Effective Sato-Tate conjecture for abelian varieties and applications

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse-Weil bound. Third, for a pair of abelian varieties defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces have opposite sign.


Introduction
Let A be an abelian variety defined over a number field k of dimension g ≥ 1.For a rational prime ℓ, we denote by the ℓ-adic representation attached to A, obtained from the action of the absolute Galois group of k on the rational ℓ-adic Tate module V ℓ (A) := T ℓ (A) ⊗ Q ℓ .Let N denote the absolute norm of the conductor of A, which we will call the absolute conductor of A. For a nonzero prime ideal p of the ring of integers of k not dividing N ℓ, let a p := a p (A) denote the trace of ̺ A,ℓ (Frob p ), where Frob p is a Frobenius element at p.
The trace a p is an integer which does not depend on ℓ and, denoting by Nm(p) the absolute norm of p, the Hasse-Weil bound asserts that the normalized trace a p := a p Nm(p) lies in the interval [−2g, 2g].Attached to A there is a compact real Lie subgroup ST(A) of the unitary symplectic group USp(2g) that conjecturally governs the distribution of the normalized Frobenius traces.More precisely, the Sato-Tate conjecture predicts that the sequence {a p } p , indexed by primes p not dividing N ordered by norm, is equidistributed on the interval [−2g, 2g] with respect to the pushforward via the trace map of the (normalized) Haar measure of the Sato-Tate group ST(A).We will denote this measure by µ.
Denote by δ I the characteristic function of a subinterval I of [−2g, 2g].Together with the prime number theorem, the Sato-Tate conjecture predicts that (1.1) where Li(x) := ∞ 2 dt/ log(t).Let L(χ, s) denote the (normalized) L-function attached to an irreducible character χ of ST(A).It is well known that (1.1) is implied by the conjectural nonvanishing and analyticity on the right halfplane ℜ(s) ≥ 1 of L(χ, s) for every nontrivial irreducible character χ.In this paper we derive an asymptotic upper bound on the error term implicit in (1.1) by further assuming the generalized Riemann hypothesis for the L-functions L(χ, s).
Our main result is a quantitative refinement of the Sato-Tate conjecture (see Theorem 3.8).In order to state it we need to introduce some notations.Let g denote the complexified Lie algebra of ST(A), and write it as s × a, where s is semisimple and a is abelian.Set (1.2) where ϕ is the size of the set of positive roots of s and q is the rank of g, and define (1.3) ν g : R >0 → R >0 , ν g (z) = max 1, log(z) 6 z 1/εg For a subinterval I of [−2g, 2g], let |I| denote its length.
Theorem 1.1 (Effective Sato-Tate conjecture).Let A be an abelian variety defined over the number field k of dimension g ≥ 1, absolute conductor N , and such that ST(A) is connected.Suppose that the Mumford-Tate conjecture holds for A and that the generalized Riemann hypothesis holds for L(χ, s) for every irreducible character χ of ST(A).Then for all subintervals I of [−2g, 2g] of nonzero length, we have (1.4) where the sum runs over primes not dividing N , the implied constant in the O-notation depends exclusively on k and g, and x 0 = O ν g (|I|) log(2N ) 2 log(log(4N )) 4 .
The dependence on g in the implied constant of (1.4) can be traced through Propositions 3.1 and 3.2 and Lemmas 3.6 and 3.7; it is highly exponential.This theorem generalizes a result of Murty [Mur85] concerning elliptic curves without complex multiplication (CM); see also [BK16,Thm. 3.1].Its proof follows the strategy envisaged in [BK16,§5] and it occupies §3.A key ingredient is the construction of a multivariate Vinogradov function; this is a continuous periodic function, with rapidly decaying Fourier coefficients, and approximating the characteristic function of the preimage of I by the trace map in the parameter space of a Cartan subgroup H of ST(A).By identifying the quotient of this space by the action of the Weyl group with the set of conjugacy classes of ST(A), one can rewrite (a Weyl average of) the Vinogradov function as a combination of irreducible characters of ST(A).One can use purely Lie algebra theoretic arguments (most notably Weyl's character dimension formula and a result due to Gupta [Gup87,Thm. 3.8] on the boundedness of the inverse of the weight multiplicity matrix) to show that the coefficients in the character decomposition of the Vinogradov function also exhibit a rapid decay.The theorem can then be obtained by using an estimate of Murty (as presented in [BK16,(2.4)])on truncated sums of an irreducible character χ over the prime ideals of k.The implied constant in the O-notation depends in principle on the exponents of the Cartan subgroup H.In order to bound these exponents purely in terms of g, we show that the Mumford-Tate conjecture implies that H is generated by the Hodge circles contained in it (see Theorem 3.5).This result may be of independent interest.
The conjectural background for Theorem 1.1 is presented in §2.We recall the Mumford-Tate conjecture and the related algebraic Sato-Tate conjecture, define the L-functions L(χ, s), and state the generalized Riemann hypothesis for them.In §4 we give three applications of Theorem 1.1.The first is what we call the interval variant of Linnik's problem for an abelian variety (see Corollary 4.1).
Corollary 1.2.Assume the hypotheses and notations of Theorem 1.1.For every subinterval I of [−2g, 2g] of nonzero length, there exists a prime p not dividing N with norm The second application concerns what we call the Frobenius sign separation problem for a pair of abelian varieties (see Corollary 4.4).
Corollary 1.3.Let A (resp.A ′ ) be an abelian variety defined over the number field k of dimension g ≥ 1 (resp.g ′ ≥ 1), absolute conductor N (resp.N ′ ), and such that ST(A) (resp.ST(A ′ )) is connected.Suppose that the Mumford-Tate conjecture holds for A (resp.A ′ ) and that the generalized Riemann hypothesis holds for L(χ, s) (resp.L(χ ′ , s)) for every irreducible character Then there exists a prime p not dividing N N ′ with norm such that a p (A) and a p (A ′ ) are nonzero and of opposite sign.Here, the implied constant in the O-notation depends exclusively on k, g, and g ′ .
We also examine what our method says about the set of primes with "maximal Frobenius trace".Let M k (x) denote the set of primes p not dividing N with norm up to x for which a p = ⌊2 Nm(p)⌋.Vaguely formulated, a natural approach to compute (at least an asymptotic lower bound on) M k (x) is to compute the number of p with norm up to x for which a p lies in a sufficiently small neighborhood I x of 2g.However, for this idea to succeed, the neighborhood I x should be sufficiently large in order for the "error term" in (1.4) to be still dominated by the "main term", which is now multiplied by the tiny quantity µ(I x ).In the case where A is an elliptic curve with CM it is possible to achieve this trade-off, yielding the following statement (see Proposition 4.9 and Corollary 4.10).
Corollary 1.4.Let A be an elliptic curve defined over k with potential CM, that is, such that A Q has CM.Under the generalized Riemann hypothesis for the L-function attached to every power of the Hecke character of A, we have This recovers a weaker version of a theorem of James and Pollack [JP17, Theorem 1], which asserts (unconditionally) that x 3/4 log(x) .
A different result in a similar spirit, concerning numbers of points on diagonal curves, is due to Duke The framework of the generalized Sato-Tate conjecture includes many additional questions about distinguishing L-functions, a number of which have been considered previously.For instance, Goldfeld and Hoffstein [GH93] established an upper bound on the first distinguishing coefficient for a pair of holomorphic Hecke newforms, by an argument similar to ours but with a milder analytic hypothesis (the Riemann hypothesis for the Rankin-Selberg convolutions of the two forms with themselves and each other).Sengupta [Sen04] carried out the analogous analysis with the Fourier coefficients replaced by normalized Hecke eigenvalues (this only makes a difference when the weights are distinct).
There is an alternative approach to the above kind of questions, which is based on the use of effective forms of Chebotaryov's density theorem conditional to the the Riemann hypothesis for Artin L-functions.This approach was introduced by Serre [Ser81], who gave an upper bound on the smallest prime at which two nonisogenous elliptic curves have different Frobenius traces.The analogue of Serre's argument for modular forms was given by Ram Murty [Mur97] and subsequently extended to Siegel modular forms by Ghitza [Ghi11] for Fourier coefficients and Ghitza and Sayer [GS14] for Hecke eigenvalues.Building on Serre's method, several recent works have explored the asymptotic number of zero Frobenius traces for abelian varieties which are either generic (see [CW22b]) or isogenous to a product of elliptic curves (see [HJS22] and [CW22a]).
Notation and terminology.Throughout this article, k is a fixed number field and g and g ′ are fixed positive integers.For an ordered set (X, ≤) and functions f, h : X → R we write f (x) = O(h(x)) to denote that there exist a real number K > 0 and an element x 0 ∈ X such that |f (x)| ≤ Kh(x) for every x ≥ x 0 .We will generally specify the element x 0 in the statements of theorems, but we will usually obviate it in their proofs, where it can be inferred from the context.We refer to K as the implied constant in the O-notation.As we did in this introduction, whenever using the O-notation in a statement concerning an arbitrary abelian variety A of dimension g defined over the number field k, the corresponding implied constant is computable exclusively in terms of g and k (in fact the dependence on k is just on the absolute discriminant | disc k/Q | and the degree [k : Q]).For statements concerning a pair of arbitrary abelian varieties A and A ′ of respective dimensions g and g ′ defined over k, the implied constant in the O-notation is computable exclusively in terms of g, g ′ , and k.Section 4.3 is the only exception to the previous convention and to emphasize the dependency on N of the implied constants in the asymptotic bounds therein, we use the notations O N and ≍ N .We write f ≍ g if f = O(g) and g = O(f ).By a prime of k, we refer to a nonzero prime ideal of the ring of integers of k.Additional notation introduced later in the paper is summarized in Table 1.
Acknowledgements.We thank Christophe Ritzenthaler for raising the question of the infiniteness of the set of primes at which the Frobenius trace attains the integral part of the Weil bound, and Jeff Achter for directing us to [JP17].This occurred during the AGCCT conference held at CIRM, Luminy, in June 2019, where Bucur gave a talk based on this article; we also thank the organizers for their kind invitation.We thank Andrew Sutherland for providing the example of Remark 4.11 and numerical data compatible with Proposition 4.9.We thank Jean-Pierre Serre for sharing the preprint of [Ser20] with us and for precisions on a previous version of this manuscript.
All three authors were supported by the Institute for Advanced Study during 2018-2019; this includes funding from National Science Foundation grant DMS-1638352.All three authors were additionally supported by the Simons Foundation grant 550033.Bucur was also supported by the Simons Foundation collaboration grant 524015, and by NSF grants DMS-2002716 and DMS-2012061.Kedlaya was additionally supported by NSF grants DMS-1501214, DMS-1802161, DMS-2053473 and by the UCSD Warschawski Professorship.Fité was additionally supported by the Ramón y Cajal fellowship RYC-2019-027378-I, by the María de Maeztu program CEX2020-001084-M, by the DGICYT grant MTM2015-63829-P, and by the ERC grant 682152.

Conjectural framework
Throughout this section A will denote an abelian variety of dimension g defined over the number field k, and of absolute conductor N := N A .We will define its Sato-Tate group, introduce the motivic L-functions attached to it, and present the conjectural framework on which §3 is sustained.
2.1.Sato-Tate groups.Following [Ser12, Chap.8] (see also [FKRS12,§2]), one defines the Sato-Tate group of A, denoted ST(A), in the following manner.Let G Zar ℓ denote the Zariski closure of the image of the ℓ-adic representation ̺ A,ℓ , which we may naturally see as lying in GSp 2g (Q ℓ ).Denote by G 1,Zar The Sato-Tate group ST(A) is defined to be a maximal compact subgroup of the group of C-points of G 1,Zar ℓ,ι .In the present paper, to avoid the a priori dependence on ℓ and ι of the definition of ST(A), we formulate the following conjecture.
The Sato-Tate group ST(A) is then a maximal compact subgroup of AST(A) × Q C. It should be noted that, following [Ser91], Banaszak and Kedlaya [BK15] have given an alternative definition of ST(A) that also avoids the dependence on ℓ and ι.However, this is rendered mostly unnecessary by Theorem 2.3 below.
The algebraic Sato-Tate group is related to the Mumford-Tate group and the Hodge group.Fix an embedding k ֒→ C. The Mumford-Tate group MT(A) is the smallest algebraic subgroup G of GL(H is the complex structure on the 2g-dimensional real vector space H 1 (A C , R) obtained by identifying it with the tangent space of A at the identity.The Hodge group Hg(A) is the intersection of MT(A) with Sp The identity component of AST(A) should thus be the Hodge group Hg(A).It follows from the definition that ST(A) has a faithful unitary symplectic representation where V is a 2g-dimensional C-vector space, which we call the standard representation of ST(A).Via this representation, we regard ST(A) as a compact real Lie subgroup of USp(2g).
Theorem 2.3 (Cantoral Farfán-Commelin).If the Mumford-Tate conjecture holds for A, then the algebraic Sato-Tate conjecture also holds for A.

Motivic L-functions.
As described in [Ser12, §8.3.3] to each prime p of k not dividing N one can attach an element y p in the set of conjugacy classes Y of ST(A) with the property that where Frob p denotes a Frobenius element at p.More in general, via Weyl's unitarian trick, any complex representation σ : ST(A) → GL(V χ ) , say of character χ and degree d χ , gives rise to an ℓ-adic representation where V χ,ℓ is a Qℓ -vector space of dimension d χ , such that for each prime p of k not dividing N one has where w χ denotes the motivic weight of χ.For a prime p of k, define where I p denotes the inertia subgroup of the decomposition group G p at p.The polynomials L p (χ, T ) do not depend on ℓ, and have degree d χ (p) ≤ d χ .Moreover, writing α p,j for j = 1, . . ., d χ (p) to denote the reciprocal roots of L p (χ, T ), we have that |α p,j | ≤ 1 .
In fact, for a prime p not dividing N , we have that d χ (p) = d χ and |α p,j | = 1.Therefore, the Euler product is absolutely convergent for ℜ(s) > 1.We will make strong assumptions on the analytic behavior of the above Euler product.Before, following [Ser69, §4.1], define the positive integer where N χ is the absolute conductor attached to the ℓ-adic representation σ A,ℓ .For j = 1, . . ., d χ , let 0 ≤ κ χ,j ≤ 1 + w χ /2 be the local parameters at infinity (they are semi-integers that can be explicitly computed from the discussion in [Ser69, §3]).Define the completed L-function Let δ(χ) be the multiplicity of the trivial representation in the character χ of ST(A).
Conjecture 2.4 (Generalized Riemann hypothesis).For every irreducible character χ of ST(A), the following holds: i) The function s δ(χ) (s − 1) δ(χ) Λ(χ, s) extends to an analytic function on C of order 1 which does not vanish at s = 0, 1. ii) There exists ǫ ∈ C × with |ǫ| = 1 such that for all s ∈ C we have where χ is the character of the contragredient representation of σ.
The following estimate of Murty [Mur85,Prop. 4.1] will be crucial in §3.We will need the formulation with the level of generality of [BK16,(2.3)].
Proposition 2.5 (Murty's estimate).Assume that Conjecture 2.4 holds for the irreducible character χ of ST(A).Then (2.2) By applying Abel's summation trick, the above gives Remark 2.6.In (2.2) and thereafter, we make the convention that all sums involving the classes y p run over primes p not dividing N .A similar convention applies for sums involving the normalized Frobenius traces a p = Trace(y p ).
Remark 2.7.We alert the reader of a small discrepancy between (2.3) and [BK16, (2.4)]: in the latter the error term stated is O(d χ √ x log(N (x + d χ ))).We make this precision here, although we note that it has no effect in the subsequent results of [BK16].Indeed, in many cases (as those of interest in [BK16] involving elliptic curves without CM) the weight w χ is bounded by the dimension d χ .
Remark 2.8.The proof of Proposition 2.5 uses the bound for every character χ of ST(A).
In order to show (2.4), let us recall the definition of N χ as a product over primes of k, where f χ (p) is the exponent conductor at p; this is a nonnegative integer whose definition can be found in [Ser69, §2], for example.If A has good reduction at p, then f χ (p) is zero and so the product is finite.Let T χ,ℓ denote a Z ℓ -lattice in V χ,ℓ stable by the action of G p .By Grothendieck [Gro70, §4], the exponent conductor can be written as for every ℓ coprime to p. Since the kernel of the action on this quotient is contained in the kernel of the action of G p on T ℓ (A)/ℓT ℓ (A), we have that σ A,ℓ factors through a finite group G χ,p whose order is O(1).Consider the normal filtration of ramification groups where e is the ramification index of p over Q, p h(G1) is the exponent of the p-group G 1 and p a is the maximal dimension among absolutely simple components of and a, because the dimension of an irreducible representation of a group is bounded by the order of the group.We deduce that from which (2.4) is immediate.

Effective Sato-Tate Conjecture
In this section we derive, from the conjectural framework described in §2, an effective version of the Sato-Tate conjecture for an arbitrary abelian variety A of dimension g defined over the number field k (see Theorem 3.8).Let I be a subinterval of [−2g, 2g].By effective we mean that we provide an upper bound on the error term in the count of primes with normalized Frobenius trace lying in I relative to the prediction made by the Sato-Tate measure.
The proof is based on the strategy hinted in [BK16,§5].The first step is the construction of a multivariate Vinogradov function aproximating the characteristic function of the preimage of I by the trace map.This is a continuous periodic function with rapidly decaying Fourier coefficients that generalizes the classical Vinogradov function [Vin54, Lem.12].This construction is accomplished in §3.2.
The core of the proof consists in rewriting the Vinogradov function in terms of the irreducible characters of ST(A) and applying Murty's estimate (Proposition 2.5) to each of its irreducible constituents.This is the content of §3.4.
In order to control the size of the coefficients of the character decomposition, we use a result of Gupta [Gup87, Thm.3.8] bounding the size and number of nonzero entries of the inverse of the weight multiplicity matrix.Gupta's result and other background material on representations of Lie groups is recalled in §3.1.
A first analysis does not yield the independence of the implied constant in the O-notation from the Lie algebra of ST(A).This independence is shown to follow from the density of the subgroup generated by the Cartan Hodge circles in the Cartan subgroup.In a result which may be of independent interest (see Theorem 3.5), this density is shown to follow from the Mumford-Tate conjecture in §3.3.

3.1.
Lie group theory background.Let s be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra h of rank h.Let Φ ⊆ h * be a root system for s, h * 0 be the real vector subspace generated by Φ, and R ⊆ h * 0 denote the lattice of integral weights of s.Fix a base S for the root system Φ.The choice of S determines a Weyl chamber in h * 0 and a partition Φ = Φ + ∪ Φ − , where Φ + (resp.Φ − ) denotes the set of positive (resp.negative) roots of s.Let C denote the set of dominant weights, that is, the intersection of the set of integral weights R with this Weyl chamber.The choice of a basis of fundamental weights {ω j } j determines an isomorphism C ≃ Z h ≥0 .For λ, µ ∈ C, the multiplicity m µ λ of µ in λ is defined to be the dimension of the space , where Γ λ is the irreducible representation of s of highest weight λ.Write ρ :=1 2 α∈Φ + α for the Weyl vector and W for the Weyl group of s.The multiplicity of µ in λ can be computed via Kostant's multiplicity formula where ǫ(w) is the sign of w, and p(v) is defined by the identity where we make a formal use of the exponential notation e α (see [FH91,Prop. 25 Here, for each v ∈ R, the integer f (v) is defined by (1 − e −α ) .
Let ϕ denote the size of the set of positive roots Φ + .
Proposition 3.1.The sum of the absolute values of the elements in each row (resp column) of (d µ λ ) λ,µ is bounded by #W • 2 ϕ .In particular d µ λ = O(1) and the number of nonzero entries at each row (resp.column) of (d µ λ ) λ,µ is O(1).Proof.The proof follows from the aforementioned result by Gupta.Indeed, the sum of the absolute values of the entries at each row (resp.column) of (d µ λ ) λ,µ is bounded by #W times the norm But this number is bounded by 2 ϕ , as one observes from (3.2).Now the other two statements are implied by the fact that ϕ, #W can be bounded in terms of g, as follows from the general classification of complex semisimple Lie algebras, and thus are O(1).
For λ ∈ C, write λ as a nonnegative integral linear combination s j=1 m j ω j of the fundamental weights and define ||λ|| fund := max j {m j } .
Proposition 3.2.The previous definition has the following properties.
Proof.For i), recall Weyl's dimension formula [Ser87, Cor. 1 to Thm. 4, Chap.VII], which states where (•, •) denotes a W-invariant positive definite form on the real vector space h * 0 spanned by the base S.This trivially implies It remains to show that (λ, α) = O(||λ|| fund ) for every α ∈ Φ + .Let α j , for j = 1, . . ., h, be the constituents of the base S, the so-called simple roots.The desired result follows from the following relation linking simple roots and fundamental weights As for ii), suppose that the expression of λ ∈ C (resp.λ ′ ∈ C) as a nonnegative linear combination of the simple roots is h j=1 r j α j (resp.h j=1 r ′ j α j ).Note that λ ′ λ implies that r ′ j ≤ r j .Therefore Part iii) is a consequence of the weight decomposition of Γ λ .

A multivariate Vinogradov function.
The main result of this section is Proposition 3.4, which is a generalization of [Vin54, Lemma 12].Let q ≥ 1 be a positive integer.We will write θ to denote the q-tuple (θ 1 , . . ., θ q ) ∈ R q (a similar convention applies to z, δ, etc).We also write m to denote (m 1 , . . ., m q ) ∈ Z q .We will say that a function h : R q → R is periodic of period 1 if it is so in each variable.
For δ = (δ 1 , . . ., δ q ) ∈ [0, 1) q , denote by R(δ) the parallelepiped q j=1 [−δ j , δ j ].Set also the multiplier Lemma 3.3.Suppose that h : R q → R admits a Fourier series expansion as For δ ∈ [0, 1) q , define Then we have that Proof.The proof follows the same lines as Vinogradov's one-dimensional version.We have Setting t = θ + z so that θ = t − z and dθ = dt in the above equation, we obtain For m j = 0 the corresponding term in the product is For m j = 0 the corresponding term becomes The desired formula follows.
(3) There exists a positive integer C > 0 such that, for every γ ∈ R, 1 ≤ j ≤ q, and ϑ ∈ [0, 1] q−1 , we have Let α, β, ∆ be real numbers satisfying Let I denote the open interval (α, β).By (3.6) we can define the disjoint sets Then for every positive integer r ≥ 1, there exists a continuous function D := D ∆,I : R q → R periodic of period 1 satisfying the following properties: iii) D(θ) has a Fourier series expansion of the form where c 0 = T −1 ((α,β))∩[0,1] q dθ and for all m = 0 we have Proof.Start by defining the function ψ 0 periodic of period 1 as Then we clearly have and for m = 0 we find the bound We next derive an alternative upper bound for c m (ψ 0 ).Let m denote max l {|m l |} and let j be such that m = |m j |.Then by Fubini's theorem we have e −2πimθj dθ j e −2πiπj (m)•πj(θ) dπ j (θ) .
Note that f), (3.7), and (3.8) imply that To conclude, take D := ψ r , and the proposition follows from f) and the fact that, for m j = 0, we have 3.3.The Cartan subgroup.As in the previous sections, A denotes an abelian variety of dimension g defined over the number field k.From now on we will assume moreover that its Sato-Tate group ST(A) is connected.
Let a denote the rank of a and let q = h + a be the rank of g.As in §3.2, write θ to denote (θ 1 , . . ., θ q ) ∈ R q .We may choose a q+1 , . . ., a g ∈ Z g−q such that the image H of the map has complexified Lie algebra isomorphic to h × a.We then say that H is a Cartan subgroup of ST(A).For notational purposes, it will be convenient to let a 1 , . . ., a q denote the standard basis of Z q .Let a l,j denote the j-th component of a l .Consider the map (3.12) In the next section, we will apply the construction of a We next reduce the general case to the previous paragraph, by arguing as in the proof of Deligne's theorem on absolute Hodge cycles.Recall that the Mumford-Tate group of A is the smallest Q-algebraic subgroup of GL(H 1 (A top C , Q)) whose base extension to R contains the action of the Deligne torus Res C/R (G m ) coming from the Hodge structure.Under our hypotheses on A, we may recover ST(A) by taking the Mumford-Tate group, taking the kernel of the determinant to get the Hodge group, then taking a maximal compact subgroup.
By the proof of [Del82, Proposition 6.1], there exists an algebraic family of abelian varieties containing A as a fiber such that on one hand, the generic Mumford-Tate group is equal to that of A, and on the other hand there is a fiber B whose Mumford-Tate group is a maximal torus in A. Using the previous paragraph, we see that the desired assertion for A follows from the corresponding assertion for B, which we deduce from the first paragraph.Lemma 3.6.Suppose that the Mumford-Tate conjecture holds for A. Then |a l,j | = O(1).
Proof.Write A for the matrix (a l,j ) l,j .Giving a Cartan Hodge circle amounts to giving a vector v ∈ {±1} q such that (3.13) Av t = u t where u ∈ {±1} g has g entries equal to 1 and g entries equal to −1 (and v t , u t denote the transposes of v, u).By Theorem 3.5, there exist q linearly independent vectors v satisfying an equation of the type (3.13).Let v j , for j = 1, . . ., q, denote these vectors, and let u j ∈ {±1} g denote the corresponding constant terms in the equation that they satisfy.Let v j,l (resp.u j,l ) denote the l-th component of v j (resp.u j ).Write V (resp.U) for the matrix (v l,j ) j,l (resp.(u l,j ) j,l ).Since V is invertible, we have The lemma now follows immediately from the fact that all the entries of V and U are ±1.
If we write q(x) = n b n x n , the above equality implies that cos(2πϑ) is a root of Since r(x) has degree ≤ 2N , we find that cos(2πϑ) is limited to 2N values.This implies that ϑ is limited to 4N values, and we conclude by applying Lemma 3.6, which shows that N = O(1).

Main theorem.
In this section we prove an effective version of the Sato-Tate conjecture building on the results obtained in all of the previous sections.Let µ be the pushforward of the Haar measure of ST(A) on [−2g, 2g] via the trace map.We refer to [Ser12, §8.1.3,§8.4.3] for properties and the structure of this measure.It admits a decomposition µ = µ disc + µ cont , where µ disc is a finite sum of Dirac measures and µ cont is a measure having a continuous, integrable, and even C ∞ density function with respect to the Lebesgue measure outside a finite number of points.Since we will assume that ST(A) is connected, we will in fact have that µ disc is trivial (see [Ser12,§8.4
Theorem 3.8.Let k be a number field and g a positive integer.Let A be an abelian variety defined over k of dimension g, absolute conductor N , and such that ST(A) is connected.Suppose that the Mumford-Tate conjecture holds for A and that Conjecture 2.4 holds for every irreducible character χ of ST(A).For each prime p not dividing N , let a p denote the normalized Frobenius trace of A at p. Then for all nonempty subintervals I of [−2g, 2g], we have where Let us resume the notations of §3.1 relative to the semisimple algebra s.Thus, h is a Cartan subalgebra for s of rank h, R ⊆ h * 0 is the lattice of integral weights, W is the Weyl group of s, C denotes the integral weights in a Weyl chamber, and ω 1 , . . ., ω h are the fundamental weights.Let a denote the rank of a, so that q = h + a.Before starting the proof we introduce some additional notations.
Recall the map ι : R q → ST(A) from (3.11).Without loss of generality, we may assume that the decomposition R q = R h × R a is such that the complexification of the Lie algebra of ι(R h ) (resp.ι(R a )) is h (resp.a).Let us write θ h (resp.θ a ) for the projection of θ onto R h (resp.R a ).
From now on we fix a Z-basis ψ 1 , . . ., ψ q of the character group Ĥ of H: for 1 ≤ j ≤ h, the character ψ j is induced by the fundamental weight ω j of s; for h + 1 ≤ j ≤ q, we set The action of W on h * 0 induces an action of W on the character group Ĥ of H.We may define an action of W on [0, 1] q by transport of structure: given w ∈ W, let w(θ) be defined by for all j = 1, . . ., q .
Of course the action of W restricts to the first factor of the decomposition [0, 1] q = [0, 1] h × [0, 1] a .Note that the map ι from (3.11) induces an isomorphism Recall the elements y p ∈ Conj(ST(A)) introduced in §2.Let θ p ∈ [0, 1] q /W be the preimage of y p by the above isomorphism.
Consider the map T : R q → [−2g, 2g] defined in (3.12).Note that T (θ p ) is well defined since T factors through [0, 1] q /W, and it is equal to the normalized Frobenius trace a p .Let K and C denote the constants of Lemma 3.7 relative to the map T .
Let the interior of I be of the form (α, β) for −2g ≤ α < β ≤ 2g.Let ∆ > 0 be any real number satisfying the constraint (3.6) relative to α, and β (arbitrary for the moment and to be specified in the course of the proof of Theorem 3.8).
Let D := D ∆,I : R q → R be the Vinogradov function produced by Proposition 3.4, when applied to α, β, ∆, and T , and relative to the choice of a positive integer r ≥ 1 (arbitrary for the moment and to be specified in the course of the proof of Theorem 3.8).Define (3.15) Notice that F (θ p ) is well-defined since F has been defined as an average over W. In consonance with Remark 2.6, we make the convention that sums involving the elements θ p run over primes p not dividing N .
Lemma 3.9.If part i) of Conjecture 2.4 holds for every irreducible character χ of ST(A), then for every ∆ satisfying (3.6) and every x ≥ 2 .
Since T satisfies property (3) of Proposition 3.4), by Lemma 3.7 the intersections of Y α , Y β with R are finite (and, in fact, even of cardinality O(1)).Let W α , W β denote the intersections of Y α , Y β with the complement of R.
We claim that W α , W β have volume O(1) as (q−1)-dimensional Riemannian submanifolds of [0, 1] q .Before showing the claim, we note that it implies the lemma.Indeed, as functions over [0, 1] q , the characteristic function of T −1 (I) and F ∆,I only differ (by construction of the latter) over the W-translates of tubular neighborhoods B(Y α , r ∆ ) and We now turn to show that W α has volume O(1) (and the same argument applies to W β ).For 1 ≤ j ≤ q, define It suffices to show that W α ∩ V j has volume O(1) for every j.By symmetry, we may assume that j = q, which will be convenient for notational purposes.Let Z α,q denote the interior of the image of W α ∩ V q by the projection map π q : [0, 1] q → [0, 1] q−1 .For ϑ ∈ Z α,q , choose θ ∈ W α ∩ V q such that π q ( θ) = ϑ.By the implicit function theorem there exist a neighborhood U ϑ ⊆ Z α,q of ϑ and a differentiable function The lifts θ can be compatibly chosen so that the functions g ϑ glue together into a differentiable function g : Z α,q → W α ∩ V q .Then Lemma 3.7 provides the following bound for the volume of which completes the proof.
Proof of Theorem 3.8.The choice of a basis of fundamental weights ω 1 , . . ., ω h gives an isomorphism by means of which, from now on, we will view integral weights of s as elements in Z h .Similarly, the choice of the basis elements of (3.14) provides an isomorphism between the lattice of integral weights of a and Z a .For a weight m ∈ Z q , let m h and m a denote the projections to Z h and Z a .For m h ∈ Z h , define where t m h denotes the size of the stabilizer of m h under the action of W.
where the sum runs over weights m h ∈ C. Equivalently, we have We remark that Proposition 3.1 ensures that, for each m h , the number of nonzero coefficients d n h m h in the above equation, as well as the size of each of them, is O(1).By taking the Fourier expansion of D, we obtain Let M ≥ 1 be a positive integer (arbitrary for the moment and to be determined later).Let C ≤M denote the subset of C × Z a made of weights m whose components have absolute value ≤ M .Note that if m ∈ C ≤M , then in particular we have that On the one hand, by invoking the bounds from part iii) of Proposition 3.4, we have (3.17) On the other hand, consider the class function (3.18) In the above expression δ(F ≤M (θ)) stands for the multiplicity of the identity representation in F ≤M (θ).Note that F ≤M is a finite linear combination of irreducible characters of ST(A) and that by Proposition 3.1 we may assume that M is large enough so that δ(F ≤M (θ)) = δ(F (θ)), which we will do from now on.The next step is to bound the virtual dimension of the nontrivial part of F ≤M (θ) in order to be able to apply Proposition 2.5.More precisely, if p n h denotes the coefficient multiplying Trace(Γ n h (θ h )) in (3.18), then we have In the above computation we have used: Proposition 3.1 to bound the size and number of nonzero entries in the inverse of the matrix of weight multiplicities; Proposition 3.2 to control the dimension of the representations of weight lower than a given one and to bound the dimension of the representation Γ m h in terms of ||m h || fund ; and part iii) of Proposition 3.4 to bound the Fourier coefficients for an unspecified (for the moment) 1 ≤ ρ ≤ r.We will now distinguish two cases, depending on whether ϕ is zero or not.
Suppose first that ϕ is nonzero.Take r = ρ = q + ϕ − 1, which we note that satisfies r ≥ 1.Then (3.19) Let L > 0 be the implied constant in the bound of part iii) of Proposition 3.2 the motivic weight, so that for m h ∈ C ≤M , we have w Γm h ≤ LM .Using the decomposition the tail (3.17) and virtual dimension (3.19) bounds, and applying Proposition 2.5, we obtain (3.20) It follows from the proof of Lemma 3.9 that Therefore, to conclude the proof, it will suffice to balance the error terms in (3.20) with O(∆ Li(x)).In the case that ϕ is nonzero, we may take where ε = ε g is as defined in (1.2).In view of Lemma 3.9, this concludes the proof, provided that we verify that this choice of ∆ satisfies the constraint (3.6).This amounts to 2∆ ≤ |I|, or equivalently to By the elementary Lemma 3.10 below, this is easily seen to be the case as soon as x ≥ x 0 , where (3.23) Suppose now that ϕ = 0. We take r = q and use the tail bound (3.17) as in the previous case.To bound the Fourier coefficients of F ≤M (θ), we use the bound |c m | = O(1/m) if q = 1 and the bound corresponding to ρ = q − 1 otherwise (as in part iii) of Proposition 3.4).We obtain (3.24) To balance the error terms in the above equation with O(∆ Li(x)), we may take ∆ := x −1/(2q) log(x) 2/q log(N x) 1/q , M = ∆ −q−1 .
Since ε g = 1/(2q) in this case, this yields precisely the error term of the statement of the theorem.Again, ∆ satisfies the constraint (3.6) as soon as x ≥ x 0 , where x 0 is as in (3.23).
We leave the proof of the following to the reader.
Lemma 3.10.For integers r, N ≥ 1, with r even, and a real number number A > 0, we have that provided that x > C log(2N ) 2 log(log(4N )) r max{1, A log(A) r+2 } for some C > 0 depending exclusively on r.
Remark 3.11.To simplify the statement of Theorem 3.8, we have assumed Conjecture 2.4 for every irreducible character χ of ST(A).It is however clear from the proof that this hypothesis can be relaxed: it suffices to assume Conjecture 2.4 for those representations Γ m with m ∈ C ≤M , where M is as in (3.22).
Remark 3.12.The choice of the exponent of x in the error term in Theorem 3.8 is dictated by the balancing of O(∆ Li(x)) with the first of the two error terms in (3.20).The balancing with the second error term only affects the logarithmic factors.

Applications
In this section we discuss three applications of Theorem 3.8.In §4.1 we consider an interval variant of Linnik's problem for abelian varieties.Given an abelian variety A defined over k of dimension g and a subinterval I of [−2g, 2g], this asks for an upper bound on the least norm of a prime p not dividing N such that the normalized Frobenius trace a p (A) lies in I.In §4.2 we consider a sign variant of Linnik's problem for a pair of abelian varieties A and A ′ defined over the number field k and such that ST(A×A ′ ) ≃ ST(A)×ST(A ′ ).This asks for an upper bound on the least norm of a prime p such that a p (A) and a p (A ′ ) are nonnegative and have opposite sign.Finally, in §4.3, when A is an elliptic curve with CM, we conditionally determine (up to constant multiplication) the asymptotic number of primes for which a p (A) = ⌊2 Nm(p)⌋.
While §4.1 is a direct consequence of Theorem 3.8, both §4.2 and §4.3 require slight variations of it.We will explain how to modify the proof of Theorem 3.8 to obtain these versions.Proof.There exist constants K 1 , K 2 > 0 such that, for x ≥ K 2 ν g (|I|) log(2N ) 2 log(log(4N )) 4 , the number of primes p such that Nm(p) ≤ x and a p ∈ I is at least where ∆ is as in (3.22).This count will be positive provided that K 1 ∆ < µ(I), or equivalently if One easily verifies that this condition is satisfied for x ≥ x 0 , for some x 0 = O(ν g (µ(I)) log(2N ) 2 log(log(4N )) 4 ), and the corollary follows.
4.2.Frobenius sign separation for pairs of abelian varieties.In this section we will provide an answer to the Frobenius sign separation problem for pairs of abelian varieties using a variation of Theorem 3.8.Resume the notations of §3.4; additionally, let A ′ be an abelian variety defined over k and let g ′ , N ′ , µ ′ , etc, denote the correponding notions.We will make the hypothesis that the natural inclusion of ST(A × A ′ ) in the product ST(A) × ST(A ′ is an isomorphism. Hypothesis 4.2.We have that ST(A × A ′ ) ≃ ST(A) × ST(A ′ ).
Theorem 4.3 shows that under the conjectures of §2, this hypothesis ensures the existence of a prime p not dividing N N ′ such that (4.1) a p (A) • a p (A ′ ) < 0 and, in fact, determines the asymptotic density of such primes.Corollary 4.4, which gives an upper bound on the least norm of such a prime, is then an immediate consequence.Note that requiring A and A ′ not to be isogenous does not guarantee the existence of a prime satisfying (4.1), as it is shown by the trivial example in which A ′ is taken to be a proper power of A.
Write the complexified Lie algebra of ST(A) (resp.ST(A ′ )) as g = s × a (resp.g ′ = s ′ × a ′ ), where s, s ′ are semisimple and a, a ′ are abelian.Throughout this section, write (4.2) ε g,g ′ := 1 2(q + q ′ + ϕ + ϕ ′ − 1) , where ϕ (resp.ϕ ′ ) is the size of the set of positive roots of s (resp.s ′ ) and q (resp.q ′ ) is the rank of g (resp.g ′ ).Define Theorem 4.3.Let k be a number field, and let g and g ′ positive integers ≥ 1.Let A (resp.A ′ ) be an abelian variety defined over k of dimension g (resp.g ′ ), absolute conductor N (resp.N ′ ), and such that ST(A) (resp.ST(A ′ )) is connected.Assume that Hypothesis 4.2 holds.Suppose that the Mumford-Tate conjecture holds for A × A ′ , and that Conjecture 2.4 holds for every product χ • χ ′ of irreducible characters χ of ST(A) and χ ′ of ST(A ′ ).For each prime p not dividing N N ′ , let a p (resp. a ′ p ) denote the normalized Frobenius trace of A (resp.A ′ ) at p. Then for all nonempty subintervals I of [−2g, 2g] and Proof.Let (α, β) and (α ′ , β ′ ) denote the interiors of I and I ′ , respectively.For a common choice of ∆ > define F ∆,I (θ) and F ′ ∆,I ′ (θ ′ ) relative to undetermined positive integers r and r ′ in a manner analogous to (3.15).Let M ≥ 1 be a positive integer (arbitrary for the moment and to be determined later).In analogy with the definition of L > 0 in the line following (3.19),let L ′ > 0 be the implied constant in the bound Corollary 4.8.Assume the same hypotheses as in Proposition 4.7.For every x ≥ 2, we have: ii) #S(y, x) = 1 πy 1/4 (Li(x) − Li(y)) + O( √ x log(N x) log(x)) for every x 2/3 ≤ y ≤ x.
In view of the above, the proposition will follow from the fact that n j=2 x 3/4 j 4 log x j 4 = O x 3/4 log(x) .
Since the first term in the right-hand side of the above equation is O(x 3/4 / log(x)), and the second term is bounded by F (x)/ log(x 3/4 ), we deduce that F (x) = O(x 3/4 / log(x)), which concludes the proof.
Corollary 4.10.Let A be an elliptic curve with potential CM (say by an imaginary quadratic field K) not defined over k.Under Conjecture 2.4 for every character of ST(A kK ) ≃ U(1), we have #M k (x) ≍ N x 3/4 log(x) as x → ∞ .
Proof.Consider the base change A kK and the set of primes of kK defined as M split kK (x) := {P ∤ N prime of kK split over k : Nm(P) ≤ x and a P (A kK ) = ⌊2 Nm(P)⌋} .Since the number of primes of kK nonsplit over k of norm up to x is O( √ x), in view of Proposition 4.9, we have that #M kK (x) ∼ #M split kK (x) as x → ∞ .On the other hand, the map M split kK (x) → M k (x) , P → P ∩ k is 2 to 1, and we thus get As noted in the introduction, it was shown unconditionally by James and Pollack [JP17, Theorem 1] that #M k (x) ∼ 2 3π x 3/4 log(x) as x → ∞.
That result, which gives a partial answer to a question of Serre [Ser20, Chap.II, Question 6.7], builds on a conditional result of James et al. [JTTWZ16]; that result is similar to ours, except that it aggregates primes for which the Frobenius trace is extremal in both directions.The added ingredient in [JP17] is the use of unconditional estimates for the number of primes in an imaginary quadratic field lying in a sector; such an estimate has been given by Maknys [Mak83], modulo a correction described in [JP17].(For the Gaussian integers, see also [Zar91].) for some constant C > 0. When d > 2, the count (4.6) is subsumed in the error term of Theorem 3.8.There is thus no hope that the method of proof of Corollary 4.8 can be extended to the case d > 2 to obtain the analogue statement.When d = 1 (in which case A is Q-isogenous to the power of a CM elliptic curve and ST(A) ≃ U(1)), it is not difficult to generalize Proposition 4.7 to show that the number of primes p such that a p (A) = ⌊2g Nm(p)⌋ is again ≍ N x 3/4 / log(x).Note that for these primes, the equality ⌊2g Nm(p)⌋ = g⌊2 Nm(p)⌋ needs to hold because of the Weil-Serre bound.
As Andrew Sutherland kindly explained to us, when d = 2 there are already examples of abelian surfaces A defined over Q for which there are no primes p of good reduction for A such that (4.7) a p (A) = 2⌊2 √ p⌋ .
Indeed, let A be the product of two elliptic curves E 1 and E 2 defined over Q with CM by two nonisomorphic imaginary quadratic fields M 1 and M 2 , respectively.Suppose there were a prime p > 3 satisfying (4.7) of good reduction for A. Then a p (E 1 ) = a p (E 2 ) = ⌊2 √ p⌋ and p would be ordinary for both E 1 and E 2 .This would force both M 1 and M 2 to be the splitting field of the local factor of E 1 (which coincides with that of E 2 ) at p, contradicting the fact that M 1 and M 2 are not isomorphic.
work of Bucur and Kedlaya [BK16, Thm.4.3], who considered the case in which A and A ′ are elliptic curves without CM.Later Chen, Park, and Swaminathan [CPS18, Thm.1.3] reexamined this case, obtaining an upper bound of the form O(log(N N ′ ) 2 ) and relaxing the generalized Riemann hypothesis assumed in [BK16, Thm.4.3].Corollary 1.2 extends [CPS18, Thm.1.8], that again applies to elliptic curves without CM.It should be noted that the aforementioned results in [CPS18] make explicit the constants involved in the respective upper bounds, a goal which we have not pursued in our work.
Weyl's integration formula [Bou03, Chap.IX, §6, Cor. 2, p. 338] together with the fact that the absolute value of Weyl's density function is O(1) (see [Bou03, Chap.IX, §6, p. 335]) imply that the Haar measure of the W-translates of B(Y α , r ∆ ) and B(Y β , r ∆ ) is O(∆).Then the lemma follows from the equidistribution of θ p implied by part i) of Conjecture 2.4 and the prime number theorem.

4. 1 .
Interval variant of Linnik's problem for abelian varieties.Theorem 3.8 has the following immediate corollary.

Table 1 .
Table of notations