Mini-courses

David Loeffler: Euler systems, syntomic regulators and $p$-adic $L$-values

The goal of the course will be to describe the relation between Euler systems and $p$-adic $L$-values via syntomic regulators, starting with the (relatively) simple setting of Beilinson-Flach classes and then describing the more recent extensions to other Euler systems for higher rank groups. The course will assume familiarity with modular curves, modular forms and their Galois representations, but will not assume any prior expertise on syntomic cohomology or Euler systems.

Alice Pozzi: Real multiplication and $p$-adic families of modular forms

The classical theory of elliptic curves with complex multiplication is a fundamental tool for studying the arithmetic of abelian extensions of imaginary quadratic fields. While no direct analogue is available for real quadratic fields, a (largely conjectural) theory of "real multiplication" relying on $p$-adic methods has been proposed. A rigid meromorphic cocycle is a class in the first cohomology of the group $\mathrm {SL}_2(\mathbb Z[1/p])$ acting on the non-zero rigid meromorphic functions on the Drinfeld $p$-adic upper half plane by Möbius transformation. The values of rigid meromorphic cocycles at real quadratic points can be thought of as analogues of singular moduli for real quadratic fields. In this minicourse, we will discuss results relating real multiplication values of rigid meromorphic cocycles and derivatives of $p$-adic families of modular forms, fitting into an emerging "$p$-adic Kudla program".

The minicourse will be structured as follows:

1. We will start by discussing the geometry of the Drinfeld $p$-adic upper half plane as a rigid analytic space. We will introduce rigid meromorphic and theta cocycles, drawing a parallel with the $p$-adic theta functions arising in the theory of uniformisation of Mumford curves. We will then explain how these cocycles can be evaluated at real multiplication points.
2. We will review aspects of the classical theory of complex multiplication. We will then describe the construction of elliptic cocycles and the Dedekind-Rademacher cocycle, and explain how their RM values are used to produce analogues of Heegner points and elliptic units.
3. We will outline the analytic proof of the factorisation of differences of singular moduli by Gross and Zagier, involving the derivative of a real analytic family of Hilbert Eisenstein series. We will discuss an analogue of this result for a $p$-adic family of Hilbert Eisenstein series.
   We will then introduce the Hilbert eigenvariety, parametrising more general $p$-adic families of modular forms.
4. We will describe how the study of the local geometry of eigenvarieties at classical weight points carries information about units in number fields. We will explain how this information can be leveraged into a proof of the algebraicity of the RM values of the Dedekind-Rademacher cocycle.

Abstracts

Giada Grossi: Mazur's main conjecture at Eisenstein primes

Let $E$ be a rational elliptic curve and p an odd Eisenstein prime of good reduction. I will talk about joint work with F. Castella and C. Skinner, in which we prove new cases of the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson–Flach classes.

Maria Rosaria Pati: On Shafarevich-Tate groups and analytic ranks in Coleman families

Let $f$ be a newform of weight $2$, square-free level $N$ and trivial character, let $A_f$ be the abelian variety attached to $f$, whose dimension will be denoted by $d_f$, and for every prime number $p\nmid N$ at which $f$ has finite slope let $\boldsymbol f^{(p)}$ be a $p$-adic Coleman family through $f$ over a suitable open disc in the $p$-adic weight space. We prove that, for all but finitely many primes $p$ as above, if $A_f(\mathbb Q)$ has rank $r\in\{0,d_f\}$ and the $p$-primary part of the Shafarevich--Tate group of $A_f$ over $\mathbb Q$ is finite, then all classical specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite $p$-primary Shafarevich--Tate group and $r/d_f$-dimensional image of the relevant $p$-adic \'etale Abel--Jacobi map. As a second contribution, assuming the non-degeneracy of certain height pairings \emph{\`a la} Gillet--Soulé between Heegner cycles, we show that, for all but finitely many $p$, if $f$ has analytic rank $r\in\{0,1\}$, then all classical specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $r$. This result provides some evidence for a conjecture of Greenberg on analytic ranks in families of modular forms.