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

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 Amri (Eur J Math, 2023. https://doi.org/10.1007/s40879-023-00682-5, Theorem 1.1). As a consequence, we provide a conditional direct proof to the generalized Lang–Trotter conjecture recently formulated and studied in Chen et al. (Ramanujan J, 2022).