The Square Sieve and a Lang-Trotter Question for Generic Abelian Varieties

TL;DR

This paper develops bounds on primes where the Frobenius endomorphism of a generic abelian variety has a specified algebraic property, extending Lang-Trotter type conjectures from elliptic curves to higher dimensions.

Contribution

It introduces new bounds for the distribution of Frobenius fields of abelian varieties, generalizing Lang-Trotter conjectures to higher-dimensional cases.

Findings

Conditional and unconditional asymptotic bounds established

Results extend Lang-Trotter conjectures to abelian varieties of any dimension

Specific bounds for primes with Frobenius fields containing a given quadratic field

Abstract

Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$. We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} = \mathbb{Q}( \pi_p )$ of the reduction of $A$ modulo primes $p$ at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which this "Frobenius field" is a specified number field and, when $A$ is two-dimensional, at which the Frobenius field contains a specified real quadratic number field. These investigations continue the investigations of variants of the Lang-Trotter Conjectures on elliptic curves.

Full text

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