Recent progress on p-adic Galois images for elliptic curves over $\mathbb{Q}$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. When $E$ does not have potential complex multiplication, Serre's open image theorem asserts that the Galois action on the torsion points of $E$ is ``as large as possible'': the image of the adelic Galois representation is open in $\operatorname{GL}_2(\widehat{\mathbb{Z}})$. In 1978, Mazur proposed a far-reaching refinement, his ``Program B'', calling for a complete classification of all possible adelic images of Galois for elliptic curves over $\mathbb{Q}$.

Over the last 15 years, there has been striking progress towards this goal, ultimately depending on our ability to find rational points on modular curves over $\mathbb{Q}$. In the first part of the talk, I will introduce the problem, survey the state of the art, explain how it splits naturally into subcases, and describe the remaining open cases. These involve in particular a class of subgroups known as normalisers of non-split Cartans.

In the second part, I will present a recent work, joint with Matthew Bisatt and Davide Lombardo, where we (almost) classify all the possible $p$-adic Galois images of $p$-adic elliptic curves with supersingular reduction, only in terms of the valuation of their $j$-invariant. Notably, this rules out a subcase of Mazur's Program B linked to the non-split Cartan problem for infinitely many primes $p$. In particular, we show that for all primes $p \gt 37$, the only possible $p$-adic images of Galois are the inverse images in $\operatorname{GL}_2(\mathbb{Z}_p)$ of a non-split Cartan subgroup modulo $p^n$ for some $n \ge 1$.

The proof requires excluding both proper subgroups of the non-split Cartan modulo $p$ and certain ``exotic'' groups arising at level $p^2$. To rule out the latter, we use tools from $p$-adic Hodge theory to obtain an explicit description of the $p^2$-torsion representation of elliptic curves over $\mathbb{Q}_p$ in the most delicate case -- bad, potentially good supersingular reduction.