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

Abstract: Let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$ without complex multiplication and pairwise non-isogenous over $\overline{\mathbb{Q}}$. For an integer $t$ and a positive real number $x$, denote by $\pi_A(x, t)$ the number of primes $p \leq x$, of good reduction for the abelian variety $A$, for which the Frobenius trace associated to the reduction of $A$ modulo $p$ equals $t$.

Based on prior approaches to the Lang-Trotter Conjecture for the Frobenius traces associated to reduction of

an elliptic curve, under the RH and the GRH for Dedekind zeta functions, we prove a non-trivial upper bound for $\pi_A(x, t)$.

This is joint work with Alina Carmen Cojocaru.

number theory

Audience: researchers in the topic

Around Frobenius distributions and related topics II