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arXiv · 2604.00822

The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group

Chihiro Ando·Shushi Harashita
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Abstract

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p 0$ is a constant depending only on $E$. While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus $2$ and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves $C_{\lambda}:y^2=x(x-1)(x-{\lambda})(x-(\lambda-1)/{\lambda})(x-1/ (1-\lambda))$. These curves are characterized as curves of genus $2$ with reduced automorphism group containing symmetric group $S_3$.

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Chihiro Ando, Shushi Harashita. 2026-04-01. The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group. https://arxiv.org/abs/2604.00822

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