On the Lang-Trotter conjecture for a class of non-generic abelian surfaces

TL;DR

This paper formulates a conjectural uniform error term for the distribution of abelian surfaces related to the Lang-Trotter conjecture and provides a direct proof for a generalized version of the conjecture.

Contribution

It introduces a conjectural uniform error term in the Chebotarev-Sato-Tate distribution for specific abelian surfaces and proves the generalized Lang-Trotter conjecture for these cases.

Findings

Formulated a conjectural uniform error term in distribution

Provided a direct proof of the generalized Lang-Trotter conjecture

Established results for abelian surfaces related to non-CM elliptic curves

Abstract

In the present article, we formulate a conjectural uniform error term in the Chebotarev-Sato-Tate distribution for abelian surfaces $\mathbb{Q}$-isogenous to a product of not $\overline{\mathbb{Q}}$-isogenous non-CM-elliptic curves, established by the author in \cite[Theorem 1.1]{Amri22}. As a consequence, we provide a direct proof to the generalized Lang-Trotter conjecture recently formulated and studied in \cite{Chen}.