# Automorphic p-adic L-functions and regulators

Europe/Paris
Salle de réunion (University of Lille)

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France
Description

The aim of this workshop is to provide an overview of recent developments in theory of p-adic L-functions associated to automorphic representations, covering both the construction of p-adic L-functions, and their relations to Euler systems in Galois cohomology via regulator maps. The workshop will consist of mini-courses, aimed at younger researchers, and more specialised individual lectures.

There will be three mini-courses, each consisting of three lectures, on the following topics:

• Construction of p-adic L-functions for automorphic forms on GL(2n), using the automorphic modular symbols introduced in work of Dimitrov (outline)
• Triple product p-adic periods and class field theory (outline)
• Construction of p-adic L-functions for GSp(4), using Pilloni’s higher Hida theory (outline)

The other talks will explore connections of these topics with other related areas of current research, such as Iwasawa theory, the theory of Hecke varieties and the theory of L-invariants.

Timetable

Titles and abstracts

### Invited speakers

 Baskar Balasubramanyam (IISER Pune, India) Daniel Barrera (Universidad de Santiago, Chile) Amnon Besser (Ben-Gurion University, Israel) Antonio Cauchi (Université Laval, Canada) Henri Darmon (McGill University, Canada) Ellen Eischen (University of Oregon, USA) Harald Grobner (University of Vienna, Austria) Michael Harris (Columbia University, USA) Andrei Jorza (University of Notre Dame, USA) David Loeffler (University of Warwick, UK) Vincent Pilloni (ENS Lyon, CNRS, France) Alice Pozzi (University College London, UK) Oscar Rivero (UPC and McGill University, Canada) Giovanni Rosso (Concordia University, Canada) Eric Urban (Columbia University, USA) Jan Vonk (University of Oxford, UK) Christopher Williams (Imperial College London, UK) Sarah Zerbes (University College London, UK)

### Organisers

 Mladen Dimitrov (Lille, France) David Loeffler (Warwick, UK) Sarah Zerbes (London, UK)

### General informations

How to come? Click here for directions to the Laboratoire Paul Painlevé.

A list of hotels is available here.

Participants
• Abhishek Saha
• Alessandro Arlandini
• Alexandre Maksoud
• Alexandros Galanakis
• Alice Pozzi
• Amnon Besser
• Andrei Jorza
• Andrew Graham
• Anne Moreau
• Antonio Cauchi
• Chao Qin
• Chris Williams
• Daniel Barrera
• David Lilienfeldt
• David Loeffler
• Di Zhang
• Dohyeong Kim
• Duc Nam Nguyen
• Elena Mantovan
• Ellen Eischen
• Emmy Duclos
• Eric Urban
• Francesc Gispert
• Francesca Gatti
• Francesco Lemma
• Giancarlo Castellano
• Giovanni Rosso
• Hao ZHANG
• Harald Grobner
• Henri Darmon
• Jacques Tilouine
• Jan Vonk
• Jeanine Van Order
• Ju-Feng Wu
• Katharina Müller
• Maria Rosaria Pati
• Mariagiulia De Maria
• Martí Roset Julià
• Matias Moya Giusti
• Michael Harris
• Nils Ellerbrock
• Pak-Hin Lee
• Peter Xu
• Pierre Charollois
• Pierre Dèbes
• Robert Rockwood
• Sarah Zerbes
• Sheng-Chi Shih
• Shin Hattori
• Syed Waqar Ali Shah
• Vincent Pilloni
• Weijia Wang
• Yan Hu
• Zeping Hao
Contact
• Monday, 14 October
• 08:30 09:30
coffee+pastries 1h Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France
• 09:30 10:30
Spin p-adic L-function for GSp(6) 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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.

Speaker: Giovanni Rosso (Concordia University)
• 10:30 11:00
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 11:00 12:00
The toric regulator 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 logarithm recovers part of the log-syntomic regulator. I expect the toric regulator to be the home of "refined" Beilinson conjectures and I will present some evidence for this claim. This is joint work with Wayne Raskind from Wayne State university.

Speaker: Amnon Besser (Ben-Gurion University)
• 12:00 14:00
Lunch break 2h R.U. Barrois

#### R.U. Barrois

Avenue Paul Langevin – Cité Scientifique 59650 Villeneuve d'Ascq France
• 14:00 15:00
Mini-course 2: Lecture 1 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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.
3. The special case of the adjoint, and a theorem of Rivero-Rotger.

Speaker: Darmon Henri (McGill University)
• 15:00 15:30
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 15:30 16:30
Mini-course 3: Lecture 1 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 outline the key statements of Pilloni's higher Hida theory for the symplectic group $\mathrm{GSp}_4$, which gives an analogous p-adic interpolation results for higher-degree coherent cohomology of Siegel threefolds, and describe how these techniques can be used to construct p-adic L-functions for $\mathrm{GSp}_4$, $\mathrm{GSp}_4\times\mathrm{GL}_2$, and $\mathrm{GSp}_4\times \mathrm{GL}_2 \times\mathrm{GL}_2$, as in the recent preprint of Loeffler --Pilloni--Skinner--Zerbes.

Speaker: David Loeffler (University of Warwick)
• 16:45 17:45
The arithmetic of the adjoint of a weight one modular form 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 logarithms of units and p-units in suitable number fields, which can be seen as a variant of Gross’ p-adic analogue of Stark’s conjecture. In a joint work with V. Rotger we prove the conjecture when h is dual to g. The proof rests on Hida’s theory of improved p-adic L-functions and Galois deformation techniques. Further, it suggests a tantalizing connection with the theory of Beilinson--Flach elements, in a setting where an exceptional vanishing of these cohomology classes emerges.

Speaker: Oscar Rivero (UPC and McGill University)
• Tuesday, 15 October
• 08:30 09:30
coffee+pastries 1h Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 09:30 10:30
Recent developments for p-adic families automorphic forms and L-functions, in the context of unitary groups 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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.

Speaker: Ellen Eischen (University of Oregon)
• 10:30 11:00
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 11:00 12:00
On the construction of elements in the Iwasawa cohomology of Galois representations for GSp(2n)xGSp(2n) 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 describe how to construct elements in the Iwasawa cohomology of Galois representations associated to a product of two cohomological cuspidal automorphic representations of the similitude symplectic group $\mathrm{GSp}_{2n}$, and, thus, p-adic L-functions using Perrin-Riou’s machinery. This construction generalises the one given by Lei-Loeffler-Zerbes when $n=1$.

Speaker: Antonio Cauchi (Université Laval)
• 12:00 14:00
Lunch 2h R.U. Barrois

#### R.U. Barrois

Avenue Paul Langevin – Cité Scientifique 59650 Villeneuve d'Ascq France
• 14:00 15:00
Mini-course 2: Lecture 2 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 at s=0, Stark-Heegner points on (modular) elliptic curves,
and singular moduli for real quadratic fields. They can often be expressed in terms of
(twisted variants of) the triple product periods covered in Lecture 1.

Speaker: Jan Vonk (University of Oxford)
• 15:00 15:30
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 15:30 16:30
Mini-course 3: Lecture 2 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 outline the key statements of Pilloni's higher Hida theory for the symplectic group $\mathrm{GSp}_4$, which gives an analogous p-adic interpolation results for higher-degree coherent cohomology of Siegel threefolds, and describe how these techniques can be used to construct p-adic L-functions for $\mathrm{GSp}_4$, $\mathrm{GSp}_4\times\mathrm{GL}_2$, and $\mathrm{GSp}_4\times \mathrm{GL}_2 \times\mathrm{GL}_2$, as in the recent preprint of Loeffler --Pilloni--Skinner--Zerbes.

Speaker: David Loeffler (University of Warwick)
• 16:45 17:45
Critical values of the Asai L-function for GL(n) over CM fields 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 $F$.

• Wednesday, 16 October
• 08:30 09:30
coffee+pastries 1h Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 09:30 10:30
Eisenstein congruences and Euler system for Siegel modular forms. 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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.

Speaker: Eric Urban (Columbia University)
• 10:30 11:00
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 11:00 12:00
Mini-course 1: Lecture 1 (Critical L-values) 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 using the cohomology of the corresponding locally symmetric space and so-called automorphic cycles.

Speaker: Andrei Jorza (University of Notre Dame)
• Thursday, 17 October
• 08:30 09:30
coffee+pastries 1h Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 09:30 10:30
Square root p-adic L-functions 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 $\pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $\mathrm{BC}$ denotes the twisted base change to $\mathrm{GL}(n) \times \mathrm{GL}(n-1)$.

Speaker: Michael Harris (Columbia University)
• 10:30 10:40

#### University of Lille

Cité Scientifique - Avenue Henri Poincaré - BP 30155 - 59653 Villeneuve d'Ascq Cedex
• 10:40 11:00
coffee 20m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 11:00 12:00
Mini-course 1: Lecture 2 (Overconvergent cohomology) 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 the distribution obtained. The $p$-adic $L$-functions are, as usual, the Mellin transform of these distributions.

Speaker: Daniel Barrera (Universidad de Santiago)
• 12:00 14:00
Lunch 2h R.U. Barrois

#### R.U. Barrois

Avenue Paul Langevin – Cité Scientifique 59650 Villeneuve d'Ascq France
• 14:00 15:00
Mini-course 2: Lecture 3 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 theory of the weight one Hilbert Eisenstein series, building on the
work of Bellaiche-Dimitrov, Darmon-Lauder-Rotger, and Betina-Dimitrov Pozzi,
is a key ingredient in some of the most important arithmetic applications.

Speaker: Alice Pozzi (University College London)
• 15:00 15:30
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 15:30 16:30
Mini-course 3: Lecture 3 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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.

Speaker: Vincent Pilloni (ENS Lyon, CNRS)
• 16:45 17:45
Rationality for Rankin-Selberg L-functions 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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.

Speaker: Harald Grobner (University of Vienna)
• 19:00 21:00
Conference dinner 2h La Terrasse des Remparts

#### La Terrasse des Remparts

Porte de Gand, 92 Rue de Gand, Lille
• Friday, 18 October
• 08:30 09:30
coffee+pastries 1h Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 09:30 10:30
An explicit reciprocity law for the GSp(4)-Euler system 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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, conditional on the nonvanishing of non-critical $p$-adic $L$-values.

Speaker: Sarah Zerbes (University College London)
• 10:30 11:00
coffee 30m Salle Kampé de Fériet

### Salle Kampé de Fériet

#### University of Lille

• 11:00 12:00
Mini-course 1: Lecture 3 (p-adic families) 1h Salle de réunion

### Salle de réunion

#### University of Lille

Cité Scientifique – Bâtiment M2 59655 Villeneuve d'Ascq Cedex France

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 a $p$-adic variation of the distributions obtained in the second lecture. Using the Mellin transform we produce $p$-adic families of $p$-adic $L$-functions.

Speaker: Chris Williams (Imperial College London)
• 12:00 14:00
Lunch 2h R.U. Barrois

#### R.U. Barrois

Avenue Paul Langevin – Cité Scientifique 59650 Villeneuve d'Ascq France