Generalizations of the Sato–Tate conjecture for genus 2 curves over Q

Establish the appropriate generalizations of the Sato–Tate conjecture for the Hasse–Weil L-function L(C,s) of the Jacobian of a genus 2 curve over Q, characterizing the equidistribution of normalized Frobenius conjugacy classes in the relevant Sato–Tate group.

Background

The authors focus on computing local factors L_p(T) for the L-function of genus 2 curves, data that underpin statistical analyses of Frobenius distributions. They highlight that L(C,s) lies at the heart of several open conjectures, including generalizations of Sato–Tate in higher dimension.

By lifting L_p(T) from modular reductions to integers rapidly, the algorithm facilitates extensive prime-by-prime computations needed to investigate equidistribution phenomena predicted by Sato–Tate-type conjectures for abelian surfaces arising from genus 2 curves.