Number theory seminar in memory of F. Momose

Computing Sato-Tate distributions of low genus curves.


Andrew V. Sutherland


The generalized Sato-Tate conjecture predicts that the distribution of normalized Euler factors of an abelian variety $A$ of dimension $g$ converges to the distribution of characteristic polynomials in a certain compact Lie group $ST_A$ (the Sato-Tate group of $A$) that is subgroup of $USp(2g)$ (the group of $2g \times 2g$ complex matrices that are both unitary and symplectic). We have developed a suite of computational tools to very efficiently compute, in the case that $A$ is the Jacobian of a curve of genus $g \leq 2$, statistics that allow one to to provisionally determine $ST_A$ , as well as techniques for then proving that this provisional identification is correct. In this talk I will describe some of these tools, which played a key role in the recently completed classification of the $52$ Sato-Tate groups that arise in genus $2$. Time permitting, I will also discuss the prospects in genus $3$, where some preliminary work has begun. This is joint work with Francesc Fité, Victor Rotger, and Kiran Kedlaya.


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