-module of finite rank, and let ρ : is isomorphic to a normal algebraic subgroup H A,B of G A .Define M B,A and H B,A in an analogous fashion.Because H 1 (A, Q) and H 1 (B, Q) have the same nonzero weight, H A,B ⊂ sG A and H B,A ⊂ sG B .Consequently, the Hodge groups also satisfy the hypotheses of Goursat's lemma, i.e., fit in a diagram a dimension count shows that equality holds if and only if (E, Φ) is nondegenerate.Under the hypotheses of the lemma, we have the following commutative diagram 0 1); and we have exact sequences1 / / SU F i (3)The restrictionδ i := det | Z i is [3] Z i ,the cubing map.Moreover, H 1 (A i , Q) is the standard representation of G i (see, e.g., [MZ99, (2.3)]).