Regulators V

Sato-Tate groups of Fermat Jacobians

7 Jun 2024, 16:00

1h

Department of Mathematics, University of Pisa

Speaker

Davide Lombardo (Università di Pisa)

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 $\mathrm{ST}(A)$, 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 $\mathrm{ST}(A)$ has particular arithmetic significance, and there exists a unique minimal extension $L/K$ such that $\mathrm{ST}(A_L)$ is connected. There is currently no general technique to determine $\mathrm{ST}(A)$, 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 $X_m^n$.