Mini-courses

Samit Dasgupta: Ribet's method

One of the central themes in number theory is the connection between special values of L-functions and certain invariants of associated global algebraic objects (such as regulators). There are many famous concrete conjectures in this direction, such as the conjectures of Stark, Birch-Swinnerton-Dyer, Beilinson, and Bloch-Kato. The main conjectures of Iwasawa theory are another important example. One of the most powerful techniques we have toward understanding some of these conjectures is a method due to Ribet, which creates a link between L-functions and global algebraic objects using modular forms and their associated Galois representations. In this course, we will describe Ribet's method starting with his original proof of the converse to Herbrand's Theorem. This result states that if $p$ is a prime and $p$ divides a certain special value of the Riemann zeta function, then the $p$-part of an associated component of the class group of $\mathbb Q(\mu_p)$ is non-trivial. We will introduce and describe all of the relevant features of the proof of Ribet's result, including Eisenstein series, congruences with cusp forms, Galois representations, Galois cohomology, class groups, and class field theory. After describing the details of Ribet's proof, we hope to spend some time describing other applications of Ribet's method, such as the Brumer-Stark conjecture.

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.

Vincent Pilloni: The integral coherent cohomology of Shimura varieties and higher Hida theory

We will describe an integral structure on the space of ordinary modular forms and more generally on the ordinary coherent cohomology of certain Shimura varieties. This integral structure arises from the cohomology of Igusa varieties.

We will explain in detail how the theory works for modular curves. We will then consider Siegel or Hilbert modular varieties.

We will try to describe (conjectural) applications of this theory, for example to modularity problems, $p$-adic $L$-functions, geometric Jaquet-Langlands correspondence...

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 `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. This is a joint work with G. Ponsinet and S. Vigni.

Peter Scholze: Analytic geometry