VACANCES à Jussieu

TBA

For an elliptic curve $E$ over a number field $K$, let $n(p)$ be the order of the group of points of $E$ over the residue field at some prime $p$ of $K$. Generically, for varying $p$, divisibility of $n(p)$ by two different primes $\ell$ and $q$ are "independent events" on the set of primes of good reduction for $E$. However, even over $K=\mathbb{Q}$ special elliptic curves do exist for which $n(p)$ can be coprime to each of $\ell$ and $q$ but not to the product $\ell q$. In this talk, I will present a joint work in progress with Francesco Pappalardi and Peter Stevenhagen where we classify the possible Galois representations for $E$ giving rise to this behaviour in the case $\ell=2$.

After the pioneering work of Minhyong Kim, J. Balakrishnan and others, recent years have seen the development of radical improvements of the classical Chabauty method for determining rational points on curves, called "non-abelian Chabauty". Whereas the first formulations relied strongly on p-adic cohomological tools, B. Edixhoven and G. Lido have subsequently proposed a version which is close in spirit to Chabauty's genuine geometric intuition.
All versions however demand explicit equations for the curves to be studied. In a large joint project with S. Hashimoto, K. Khuri-Makdisi, G. Lido, D. Lombardo and N. Mascot, we endeavour to formulate the geometric method in moduli terms only, in order to study new examples of modular curves, hopefully with increased efficiency, and better understanding of some of the method's limits (or possibilities).

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 $k(\varepsilon A)/k(\operatorname{End} A)$ 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.