Arithmetic Properties of the Frobenius Traces Defined by a Rational Abelian Variety (with two appendices by J-P. Serre)

Let A be an abelian variety over Q of dimension g such that the image of its associated absolute Galois representation ρ A is open in GSp 2 g ( ˆ Z ) . We investigate the arithmetic of the traces a 1, p of the Frobenius at p in Gal ( Q / Q ) under ρ A . In particular, we obtain upper bounds for the counting function # { p ≤ x : a 1, p = t } and we prove an Erdös–Kac-type theorem for the number of prime factors of a 1, p . We also formulate a conjecture about the asymptotic behaviour of # { p ≤ x : a 1, p = t } , which generalizes a well-known conjecture of Lang and Trotter from 1976 about elliptic curves.


Introduction
Given an abelian variety A/Q, its reductions A p /F p modulo primes encode deep arithmetic global information.A primary question related to these reductions concerns their p-Weil polynomials, in particular the coefficients of these polynomials.
In the simplest case when A has dimension 1, that is, when A is an elliptic curve over Q, for each prime p of good reduction the p-Weil polynomial is P A,p (X ) = Briefly, the Lang-Trotter Conjecture [35] on the behaviour of a p predicts that for every elliptic curve A/Q and every integer t ∈ Z, if End Q (A) Z or t = 0, and if we write N A for the product of the primes of bad reduction for A, then either there are at most finitely many primes p such that a p = t or there exists a constant c(A, t) > 0 such that, as x → ∞, π A (x, t) := # p ≤ x : p N A , a p = t ∼ c(A, t) The constant c(A, t) has a precise heuristic description derived from the Chebotarev Density Theorem, combined with the Sato-Tate Conjecture when End Q (A) Z and with a prime distribution law arising from works of Deuring and Hecke when End Q (A) Z.
While the Lang-Trotter Conjecture remains open, several remarkable related results have been proven.When End Q (A) Z (the CM case) and t = 0, upper bounds of the right order of magnitude can be proved using sieve methods.When End Q (A) Z and t = 0, weaker upper bounds, unconditional or conditional (upon the Generalized Riemann Hypothesis, GRH), can be proved using effective versions of the Chebotarev Density Theorem; such bounds were first obtained by Serre [42,Theorem 20].The currently best unconditional upper bound, π A (x, t) A x(log log x) 2   (log x) 2 , was obtained by V. K. Murty [38,Theorem 5.1] (see [48] for an earlier result), while the currently best upper bound under GRH, π A (x, t) Frobenius Traces Defined by a Rational Abelian Variety 3559 π A (x, 0) x 3 4 were obtained by Fouvry and M. R. Murty [22, Theorem 1] and, respectively, by Elkies [19] using, as a key tool, Deuring's characterization of supersingular primes [17].
Inspired by these works, the main goal of our article is to investigate the arithmetic of the Frobenius traces of a generic higher-dimensional abelian variety A/Q; in particular: (i) we will prove upper bounds for the generalization of the counting function π A (x, t) and deduce results on the growth of the Frobenius traces; (ii) we will determine the normal order of the sequence defined by the prime divisor function of the Frobenius traces, and, more generally, we will prove an Erdös-Kac-type result for this sequence; (iii) under suitable hypotheses, we will formulate a generalization of (1).
Our main results mark only the beginning of such investigations in higher dimensions and we hope shall stimulate further research.
Our main setting and notation are as follows.Let A/Q be a principally polarized abelian variety of dimension g.Let Q denote an algebraic closure of Q and let End Q (A) denote the endomorphism ring of A over Q.Let N A be the product of primes of bad reduction for A.

We denote by
the absolute Galois representation defined by the inverse limit of the representations For each prime we denote by For each prime p N A , we consider the p-Weil polynomial P A,p (X ) of A, which is uniquely determined by the property that for any prime = p.In particular, we have for any integer m coprime to p.We write where the integers a i,p , 1 ≤ i ≤ g − 1, are independent of .
For any integer t ∈ Z, we consider the function The reason we usually impose the restriction that our abelian varieties be principally polarized is for ease of notation.When the abelian variety is principally polarized, the image of the -adic representation ρ A, lies in GSp 2g (Z ).Without the restriction on the polarization, the image lies in a group that can be defined by replacing the matrix J 2g of Section 2.1 below with a matrix that has a more complicated description, and our results could be modified accordingly; see, for example, Section 2.3 of [43] for the group of symplectic similitudes in this general setting.

Theorem 1.
Let A/Q be a principally polarized abelian variety of dimension g and let For any ε > 0 we have: i1) and (i2) hold with α replaced by β; (iii) if t = 0, then (i1) and (i2) hold with α replaced by γ .
We will actually prove a more general result, stated as Theorem 14 in Section 4, and that the case g = 1 of Theorem 1 is [42,Theorem 20,p. 189].
An immediate consequence of Theorem 1 concerns the non-lacunarity of the sequence (a 1,p ) p : Corollary 2. We keep the setting and notation of Theorem 1.For any ε > 0 we have: Recall that ν(n) denotes the number of distinct prime factors of a positive integer n and that an arithmetic function f (•) is said to have normal order for all but a zero density subset of positive integers n.It is a classical result of Erdös, originating in work of Hardy and Ramanujan [25], that ν(p − 1) has normal order log log p.More generally, Erdös and Kac [20] proved that ν(p − 1) has a normal distribution.Variations of these results have also been obtained in arithmetic geometric contexts, including that of modular forms [36].We now prove such results in the context of abelian varieties: In particular, ν(a 1,p ) has normal order log log p.
The case g = 1 not only recovers but also generalizes the main theorem of [36] for weight 2 newforms that are not of CM type.
Finally, in Conjecture 4 below we propose a generalization of (1) to the case of higher-dimensional abelian varieties for which Im ρ A is open in GSp 2g ( Ẑ) and for which the following holds: Equidistribution assumption: the normalized traces The assumption that Im ρ A is open in GSp 2g ( Ẑ) gives rise to an integer m A ≥ 1 that is the smallest positive integer m such that with : GSp 2g ( Ẑ) −→ GSp 2g (Z/mZ) the natural projection.
The Equidistribution Assumption gives rise to a continuous function : ), nonzero at 0, with the property that for every interval we have We propose: Conjecture 4. Let A/Q be a principally polarized abelian variety of dimension g and let t ∈ Z, t = 0. Assume that Im ρ A is open in GSp 2g ( Ẑ) and that the Equidistribution Assumption holds.Then, as x → ∞, the integers v (t) ≥ 0 are defined by v (t) |t, v (t)+1 t, and If c(A, t) = 0, we interpret the asymptotic as saying that there are at most finitely many primes p such that a 1,p = t.
For a discussion about the possible growth of π A (x, 0), see Section 5.

Remark 5. The image of ρ
A is open in GSp 2g ( Ẑ) for a large class of abelian varieties.
Indeed, in [43,44] Serre showed that this holds whenever End Q (A) Z and the dimension The paper is structured as follows.In Section 2 we present some of the key results needed for proving Theorem 1, Corollary 2, and Theorem 3, and for arguing towards Conjecture 4. In Section 3 we prove Theorem 1 and Corollary 2 using the strategy of [42, and also with the help of the main result of Serre's Appendix 1 of this aricle.In Section 4 we prove Theorem 3 following a general strategy of [5].In Section 5 we provide our heuristic reasoning towards Conjecture 4 and address some connections with existing works.In Section 6 we provide computational data related to our theoretical investigations.J-P.Serre supplied two appendices: the first gives a result on the dimension of conjugacy classes in symplectic groups, while the second gives properties of a certain density function for unitary symplectic groups.

Basic notation
Along with the standard analytic notation O, , , o, ∼, we use p and to denote rational primes; we write n|m ∞ to mean that all the prime divisors of n occur among the prime divisors of m, possibly with higher multiplicities; we write n||m to mean that n|m, but n 2 m; we write v (n) for the valuation of n at .
For a commutative, unitary ring R and a positive integer g, we denote by R × its group of units, by I g ∈ M g (R), I 2g ∈ M 2g (R) the identity matrices, and by We recall that the general symplectic group on R is defined by where M t denotes the transpose of M, while We note that GSp 2 (R) = GL 2 (R).We recall that GSp 2g (R) has centre {μI 2g : μ ∈ R × } and that, as an algebraic group, it has dimension 2g 2 + g + 1.
For R = C, we recall that the unitary symplectic group is defined by

Finite extensions of a number field
Let L/K be a finite Galois extension of number fields and let G be its Galois group.
Let C be a non-empty subset of G that is stable under conjugation.For any x > 0, let The Chebotarev Density Theorem states that We will use the following conditional effective version of this theorem: 34]; for this version see [42,Theorem 4,p. 133]).Keep the above setting and notation.Assume GRH for the Dedekind zeta function of L. Then there exists an absolute constant c > 0 such that In order to apply this theorem, the following variation of a result of Hensel [26], proved in [42], is useful: Proposition 9 ([42, Proposition 5, p. 129]).Keep the above setting and notation.Then where P(L/K) := {primes p : there is a place p of K, ramified in L/K, with p|p}.

-adic extensions of a number field
In [42], Serre used the effective versions of the Chebotarev Density Theorem of Lagarias and Odlyzko [34] to deduce upper bounds for π C (x, L/K) in the case of an -adic Galois extension L/K of a number field K.We recall his main results below.
Let K be a number field.Let be a rational prime and G a compact -adic Lie Following [42, p. 151], we define and R (x) := x (i) Unconditionally, we have In particular, for any ε > 0, we have (ii) Under GRH for Dedekind zeta functions, we have In particular, for any ε > 0, we have Serre obtains the following improvement in special cases: , where Z G (M) denotes the centralizer of M in G. Define Then (i) and (ii) of Theorem 10 hold with β C in place of α.Note that r C ≥ 0, hence β C ≥ α and so Theorem 11 is Theorem 10 when β C = α.

Abelian varieties
Let A/Q be an abelian variety of dimension g and let p be a prime of good reduction.
Recall that for any root π ∈ C of P A,p (X ) we have |π | = √ p, hence Property ( 2) links the p-Weil polynomial P A,p (X ) to the division fields of A, in particular to the Galois representation defining ρ A .
For arbitrary integers m ≥ 1 and t, we set We recall that: • by the Néron-Ogg-Shafarevich criterion, • by the injectivity of the restriction of ρA,m to Gal(Q In many cases, the image of the representation ρ A is better understood.For example, as already mentioned in Remark 5 of Section 1, for several classes of abelian varieties In particular, for such A we have that: ) for all but finitely many rational primes .
Lemma 12 below gives further consequences of the openness of Im ρ A in GSp 2g ( Ẑ).
To state the lemma, we introduce the following notation: Lemma 12. Let A/Q be a principally polarized abelian variety of dimension g such that , where we recall that is the natural projection.Denote by m A the least such integer.
(ii) For all positive integers m 1 , (iii) For all t ∈ Z we have In particular, if t = 0, then (iv) For all t ∈ Z, t = 0, we have Proof.Parts (i) and (ii) are clear from the openness assumption on Im ρ A .For part (iii), let m A and t be fixed.First, we will show that Recall that the multiplicator of GSp 2g (Z/ Z) is the character of GSp 2g (Z/ Z) with kernel Sp 2g (Z/ Z); we denote it by mult.Let char(M) denote the characteristic polynomial of a square matrix M.
Next we will prove that This ensures the convergence of the infinite product H t ( ), proving (iii).
We first prove (9) for t = 0.For this, observe that for any t 1 , t 2 ∈ Z we have and Indeed, the first assertion is trivial, while the second assertion follows by noting that, if t 1 ≡ 0(mod ) and t 2 ≡ 0(mod ), then the endomorphism From the above observations, It is now easy to show that (9) follows from this along with (8) for |C( , 0)|.Now we prove (9) for t = 0.When g = 1, a straightforward calculation gives that When g ≥ 2, we proceed as follows.By [32, Theorem 5.3, p. 170], for some explicit polynomials f ( ) and g( ) in .Of relevance to us is that the degree d g( ) of the leading term of g( ) in satisfies and that the degree d f ( ) of the leading term of f ( ) in , while less explicit, can be shown to satisfy Before justifying this bound, let us complete the proof of (9) for t = 0, g ≥ 2.
From (10), we see that for any t = 0, Since we already know (9) for t = 0, it suffices to show that This follows from (11) and (12), as well as the assumption that g ≥ 2; indeed, Consequently, (9) holds for t = 0, g ≥ 2.
Finally, let us justify (12).The expression for f ( ) is rather delicate; indeed, Kim showed that where K(α) is the ordinary Kloosterman sum for any non-trivial additive character λ of F q .
To find the leading term, we first focus on for an arbitrary integer r ≥ 0.
When r = 0, the sum is simply − 1.When r = 1, by Weil's estimate on Kloost- where M 0 := 1 and for any integer s ≥ 1, Note that M 1 = 1 and that for s ≥ 2, the first of the two conditions defining M s gives α 1 linearly in terms of the other α i , while the second gives α 2 as a root of a quadratic in the remaining terms.Thus, if s ≥ 2, then M s ≤ 2( − 1) s−2 .It follows that when r = 2, the sum α∈F × K(α) r is bounded by an expression of leading degree at most 2 in (by direct computation using M 1 ), and when r ≥ 3, by an expression of leading degree at most r − 1 in .
Using the above estimates, we now focus on the degree d f ( ) of the leading term in (13); we deduce that The quadratic function above is maximized when b = g 2 and k = g−2b+2 2 = 1, with maximal value 3g 2 2 + g 2 + 1; the bound (12) follows.This proves (9), and therefore the first part of (iii).
To prove the second part of (iii), observe that t = 0 is divisible by at most finitely many primes, and so , where m 0 , t 0 ∈ Z satisfy m 0 , t 0 , and note that v (m) ≥ 1.For any s ∈ Z Therefore we may write s = s 0 v (t) with s 0 ∈ Z and s 0 .By the Chinese Remainder Lemma, there exists u ∈ Z such that u ≡ t −1 0 s 0 (mod ) and u ≡ 1(mod m 0 ), hence such that We have since if (m, m A ) = 1 then by (ii) we have G(m) = GSp 2g (Z/mZ) and so uI 2g ∈ G(m), while if m A |m then u ≡ 1(mod m A ) (by ( 15)) and (by the definition of m A ) and thus uI 2g ∈ G(m).
Using ( 14) and ( 16), we deduce that the multiplication by uI 2g map Now consider the natural projection and observe that Letting and using (17), we obtain Putting together ( 18) and ( 19), we deduce that for all positive integers m such that (m, m A ) = 1 or m A |m, and for all primes | m, we have and thus Therefore for all d | m A we have and for all k ≥ 1 and all primes m A we have Now for any positive integer m consider its unique factorization By (ii), Using (20) for the second line below and ( 21) for the third line, we have which gives (iv).
Remark 13.As in the case g = 1, when g = 2 it is possible to derive closed formulae and We sketch a proof of the latter using arguments from [8]; we leave it as an exercise to the reader to derive these formulae using the aforementioned results of [32].We will use these formulae in Remark 24. Define It follows from the proof of [8,Theorem 12] that We will now show that which in turn confirms (22).
Note that and since, for any given x, the defining conditions of these sets determine y uniquely, provided that δ = −y.Putting the two together, we obtain that Dividing by − 1, we deduce that N ,t = ( − 2) 2 for any fixed nonzero t; this completes the proof of (22).

Proof of Theorem 1
For a prime and an integer t, define: .
We will deduce Theorem 1 from the following more general result: Theorem 14.Let A/Q be a principally polarized abelian variety of dimension g and let t ∈ Z.
(i) Assume that there exists a prime such that: Then for any ε > 0 we have: for, otherwise, recalling that Z(G ) = {μI 2g : μ ∈ Z × }, we would have that the -adic valuation of t 2g satisfies v t 2g = 0, a contradiction.In particular, for any M ∈ C (t), Centralizers are closed subgroups, hence Lie subgroups, and Z G (M) has a well-defined dimension.Since GSp 2g is connected as an algebraic group, (25) Therefore we can improve upon the result of (i) by applying Theorem 11 to the extension L/Q and the conjugacy set C (t) with D := dim G .
(iii) If t = 0, we choose as in the hypothesis of (iii) and with ρA, := • ρ A, we consider a Galois extension of Q with Galois group G .Observing that it remains to estimate the right-hand side.
Since G is open in G , we have that G is open in PG and so dim In particular, as in the proof of part (ii), for any M ∈ C (0), Therefore we can improve upon the result of (i) by applying Theorem 11 to the extension Recalling that dim GSp 2g = 2g 2 + g + 1, we see that α = 1 2g 2 +g+1 .If g = 1, then r C (t) and r C (0) are calculated as in [42, pp. 189-190], giving rise to β = 1 3 and γ = 1 2 .If g ≥ 2, then r C (t) and r C (0) are estimated using Serre's Theorem A.1 in Appendix 1. Indeed, by this theorem and Remark 5 that follows its statement, for To improve upon this bound when t = 0, we focus on estimating γ and use If g = 2, we use (26) and once again the first part of Theorem A.1 in Appendix 1 If g ≥ 3, we use (26) and the last part of Theorem A.1 in Appendix 1 to deduce This completes the proof of Theorem 1.
Proof of Corollary 2. The proof of Corollary 2 is deduced easily from part (i) of Theorem 1 and the Prime Number Theorem, as follows.Unconditionally, The uniformity in t of the bounds for π A (x, t) provided by Theorem 1 was crucial in the above estimates.

Proof of Theorem 3
Let A/Q be a principally polarized abelian variety of dimension g such that Im ρ A is open in GSp 2g ( Ẑ).We will investigate ν(a 1,p ) via the method of moments, with the goal of proving: for each integer k ≥ 1, where is the kth moment of the standard Gaussian.
With this, by adapting to our context the proof of the Erdös-Kac Theorem due to Billingsley [5] (see also [4] and the references therein for an accessible exposition), Theorem 3 is proved.
The core ingredient in our proof is the following application of ( 6)-( 7), Theorem 8 (under GRH) and Proposition 9: for any positive integer m and any x > 0 (to be thought of as approaching infinity), we have Related to this, remark that by the openness assumption of Im ρ A in GSp 2g ( Ẑ) and by (8) from the proof of part (iii) of Lemma 12, we have for all m A .In particular, for any y > 0, and, after using ( 29) and ( 7) Crucial to the method is also the following simple observation.Let x > 0 and 0 < δ < 1 be fixed and let y := x δ .For any integer m ≥ 1, we have where ν y (m) denotes the number of distinct prime divisors ≤ y of m.
We now proceed with the proof of (27).For each prime , we define a random variable R to be 1 with probability 1 and 0 with probability 1 − 1 .Upon taking y := x δ for some fixed 0 < δ < 1 and x → ∞, R(y) := ≤y R becomes normally distributed with mean and variance each equal to log log x; by the Central Limit Theorem, for any integer k ≥ 1 we have By (29), R models the event that |a 1,p for some p.Our strategy then is to prove (27) by We fix x > 0 and k ≥ 1, choose a parameter δ = δ(g, k) such that and define y := x δ .In what follows, our O-estimates will reflect the growth of various functions as x → ∞.
For each and each p N A , we define Then, for each integer 1 ≤ j ≤ k, upon applying ( 28)-( 31) and (33), we obtain By the binomial theorem and the above, we deduce Recalling the choice of δ given in (34) and using part (i2) of Theorem 1, we see that the Then, upon applying the binomial theorem once again in order to rewrite the first term, we deduce Finally, recalling ( 5) and (32) and applying ( 33) and ( 35) several times, we deduce This completes the proof of Theorem 3.
Remark 16.The first and second moments of ν(a 1,p ) may be estimated directly, without any comparison with the model defined by R .The strategy originates in Turán's proof of the Hardy-Ramanujan Theorem, [47], and is summarized below.
We choose 0 < δ < 1 8g 2 +4g+1 and let y = x δ .Then, proceeding as in the proof of Theorem 3 but without the model R , we obtain In turn, this is obtained by remarking that  Proof.This follows by viewing the expression inside the limit as a Riemann sum approximation of the integral     x/ log x for various Jacobians of hyperelliptic curves.

Converging products of Lemma 12
In part (iii) of Lemma 12, we showed that the following infinite product converges for all integers t and all integers g ≥ 1:

X 2 −
a p X + p ∈ Z[X ], where a p := p + 1 − |A p (F p )|.The coefficient a p satisfies the Weil bound |a p | < 2 √ p and is of major significance in number theory.For example, it appears as the pth Fourier coefficient in the expansion of the weight 2 newform associated to A. The study of a p comes in several flavours, some having led to well-known problems in arithmetic geometry, such as the Sato-Tate Conjecture from the 1960s (now a theorem) and the Lang-Trotter Conjecture on Frobenius traces from the 1970s (still open).
, 2g] with respect to the projection by the trace map of the (normalized) Haar measure of the unitary symplectic group USp(2g).
group of dimension D. Denote by Z(G) the centre of G. Let C ⊆ G be a non-empty closed subset of G that is stable under conjugation.In [42, Section 3] Serre explains what it means for the Minkowski dimension dim M C of C to be ≤ d.Let L/K be an infinite Galois extension, with Galois group G.For any x > 0, let

Theorem 11 (
[42, Theorem 12, p. 157]).Keep the above setting and notation.Let 0 ≤ d < D be such that the Minkowski dimension of C satisfies dim M C ≤ d.Define

1 .of Theorem 1 .
and the conjugacy set C (0) with D := dim G − Proof In our setting, by the openness assumption on Im ρ A , hypothesis (a) of Theorem 14 holds for any prime .It remains to verify hypothesis (b) and to compute the values of α, β, and γ .To verify hypothesis (b) of either parts (i) or (ii), observe that C (t) is a closed subvariety of the algebraic group GSp 2g and so C (t) has a well-defined dimension strictly smaller than dim G .The bound applies to the Minkowski dimension dim M C (t) also by [42, Theorem 8].Part (b) follows with d := dim G − 1.To verify hypothesis (b) of part (iii), observe that (C (0)) is a closed subvariety of the algebraic group PG and so C (0) has a well-defined dimension strictly smaller than dim PG .The bound applies to the Minkowski dimension dim M C (0) also by [42, Theorem 8].Part (b) follows with d := dim G − 2.

Lemma 20 .
For all integers m ≥ 1 and τ 0 ∈ Z we have lim

Fig. 4 .
Fig.4.Histograms of ν(a 1,p ) for J 1 ; on the left, the data are for primes p < 220 , and on the right, the data are for primes 220 < p < 221 .Primes of bad reduction and primes for which the trace is zero are excluded.

Table 1 .
(38)res1 and 2show the values of π A (x, t) graphed versus √ x/ log x for t ∈ {0, 1} and A ∈ {J 1 , J 2 , J 3 }, where J 1 , J 2 , J 3 are the Jacobians of the hyperelliptic curves listed in Prediction(38)would imply that these graphs approximate a straight line, whose slope is determined by the constant in front of √ x/ log x; the graphs are indeed consistent with this implication.