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

Published online by Cambridge University Press:  07 March 2022

Alina Carmen Cojocaru*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, IL 60607, USA Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei St, Bucharest 010702, Sector 1, Romania
Tian Wang
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, IL 60607, USA e-mail: twang213@uic.edu
*

Abstract

Let $g \geq 1$ be an integer and let $A/\mathbb Q$ be an abelian variety that is isogenous over $\mathbb Q$ to a product of g elliptic curves defined over $\mathbb Q$ , pairwise non-isogenous over $\overline {\mathbb Q}$ and each without complex multiplication. 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 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 that $\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$ and $\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$ if $t \neq 0$ . These bounds largely improve upon recent ones obtained for $g = 2$ by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for $g=1$ by Murty, Murty, and Saradha, combined with a refinement in the power of $\operatorname {log} x$ by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying $|a_{1, p}(A)|>p^{\frac {1}{3 g + 1} - \varepsilon }$ for any fixed $\varepsilon>0$ .

Type
Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

A.C.C. was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation under Award No. 709008.

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