Abstract

Let $A$ be an abelian variety defined over a number field $k$. The connected monodromy field $k(\varepsilon A)$ is the minimal extension of $k$ over which every $\ell$-adic Galois representation attached to $A$ has connected image. Equivalently, it is the smallest field over which all Tate classes on self-products $A^r$ are defined. When the extension $k(\varepsilon A)/k(\mathrm{End} A)$ has positive degree, one finds “exotic” Tate classes on certain powers $A^r$.

In this talk, I will explain how to compute the connected monodromy field for the Jacobian $A$ of a curve with complex multiplication. After computing the endomorphism ring of $A$, we use CM theory to describe the algebra of Tate classes on all powers of $A$. We make the Galois action on this algebra explicit in terms of periods – suitable integrals of algebraic differential forms. Although periods are generally transcendental, those attached to Tate classes are algebraic, hence computing $k(\varepsilon A)$ amounts to identifying these periods as exact algebraic numbers. This can be done numerically and, in the case of Fermat curves, via explicit algebraic identities.