The Lang-Trotter conjecture on average for genus-$2$ curves with Klein-$4$ reduced automorphism group

TL;DR

This paper extends the Lang-Trotter conjecture on average to genus-2 curves with Klein-4 automorphism group, providing asymptotic estimates for supersingular reductions similar to elliptic curves.

Contribution

It generalizes the average case of the Lang-Trotter conjecture from elliptic curves to a specific family of genus-2 curves with Klein-4 automorphism group.

Findings

Derived an average estimate for supersingular primes for genus-2 curves

Established asymptotic behavior similar to elliptic curve case

Focused on curves characterized by Klein-4 automorphism group

Abstract

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjecture \[ \# \{ p<X \mid E \text{ has a supersingular reduction at } p \} \sim \frac{c\sqrt{X}}{\log X} \] as $X \rightarrow \infty$, where $c>0$ is a constant depending only on $E$. Fourvy and Murty obtained an average estimation related to the Lang-Trotter conjecture, called the Lang-Trotter conjecture on average. We considered the Lang-Trotter conjecture for curves of genus 2, and obtained a similar result to the Lang-Trotter conjecture on average for the family of curves $C_{\lambda}:y^2=x(x-1)(x+1)(x-{\lambda})(x-1/ \lambda)$. Such curves are characterized as curves of genus two with reduced automorphism group containing the Klein $4$-group.