Paramodular conjecture for L-functions of genus 2 curves over Q

Establish the paramodular conjecture for the Hasse–Weil L-function L(C,s) associated to the Jacobian of a genus 2 curve C over Q, verifying that L(J,s) coincides with the L-function of a corresponding paramodular Siegel modular form.

Background

The paper considers a genus 2 curve C over Q and defines its Hasse–Weil L-function L(C,s) via an Euler product whose local factors are polynomials L_p(T). The authors note that L(C,s) is central to several major open conjectures in arithmetic geometry and emphasize that explicit computation of the local polynomials L_p(T) up to a bound is essential to investigate these conjectures.

The work presents an algorithm that efficiently lifts L_p(T) mod p to an integral polynomial in Z[T], enabling large-scale computation of L_p(T) across many primes. This computational capability directly supports empirical tests and data gathering relevant to the paramodular conjecture in the genus 2 setting.