Speaker
Description
The arithmetic of abelian varieties is often studied through the lens of their Galois representations. Given an abelian variety $A$ over a number field $K$, an important invariant is the so-called Sato-Tate group $\operatorname{ST}(A)$, a compact Lie group which conjecturally describes the asymptotic distribution of the characteristic polynomials of Frobenius acting on the Tate modules of $A/K$. The group of connected components of $\operatorname{ST}(A)$ has particular arithmetic significance, and there exists a unique minimal extension $L/K$ such that $\operatorname{ST}(A_L)$ is connected. There is currently no general technique to determine $\operatorname{ST}(A)$, nor the extension $L/K$. In this talk I will describe how to compute these two invariants for the Jacobian of the curve $y^2=x^m+1$ by relating them to the cohomology of (several) Fermat hypersurfaces $X_m^n : Y_0^m + \cdots + Y_{n+1}^m=0$. The structure of this cohomology has been studied extensively by Deligne; our application, however, will require a more detailed analysis of the action of the absolute Galois group of $\mathbb{Q}$ on the étale cohomology of $X_m^n$.
