Paramodular conjecture for genus 2 curve L-functions

Establish the paramodular conjecture for the L-function L(C,s) of a genus 2 curve C over Q, determining its predicted modularity relationship with paramodular Siegel modular forms.

Background

The paper considers the L-function L(C,s) of a genus 2 curve C over Q, defined by an Euler product whose local factors are L_p(T). The authors note that L(C,s) is central to several major conjectures in arithmetic geometry.

Efficient computation of L_p(T) across many primes enables empirical investigation and, in some cases, proofs of instances of these conjectures. The paramodular conjecture, specifically for abelian surfaces (including Jacobians of genus 2 curves), remains open and motivates fast algorithms for determining L_p(T) to gather evidence.