Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

Abstract

Let g ≥ 1 be an integer and let A/Q be an abelian variety that is isogenous over Q to a product of g elliptic curves defined over Q, pairwise non-isogenous over Q and each without complex multiplication. For an integer t and a positive real number x, denote by π A (x, t) the number of primes p ≤ x, of good reduction for A, for which the Frobenius trace a 1,p (A) associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove thatThese bounds largely improve upon recent ones obtained for g = 2 by H. Chen, N. Jones, and V. Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for g = 1 by M.R. Murty, V.K. Murty, and N. Saradha, combined with a refinement in the power of log x by D. Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying |a 1,p (A)| > p 1 3g+1 −ε for any fixed ε > 0.the Frobenius traces defined by the product of non-isogenous elliptic curves defined over Q and having no complex multiplication, as explained below.Let g ≥ 1 be an integer and let A/Q be an abelian variety that is isogenous over Q to a product of g elliptic curves defined over Q, pairwise non-isogenous over Q and each without complex multiplication.

Quantitative upper bounds related to an isogeny criterion for elliptic curves

“…Finally, from part (iii) of Lemma 9 and part (i) of the current lemma, we deduce that 5 . This completes the proof of (iii).…”

“…We follow the methods developed in [6] and [25]. These methods already give rise to the stated conditional estimates for  1,1 𝐸 1 ,𝐸 2 (𝑥), but need to be adjusted for the general conditional and unconditional bounds, as we explain below.…”

“…The second modification is a variation of the argument in the proof of the second part of [17, Corollary 1.6, p. 406], which we make in order to improve the resulting bound 𝑥 , as follows. Letting 𝓁(𝑥) be as in (6), instead of as in [17, p. 422], we deduce that each of the terms on the right-hand side of inequality ( 14) is bounded from above by 𝜅 ′′ 1 (𝐸 𝑗 , 𝐾)𝑥 2 3 (log 𝑥) 1 3 for some positive constant 𝜅 ′′ 1 (𝐸 𝑗 , 𝐾), which depends on 𝐸 𝑗 and 𝐾. Putting everything together gives part (iii) of Theorem 1.…”

“…Remark 5. Theorem 3 may be viewed under the general Lang-Trotter theme of results about the number of primes for which the Frobenius trace of an abelian variety is fixed, such as those proven in [2,5,6,16,17,19,22,23], and [27]. The connection between Theorem 3, and thus Theorem (𝑥) for 𝐸 1 , 𝐸 2 , 𝛼 1 , and 𝛼 2 as in the setting of Theorem 3.…”