Abstracts

Marco Baracchini (Università di Padova)

Title - On (de-)linearization and BGG decomposition 

Abstract - In this talk we will discuss the (de-)linearization of modules over smooth varieties, introduced by A. Grothendieck, and we will introduce the classic Berstein—Gelfand—Gelfand (BGG) theory. The two topics arise in different fields of mathematics, but together could be used in order to compute the de Rham cohomology of certain sheaves over varieties of the form G/P, where G is a reductive algebraic group and P is a parabolic subgroup; getting a geometrization of the BGG theory. Finally we will see an application in arithmetic number theory: we will compute the de Rham cohomology of some sheaves that p-adic interpolate de Rham classes of modular forms. This talk is based on a joint work with Fabrizio Andreatta e Adrian Iovita.

Adel Betina (Universität Wien)

Title - Eigenvarieties for non-cuspidal automorphic forms over PEL Shimura varieties

Abstract - We construct eigenvarieties parametrizing locally analytic overconvergent p-adic automorphic forms including Eisenstein families for unitary and symplectic groups. This is achieved by refining the method of Andreatta—Iovita—Pilloni for cuspidal automorphic forms and a deep study of the Hodge—Tate period map at the boundary of a toroidal compactification. The main novelty is the construction of subsheaves of the p-adic automorphic sheaves of Andreatta—Iovita—Pilloni by imposing a condition on the sections at the boundary. This talk is based on a joint work with Riccardo Brasca and Giovanni Rosso. 

Andrea Conti (Universität Heidelberg)

Title - Bogomolov property for Galois representations with big local image 

Abstract - An algebraic extension of Q is said to have the Bogomolov property if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation ρ of the absolute Galois group G_Q, one can ask whether the field fixed by ker(ρ) has the Bogomolov property (in short, we say that ρ has (B)). In a joint work with Lea Terracini, we prove that, if ρ:G_Q ---> GL_N(Z_p) maps an inertia subgroup at p surjectively onto an open subgroup of GL_N(Z_p), then ρ has (B). More generally, we show that if the image of a decomposition group at p is open in the image of G_Q, and a certain condition on the center of the image is satisfied, then ρ has (B). In particular, no assumption on the modularity of ρ is needed, contrary to previous work of Habegger and Amoroso—Terracini.

Michael Daas (Max Planck Institute for Mathematics)

Title - A p-adic analogue of a formula by Gross and Zagier 

Abstract - In their 1984 paper “On singular moduli”, Gross and Zagier proved an explicit factorisation formula for the norm of the difference between two CM-values of the classical j-function. In 2022, it was conjectured by Giampietro and Darmon that the CM-values of certain p-adic theta-functions on Shimura curves should obey similar factorisation patterns. In this talk, we explore the classical result about the j-function, discuss its proofs and outline how the study of infinitesimal deformations of p-adic Hilbert Eisenstein series was used to settle the conjectures about the theta-function. This p-adic analytic approach bears resemblance to some of the newly developed methods in modern RM-theory.

Michele Fornea (CRM Barcelona)

Title - A unified framework for plectic Heegner classes 

Abstract - Nekovář and Scholl's plectic conjectures suggest leveraging CM points on higher dimensional quaternionic Shimura varieties to study the arithmetic of elliptic curves of higher rank. Inspired by that insight, in a series of joint works with Darmon, Gehrmann, Guitart and Masdeu, we proposed unconditional (albeit not completely satisfying) constructions of special elements conjecturally controlling Mordell—Weil groups of higher rank. In this talk we will present a construction of plectic Heegner classes generalizing simultaneously that of plectic Heegner and mock plectic points. This is joint work with Henri Darmon.

Daniel Kriz (Laboratoire de Mathématiques d'Orsay)

Title - An overconvergent weight 2 Eisenstein series and quasi-overconvergent modular forms 

Abstract - A prototypical example of a Katz p-adic modular form which is not classical or even overconvergent is Katz's p-adic weight 2 Eisenstein series E_2. The non-overconvergence of Katz's E_2 is closely related to the non-overconvergence of the unit root splitting and has motivated several definitions of nearly-overconvergent modular forms in the past decade, for example those of Urban and Andreatta—Iovita—Pilloni. In this talk we introduce a new period sheaf defined on infinite-level Hodge-type Shimura varieties containing certain geometric periods which extend Katz's E_2 in the modular curve case and thus can be viewed as overconvergent weight 2 Eisenstein series. Over this period sheaf we construct a natural splitting of the p-adic Hodge filtration of universal de Rham cohomology that analytically continues the unit root splitting on the ordinary Igusa tower to a large open neighborhood containing (most of) the supersingular locus. In this sheaf we define a space of "quasi-overconvergent modular forms" as well as weight-raising operators acting on it which specialize to classical theta operators over the ordinary Igusa tower. We will focus on the Shimura curve case and address the general Hodge-type case if time permits. 

Fırtına Küçük

Title - Factorization of algebraic p-adic adjoint L-functions 

Abstract - In this talk, I will briefly review Artin formalism and its p-adic variant. Artin formalism provides a factorization of L-functions whenever the associated Galois representation decomposes. I will explain why the p-adic Artin formalism is a non-trivial problem when there are no critical L-values. In particular, I will focus on the case where the Galois representation arises from a Rankin—Selberg product of a newform with its conjugate form. I will present the results in this direction including the one I obtained in my PhD thesis, and discuss the future directions.

Alexandre Maksoud (Universität Paderborn)

Title - The geometry of the eigencurve at classical weight one modular forms 

Abstract - The study of p-adic deformations of automorphic forms was initiated by Hida in the 1980s, after he discovered the existence of systematic congruences between the Fourier coefficients of modular forms. The eigencurve is a geometric incarnation of these congruences, introduced by Coleman and Mazur, which is proving to be a fundamental tool in the study of number-theoretic conjectures such as the Birch and Swinnerton-Dyer conjecture. In this talk, I will explain a new method for studying the local geometry of the eigencurve at weight 1 forms in a case where the classical R=T arguments fall short. If time allows, I will then discuss the construction of two-variable p-adic L-functions in the neighbourhood of such forms. This is a joint work with Adel Betina and Alice Pozzi.

Luca Marannino (CNRS — Institut de Mathématiques de Jussieu)

Title - Anticyclotomic Iwasawa theory of modular forms at inert primes via diagonal classes 

Abstract - In this talk, we try to outline an approach to the study of anticyclotomic Iwasawa theory of modular forms when the fixed prime p is inert in the relevant quadratic imaginary field. Following ideas of Castella—Do for the "p split" case, one can envisage a construction of an anticyclotomic Euler system arising from a suitable manipulation of diagonal cycles. We will report on this work in progress, trying to underline the main difficulties arising in the "p inert" setting.

Maria Rosaria Pati (Università di Genova)

Title - Perrin-Riou's main conjecture for modular forms

Abstract - In this talk, I will state and sketch a proof of the counterpart for a higher (even) weight newform f of Perrin-Riou's Heegner point main conjecture for elliptic curves ("Heegner cycle main conjecture" for f). Our strategy of proof builds on ideas of Bertolini and Darmon, as elaborated by Howard, and consists in the construction of a so-called bipartite Euler system, which is a collection of cohomology classes and p-adic L-functions attached to congruent modular forms. This is joint work with Matteo Longo and Stefano Vigni.

Gautier Ponsinet (Universität Duisburg—Essen)

Title - On a characterisation of perfectoid fields by Iwasawa theory  

Abstract - With a p-adic representation of the Galois group of a p-adic field are associated the Bloch—Kato groups defined via p-adic Hodge theory. Iwasawa theory motivates the study of these Bloch—Kato groups over infinite algebraic extensions of the field of p-adic numbers. Over perfectoid fields, several results state that the Bloch—Kato groups admit a simple description. In this talk, we will present a reciprocal statement: the structure of the Bloch—Kato groups associated with certain de Rham representations characterises the algebraic extensions of the field of p-adic numbers whose completion are perfectoid fields. In particular, we will recover, via a different method, results by Coates and Greenberg for abelian varieties, and by Bondarko for p-divisible groups.

Óscar Rivero (Universidad de Santiago de Compostela)

Title - Eisenstein degeneration of Beilinson—Kato classes and circular units

Abstract - In this talk, I will present joint work with Javier Polo, where we investigate the Euler system of Beilinson—Kato elements in families passing through the critical p-stabilization of an Eisenstein series. Within this framework, we establish an explicit link with the system of circular units, making use of factorization formulas in a setting where several p-adic L-functions vanish. We also discuss analogous results in the context of Beilinson—Flach classes, previously studied in collaboration with David Loeffler.

Ju-Feng Wu (University College Dublin)

Title - Eigenstacks 

Abstract - Eigenvarieties are geometric objects that parametrise finite-slope p-adic  automorphic forms. It is discovered by mathematicians that the geometry of eigenvarieties encodes interesting arithmetics of finite-slope automorphic forms. It is natural to ask what happens if one is interested in more general automorphic forms (not necessarily the finite-slope ones). In this talk, I will report on an ongoing joint work with Andrew Graham, where we construct eigenstacks, which see beyond finite-slope families of automorphic forms. I will also discuss its Galois counterpart and some open questions. 

Francesco Zerman (FernUni Schweiz)

Title - Euler and Kolyvagin systems in the anticyclotomic setting 

Abstract - One main concern in algebraic number theory is to find a relation between special values of L-functions and the rank of some arithmetic groups. In the case of elliptic curves, the work of Kolyvagin in the late '80s linked this problem to the existence of anticyclotomic Euler systems, that are sets of cohomology classes with some rigid compatibility properties. This idea has been generalized in many fruitful ways to other Galois representations, coming from modular forms and p-adic families of modular forms. In this talk, I will present how one can formalize Kolyvagin’s method in order to deal with all known examples all at once. This comes partly from my PhD thesis and partly from a joint work with Luca Mastella.

Xiaoyu Zhang (Universität Duisburg—Essen)

Title - Modularity of Galois representations valued in dual group of SO(n) and applications 

Abstract - One of the basic ingredients in Wiles' proof of Fermat's Last Theorem is modularity lifting theorems of Galois representations valued in GL(2). This was further generalised by Clozel—Harris—Taylor to definite unitary groups. In this talk, I will discuss the case of definite special orthogonal groups and then talk about some potential applications in Bloch—Kato conjectures.

Luochen Zhao (Morningside Center of Mathematics)

Title - On the structure of anticyclotomic Selmer groups of modular forms 

Abstract - I will report the recent work with Antonio Lei and Luca Mastella, in which we determine the structure of the Selmer group of a modular form over the anticyclotomic Z_p extension, assuming the imaginary quadratic field satisfies the Heegner hypothesis, that p splits in it and at which the form has good reduction, and that the bottom generalized Heegner class is primitive. Here the last assumption springs from Gross's treatment of Kolyvagin's bound on Shafarevich—Tate groups, and was put in the Iwasawa theoretic context by Matar—Nekovář and Matar for elliptic curves. I will explain our use of the vanishing of BDP Selmer groups in proving the result, which allows us to treat both ordinary/supersingular reduction types uniformly.