In this talk we shall present the construction of the Spin $p$-adic $L$-function for $p$-ordinary Siegel modular forms of genus $6$, using an integral expression due to Pollack. Joint work (still in progress) with E. Eischen and S. Shah.

I will present a map - The toric regulator, from motivic cohomology of algebraic varieties over p-adic fileds with "totally degenerate reduction", e.g., $p$-adically uniformized varieties, to "toric intermediate Jacobians" which are quotients of algebraic toruses by a discrete subgroup. The toric regulator recovers part of the $\ell$-adic etale regulator map for every prime $\ell$, and its...

General overview of triple product periods.

This lecture will describe the general conjectures on triple product periods formulated over

the years in joint work with Alan Lauder and Victor Rotger, and discuss a few of their

ramifications, including:

1. The connection with generalised Kato classes and their arithmetic applications.

2. Tame variants and the Harris-Venkatesh conjecture....

In the first two lectures, Loeffler will recall Hida's theory of ordinary p-adic families of modular forms, and how it was used to construct p-adic Rankin--Selberg L-functions for $\mathrm{GL}_2\times \mathrm{GL}_2$ (by Hida and Panchishkin), and triple-product L-functions for $\mathrm{GL}_2\times \mathrm{GL}_2\times\mathrm{GL}_2$ (by Harris--Tilouine and Darmon--Rotger).

Then he will...

Darmon, Lauder and Rotger have formulated different conjectures involving the so-called p-adic iterated integrals attached to a triple (f,g,h) of classical eigenforms of weights (2,1,1). When f is a cusp form, it involves the p-adic logarithm of distinguished points on the modular abelian variety attached to f. However, when f is Eisenstein, they conjecture a formula involving the p-adic...

The p-adic theory of modular forms plays a key role in modern number theory. Geometric developments have enabled vast expansion of Serre's original notion of p-adic modular forms, including by Hida to the case of automorphic forms on unitary groups. This talk will introduce some challenges that arise in the setting of unitary groups, recent efforts to overcome them, and applications.

The study of arithmetic invariants associated to Galois representations has often relied on the construction of a special family of elements in their Galois cohomology groups. For instance, it has been a crucial ingredient in the work of Kato in the proof of special cases of the conjecture of Birch and Swinnerton-Dyer and the Iwasawa main conjecture for modular forms.

In this talk, we...

Rigid meromorphic cocycles and their RM values.

This lecture will introduce the basic structures that arise in a p-adic approach

to explicit class field theory based on the values at real quadratic arguments

of rigid meromorphic cocycles.

These values comprise as special cases the

Gross-Stark units arising in Gross’s p-adic analogue of the Stark conjecture

on p-adic Artin L-series...

In the first two lectures, Loeffler will recall Hida's theory of ordinary p-adic families of modular forms, and how it was used to construct p-adic Rankin--Selberg L-functions for $\mathrm{GL}_2\times \mathrm{GL}_2$ (by Hida and Panchishkin), and triple-product L-functions for $\mathrm{GL}_2\times \mathrm{GL}_2\times\mathrm{GL}_2$ (by Harris--Tilouine and Darmon--Rotger).

Then he will...

Let $F$ be a totally real field, and $K$ a CM extension. For a cuspidal, automorphic, cohomological representation $\pi$ over $\mathrm{GL}_n/K$, I will talk about the special values at critical points of the Asai L-function associated to $\pi$. I will also talk about the special values of the Asai L-function twisted by Hecke characters of $F$.

I will discuss on some works in progress for the construction of Euler systems attached to the Standard p-adic L-function

attached to ordinary Siegel modular forms using congruences between Klingen-type Eisenstein series and cusp forms.

We recall general conjectures about the existence of $p$-adic $L$-functions attached to motives and automorphic representations. Then the lecture is devoted to the study of the critical values of the complex $L$-function of cuspidal automorphic representations of $\mathrm{GL}(2n)$ admitting a Shalika model. In particular we describe such $L$-values in terms of classical evaluations constructed...

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for L-values of the form $L(1/2,\mathrm{BC}(\pi)\times\mathrm{BC}(\pi'))$, where $\pi$ and...

We introduce and study the overconvergent cohomology adapted to the

Shalika setting. Then we describe how to evaluate this cohomology in order to produce distributions over the expected Galois group. Moreover, we verify that this overconvergent evaluation interpolates the classical evaluations explained in the first lecture. Another consequence of this method is the control of the growth of...

Diagonal restrictions of Hilbert Eisenstein series.

This last lecture explains how the diagonal restrictions of the p-adic family of

Hilbert modular Eisenstein series for a real quadratic field can be related to

RM values of certain rigid analytic cocycles, leading to an interpretation of

Gross-Stark units and Stark-Heegner points as triple product periods. The

p-adic deformation...

In the third lecture, Pilloni will outline the proofs of the main theorems of higher Hida theory for $\mathrm{GSp}_4$, and describe work in progress to generalise these results to higher-rank symplectic groups.

Investigating critical values of Rankin-Selberg L-functions has a long history, both, on the side of results as well as on the side of conjectures. While most of the known results treat the case of $\mathrm{GL}(n) \times\mathrm{GL}(n-1)$, in this talk we will shade some light on what can be said in the general case $\mathrm{GL}(n) \times\mathrm{GL}(m)$, when the ground field is CM.

I will report on work in progress with David Loeffler and Chris Skinner. I will sketch a proof for of an explicit reciprocity law for the Euler system attached to the spin representation of genus $2$ Siegel modular forms, relating the Euler system to the spin $p$-adic $L$-function that we constructed in joint work with Vincent Pilloni. As an application, we obtain bounds on Selmer groups,...

The correct eigenvarieties to be considered in the Shalika setting are constructed using the parabolic subgroup of $\mathrm{GL}(n)$ having Levi subgroup $\mathrm{GL}(n)\times \mathrm{GL}(n)$. After the introduction of these parabolic eigenvarieties the talk is devoted to the study of the local properties of them and the existence of Shalika components. We use such results in order to perform...