Generalized Sato–Tate conjectures for genus 2 curve L-functions

Establish generalizations of the Sato–Tate conjecture for the L-function L(C,s) of a genus 2 curve C over Q, characterizing the expected equidistribution of Frobenius data associated to its Jacobian.

Background

The Sato–Tate conjecture generalizes to higher-dimensional abelian varieties, including Jacobians of genus 2 curves, predicting statistical distributions of Frobenius eigenvalues. Such results rely on extensive computations of L_p(T) for many primes.

The authors highlight that L(C,s) is linked to these generalized Sato–Tate conjectures, which remain open in many cases, motivating their algorithm for rapidly lifting L_p(T) from its reduction modulo p.