Conference 

Monday, June 30, 2025

9:40- 10:40

Speaker: Massimo Bertolini (University of Duisburg-Essen)

Title: Anticyclotomic main conjectures

Abstract: This talk is based on a joint work with Matteo Longo and Rodolfo Venerucci. It plans to report on recent (and not so recent) results by several authors on the anticyclotomic Main conjectures of Iwasawa theory for elliptic curves.

10:50- 11:50

Speaker: Francesc Castella (University of California, Santa Barbara)

Title: Bloch-Kato conjecture for CM modular forms and Rankin-Selberg convolutions 

Abstract: Let E/F be an elliptic curve with CM by an imaginary quadratic field K, and assume that the extension of F generated by the torsion points of E is abelian over K. In this talk I will outline the proof of the p-part of the Birch-Swinnerton-Dyer formula for E in analytic rank 1 for primes p > 3 of ordinary reduction. For F = Q, this was originally proved by Rubin in 1991 as a consequence of his proof of the Iwasawa main conjecture for K. In contrast, our approach to the problem is based on the study of an auxiliary Rankin-Selberg convolution, and extends to CM abelian varieties A/K and higher weight CM modular forms.

14:00- 15:00

Speaker: Marco Sangiovanni-Vincentelli (Columbia University)

Title: A base change of Kato’s Euler system

Abstract: In this talk, I will present joint work with A. Burungale on the construction of a new Euler system for the base change of an elliptic modular form to a quadratic imaginary field. This Euler system exhibits remarkably good p-adic deformation properties and specializes in the cyclotomic direction to Beilinson–Kato’s classes: this relation exhibits many features parallel to the case of cyclotomic units and elliptic units. I will argue that it can be viewed as a “base change” of Kato’s Euler system, as anticipated by the analytic side of the Bloch-Kato conjecture.

15:30- 16:30

Speaker: Takamichi Sano (Osaka Metropolitan University)

Title: On derivatives of p-adic L-functions for motives

Abstract: We give a generalization of various p-adic analogues of the Birch and Swinnerton-Dyer conjecture for a motive. We first generalize the conjecture of Coates–Perrin-Riou on the existence of a p-adic L-function for a motive. We also formulate the Iwasawa main conjecture for the general p-adic L-function. We then formulate a conjecture on derivatives of the p-adic L-function, which generalizes the conjectures of Mazur– Tate–Teitelbaum, Bernardi–Goldstein–Stephens, Bertolini–Darmon, and Agboola–Castella. We study its relation with the Iwasawa main conjecture and give a generalization of classical results of Perrin-Riou and Schneider on Iwasawa L-functions. We prove that, under the validity of Kato’s local epsilon conjecture, our conjecture on derivatives of the p-adic L-function is equivalent to a conjecture on derivatives of the corresponding Euler system, which was formulated earlier by the speaker. We also give an application to the Tamagawa number conjecture.

Tuesday, July 1, 2025

9:30- 10:30

Speaker: Mahesh Kakde (Indian Institute of Science)

Title: On the Brumer-Stark Conjecture and refinements.

10:40- 11:40

Speaker: Adel Betina (University of Vienna)

Title: The eigencurve at crystalline points with scalar Frobenius and Gross–Stark regulators

Abstract: We present a complete description of the local geometry of the p-adic eigencurve at p-irregular classical weight one cusp forms, assuming the non-vanishing of certain p-adic Gross–Stark regulators (a condition predicted by conjectures in classical p-adic transcendence theory). As an application, we establish several results concerning the Hecke structure of the ordinary p-adic  ́etale cohomology of the tower of modular curves. This talk is based on joint work with Maksoud–Pozzi.

14:00- 15:00

Speaker: Muhammad Manji (Concordia University)

Title: Lubin–Tate Iwasawa Theory for CM elliptic curves with p inert

Abstract: The Iwasawa theory for CM elliptic curves is a rich problem which has been studied over decades. In the 1990s Rubin proved a 2-variable main conjecture without p-adic L-functions independent of splitting behaviour of p, leading to a main conjecture with p-adic L-functions when p splits. However, a 2-variable main conjecture for p inert remains out of reach. In this work, joint with A.Lei, we prove a main conjecture in the Lubin–Tate direction (utilising the full Z_2 p -extension of the CM field). We also extend the interpolation region of the attached p-adic L-function of Schneider-Teitelbaum and demonstrate the relationship between the p-adic periods involved.

15:30- 16:30

Speaker: Kazuto Ota (Osaka University)

Title: Local sign decomposition for symplectic self-dual Galois representations of rank two

Abstract: Around 1987, Rubin envisioned a signed Iwasawa theory for CM elliptic curves at supersingular primes p over the anticyclotomic Zp-extension of the CM field, and he formulated a fundamental conjecture on sign decomposition of the local Iwasawa cohomology. In a joint work with Ashay Burungale, Shinichi Kobayashi and Kentaro Nakamura, we generalize and prove the Rubin conjecture for generic symplectic self-dual families of local Galois representations of rank two.

Wednesday, July 2, 2025

9:00- 10:00 *Online Talk

Speaker: Liang Xiao (Peking University)

Title: Cohomological dimension of Shimura varieties, Beilinson’s conjecture, higher Chow group arithmetic theta lifts, and higher Borcherds products 

Abstract: I will report on a series of recent joint works on a ”higher Chow group” version of arithmetic theta lifting. I will start by discussing conjectures on the cohomological dimension of Shimura varieties (as an unexpected technical input later), and then motivated by the Beilinson’s conjecture, we explain a program that aims to construct a higher Chow group version of arithmetic theta lifting, and perhaps gives nontrivial applications to Beilinson’s conjecture in the corresponding situation. If time permits, I will discuss interesting predictions of this program, including a conjectural generalization of Borcherds product to the Siegel case. This is an ongoing joint work with Haocheng Fan, Wenxuan Qi, Linli Shi, Peihang Wu, and Yichao Zhang.

10:10- 11:10

Speaker: Andrew Graham (University of Oxford) 

Title: Local-global compatibility and the exceptional zero conjecture for GL(3) (I)

Abstract: If E is a rational elliptic curve with split multiplicative reduction at p, then the associated p-adic L-function Lp(E, s) has an ”exceptional zero” at s = 1 regardless of the vanishing of the complex L-value L(E, 1). A seminal result of Greenberg and Stevens gives a precise formula for the first derivative of the p-adic L-function at s = 1 in terms of the complex L-value L(E, 1) and the so-called L-invariant. This L-invariant can be interpreted in several ways – on the automorphic side for example, L-invariants parameterise part of the p-adic local Langlands correspondence for GL2(Qp). 

In these two talks, we will explain our joint work with Chris Williams on a generalisation of this exceptional zero formula to regular algebraic, cuspidal automorphic representations of GL(3) which are Steinberg at p. The proof is divided into two parts: an automorphic argument and a Galois argument. The automorphic part establishes a formula relating the relevant p-adic L-function to the automorphic L-invariants defined by Gehrmann, and the Galois argument follows the strategy of Greenberg–Stevens and Gehrmann–Rosso, employing techniques from the p-adic deformation of Galois representations. A key ingredient is a local-global compatibility result for p-adic families of ordinary automorphic representations, for which we follow the arguments of the ”10-author paper.”

11:20- 12:20

Speaker:Daniel Barrera Salazar (Universidad de Santiago de Chile)

Title: Local-global compatibility and the exceptional zero conjecture for GL(3) (II)

Abstract: If E is a rational elliptic curve with split multiplicative reduction at p, then the associated p-adic L-function Lp(E, s) has an ”exceptional zero” at s = 1 regardless of the vanishing of the complex L-value L(E, 1). A seminal result of Greenberg and Stevens gives a precise formula for the first derivative of the p-adic L-function at s = 1 in terms of the complex L-value L(E, 1) and the so-called L-invariant. This L-invariant can be interpreted in several ways – on the automorphic side for example, L-invariants parameterise part of the p-adic local Langlands correspondence for GL2(Qp). 

In these two talks, we will explain our joint work with Chris Williams on a generalisation of this exceptional zero formula to regular algebraic, cuspidal automorphic representations of GL(3) which are Steinberg at p. The proof is divided into two parts: an automorphic argument and a Galois argument. The automorphic part establishes a formula relating the relevant p-adic L-function to the automorphic L-invariants defined by Gehrmann, and the Galois argument follows the strategy of Greenberg–Stevens and Gehrmann–Rosso, employing techniques from the p-adic deformation of Galois representations. A key ingredient is a local-global compatibility result for p-adic families of ordinary automorphic representations, for which we follow the arguments of the ”10-author paper.”

Thursday, July 3, 2025

9:30- 10:30 

Speaker: Eric Urban (Columbia University)

Title: TBA

Abstract: TBA

10:40- 11:40

Speaker: Rustam Steingart (ENS Lyon)

Title: Derived analytic vectors of Be

Abstract: I will explain how to compute the derived analytic vectors of the invariants under the kernel H of the cyclotomic character of the period ring Be using condensed mathematics. Arithmetically, these derived analytic vectors appear in the context of the Bloch-Kato exponential map when passing from the cohomology of H with values in Qp(n) to its analytic vectors. This picture is dual to the passage from the Iwasawa cohomology to its base change to the distribution algebra. Even though this passage is exact, since the cohomology groups are admissible. The sections of the corresponding vector bundle on the punctured Fargues–Fontaine curve are not admissible, which explains why their derived analytic vectors intervene.

14:00- 15:00

Speaker: Chi-Yun Hsu (Santa Clara University)

Title: p-adic companion forms for Yoshida lifts

Abstract: Coleman showed that the (k − 1)st power of the theta operator q d/dq defines a map from overcon-vergent forms of weight 2−k and slope 0 to weight k and slope k−1. Moreover, the critical p-stabilization of a classical CM form is the image of a p-adic CM form, strengthening the fact that its Galois representation splits locally at p. In the GSp(4) setting, the Galois representation of a Yoshida lift splits locally into two 2-by-2 blocks at p. In joint work in progress with Bharathwaj Palvannan, we aim to prove an analogous strengthening. The relevant theta operator arises from the last differential of the dual BGG complex. We computed its explicit effect on q-expansions for weight (k, 3), and expect that the effect for general weights to be a power of this. Using the explicit Fourier coefficients of Yoshida lifts by Hsieh–Namikawa, we show that Yoshida lifts lie in the image of this theta operator.

15:30- 16:30

Speaker: Kenichi Namikawa (Tokyo Denki University)

Title: On p-adic L-functions for GL(3) × GL(2)

Abstract: Based on C.G. Schmidt, Januszewski constructed a p-adic analogue of the Rankin-Selberg L-functions of GL(n)×GL(n−1). However, in his construction, some expected properties for p-adic L-functions still remain to prove. In this talk, we introduce a refined construction of p-adic L-functions for GL(3)×GL(2) which is compatible with Coates-Perrin-Riou’s conjecture. We also discuss its variations on p-adic families of automorphic representations and an application to the trivial zero conjecture. This is a joint work with Takashi Hara at Tsuda university.

16:40- 17:40

Speaker: Mladen Dimitrov (University of Lille)

Title: Improved Shalika models and p-adic L-functions for GL(N)

Abstract: The arithmetic of L-functions has long been a topic of intense interest in number theory. Via the Bloch? Kato Conjecture and its p-adic avatars, special values of L-functions are expected to carry deep algebraic data, and good understanding of p-adic L-functions, eigenvarieties, and p-adic L-functions over eigenvarieties have been instrumental in most recent progress towards these conjectures via Iwasawa theoretic methods. 

In this talk, I will present a recent work with Andrei Jorza on p-adic L-functions attached to parahoric representations of GL(N) of symplectic type with a view towards the trivial zero conjecture. Naive general-izations of previous constructions for spherical representations, based on Friedberg-Jacquet integral formulas and the Ash-Ginzburg functional, produce the zero function. Inspired by the construction of improved p-adic L-functions, we resolve this problem by constructing an ”improved” local Shalika functional.

Friday, July 4, 2025

9:30- 10:30 

Speaker: Antonio Lei (University of Ottawa)

Title: Iwasawa invariants of Mazur-Tate elements at non-ordinary primes

Abstract: Let f be a modular form and p a prime at which f is crystalline and non-ordinary. When the Serre weight of f at p is 2, Pollack and Weston established, under certain hypotheses, explicit formulae for the Iwasawa invariants of the Mazur-Tate elements associated with f. In this talk, we discuss extensions of these results to modular forms of higher Serre weights. In particular, we show that when the weight or the slope of the Hecke-eigenvalue at p is sufficiently small, the Iwasawa invariants can be described explicitly. This is joint work in progress with Rylan Gajek-Leonard.

10:40- 11:40 

Speaker: Pierre Colmez (CNRS)

Title: Completed cohomology and group cohomology of arithmetic groups

Abstract: Using Shapiro’s lemma one can express Iwasawa cohomology groups as Galois cohomology of the base field with values in a big module with a lot of structures; these extra structures makes it possible to define a multitude of operators on these Iwasawa modules. In this vein, I will explain that Emerton’s completed Cohomology has a natural definition as the group cohomology of arithmetic groups with values in spaces of functions on adelic groups.