Abstracts

Fabrizio Andreatta (Università di Milano)

Title: Katz type p-adic L-functions for primes p non-split in the CM field

Abstract: I will discuss a way to construct anticyclotmic p-adic L-functions attached to an elliptic eigenform f and an imaginary quadratic field K, interpolating p-adically the central critical values of the Rankin L-functions of f twisted by anticyclotomic characters of higher infinity type. I will also provide p-adic Gross-Zagier formulae for these p-adic L-functions. Here the prime p is assumed to be inert or ramified in K. The case that p is split is due to Katz (for Eisenstein series) and Bertolini-Darmon-Prasana (for cuspidal eigenforms). This is joint work with Adrian Iovita.

Kazim Büyükboduk (University College Dublin)

Title: On the non-critical exceptional zeros of Katz's p-adic L-functions

Abstract: We address a question of Hida and Tilouine that concerns the exceptional zeros of Katz's p-adic L-function for CM fields at the trivial character (which lies off the interpolation range), via the leading term formulae we prove along all Z_p-towers of the base CM field. Our formulae involves the L-invariants which we associate to each one of these Z_p-extensions (which are not necessarily Lubin-Tate). This is joint work with R. Sakamoto.

Francesc Castella (UC Santa Barbara & Princeton University)

Title: Elliptic curves of rank 2 and generalized Kato classes

Abstract: Let E be an elliptic curve over Q, and let p be a prime of good ordinary reduction for E. In this talk, I will explain the proof of a special case of a conjecture of Darmon-Rotger relating the non-vanishing of generalized Kato classes attached to (E,p) to the rank 2 property of E(Q). If time permits, I will also describe progress toward the extension of our result to supersingular primes p. All based on joint work with Ming-Lun Hsieh.

Andrea Conti (Universität Heidelberg)

Title: Families of potentially trianguline Galois representations

Abstract: We characterize the p-adic families of local Galois representations that are potentially trianguline in the sense of the theory of (φ,Γ)-modules. We present some work in progress towards deducing results on the (non-) existence of p-adic families of modular forms of infinite p-slope.

Christophe Cornut (CNRS, Sorbonne Université)

Title: Horizontal distribution relations for the Euler System attached to U(n-1,n) inside SO(2n-1,2)

Abstract: I will sketch the construction of the Euler System, and then give a detailed account of the proof of the distribution relations.

Fred Diamond (King's College London)

Title: The Serre filtration on mod p Hilbert modular forms of level p

Abstract: A result of Serre relates the space of mod p modular forms of level Γ_1(Np) and weight 2 to the spaces of mod p modular forms of level Γ_1(N) and weight between 2 and p+1. I will explain a generalization of this to the context of Hilbert modular forms which is motivated by the interplay between "algebraic" and "geometric" Serre weight conjectures for mod p Galois representations. The resulting filtration on mod p Hilbert modular forms of parallel weight 2 and pro-p Iwahori level mirrors, via a mod p geometric Jacquet-Langlands correspondence, the more evident filtration on cohomology coming from the mod p representation theory of GL_2. This is joint work with P. Kassaei and S. Sasaki.

Daniel Disegni (Ben-Gurion University of the Negev)

Title: p-adic equidistribution of CM points and applications

Abstract: Consider a sequence of CM points of increasing p-adic conductor on a modular curve X. What is its limiting distribution in any of the geometric incarnations of X? Works from the 2000s provide the answer for the Riemann surface X(C), and for the reduction of X modulo primes different from p. I will describe the answer in the p-adic (Berkovich) analytic setting. A weak generalisation of this result has an application to the arithmetic of elliptic curves.

Wojciech Gajda (Adam Mickiewicz University in Poznań)

Title: Semisimplicity and adelic openness

Abstract: We will discuss key properties of compatible systems of Galois representations over fields of positive characteristic, including a result towards the conjecture of Tate on semisimplicity of such representations (joint work with Gebhard Böckle and Sebastian Petersen).

David Helm (Imperial College London)

Title: Moduli of Langlands parameters

Abstract: Let G be a quasi-split reductive group over a p-adic field F, and assume that G splits over a tamely ramified extension of F. We construct moduli spaces of Langlands parameters for G with fixed "wild inertial type" and establish their basic geometric properties. This is joint work with Jean-Francois Dat, Robert Kurinczuk, and Gil Moss.

Ming-Lun Hsieh (Academia Sinica)

Title: Anticyclotomic p-adic spinor L-functions

Abstract: I will talk about a construction of anticyclotomic p-adic spin L-functions for degree two Siegel modular forms via the p-adic interpolation of the Bessel periods of Siegel modular forms. This is a joint work with Shunsuke Yamana.

Yukako Kezuka (Universität Regensburg)

Title: Vanishing of certain μ-invariants and its application to elliptic curves

Abstract: Take q to be any prime number congruent to 7 modulo 8, and let K=Q(√-q). The prime 2 splits in K, and we take p to be one of the primes of K above 2. Let H be the Hilbert class field of K and write K_∞ for the unique Z_2-extension of K unramified outside p. I will first describe a joint work with J. Choi and Y. Li, in which we show an analogue of Iwasawa's μ=0 conjecture. Then I will discuss how this result can be applied to study the 2-part of the conjecture of Birch and Swinnerton-Dyer for a large class of elliptic curves with complex multiplication by the ring of integers of K.

Guido Kings (Universität Regensburg)

Title: Algebraicity of critical values of Hecke L-functions for arbitrary number fields

Abstract: The equivariant polylogarithm allows to construct in a very general setting cohomology classes of arithmetic groups with values in motivic cohomology. Using the regulator to algebraic de Rham cohomology gives interesting algebraic Eisenstein classes. We use this theory to generalize the results of Damerell, Shimura and Katz on the algebraicity of special values of L-functions for Hecke characters for CM fields K to the case of finite extensions L/K over CM fields K.

Jaclyn Lang (Université Paris 13)

Title: Image of two-dimensional pseudorepresentations

Abstract: There is a general philosophy that the image of a Galois representation should be as large as possible, subject to its symmetries. This can be seen in Serre's open image theorem for non-CM elliptic curves, Ribet and Momose's work on Galois representations attached to modular forms, and recent work of the speaker and Conti-Iovita-Tilouine on Galois representations attached to p-adic families of modular forms. Recently, Bellaïche developed a way to measure the image of an arbitrary pseudorepresentations taking values in a local ring A. Under the assumptions that A is a domain and the residual representation is not too degenerate, we explain how the symmetries of such a pseudorepresentation are reflected in its image. This is joint work with Andrea Conti and Anna Medvedovsky.

Antonio Lei (Université Laval)

Title: Codimension two cycles and tensor products of Hida families

Abstract: In a recent work of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor, certain codimension two cycles are defined to study pseudo-null Iwasawa modules. In particular, they relate a pair of Katz p-adic L-functions to a pseudo-null Selmer group. We show that their result can be extended to tensor products of Hida families. Given two Hida families, there are two p-adic L-functions attached to their rank-Selberg convolution, depending on which family is "dominant". We show that if these two p-adic L-functions are coprime, then we may relate them to a pseudo-null Selmer group. If time permits, we will also talk about generalizations of this result for triple products. This is joint work with Bharath Palvannan.

Daniel Macias Castillo (Universidad Autónoma de Madrid)

Title: On refined conjectures of Birch and Swinnerton-Dyer type

Abstract: Let A be an abelian variety defined over a number field k. The conjecture of Birch and Swinnerton-Dyer ("BSD") predicts an explicit formula for the leading term at z=1 of the Hasse-Weil L-series L(A,z) of A over k. Deligne and Gross have also predicted the order of vanishing at z=1 of the Hasse-Weil-Artin L-series L(A,ψ,z) associated to finite dimensional characters ψ of the absolute Galois group of k. In certain settings a conjecture of Mazur and Tate predicts integral congruence relations between the values at z=1 of the functions L(A,ψ,z) as ψ ranges over certain families of characters. It has however proved more difficult to formulate explicit refinements of BSD that take into account any connections that might exist between the leading terms of L(A,ψ,z) for varying characters ψ of arbitrary order of vanishing. Let F be a finite Galois extension of k. In this talk we present a completely general "refined BSD conjecture" for (A,F/k) which provides an appropriate framework for the investigation of such connections as ψ ranges over the irreducible characters of Gal(F/k). Our refined BSD conjecture is consistent with the relevant case of the equivariant Tamagawa number conjecture and in particular renders the latter conjecture amenable to theoretical or numerical verifications in general situations. This is joint work with D. Burns.

Stefano Morra (Université Paris 13)

Title: Local models for potentially crystalline deformation rings and the Breuil-Mézard conjecture

Abstract: The Breuil-Mézard conjecture is a broadening of the Serre weight conjecture for 2-dimensional mod p-representations of the absolute Galois group of Q, and describes certain invariants of the special fiber of local Galois deformation rings with p-adic Hodge theory conditions in terms of modular representation theory of GL_n(F_p). It was the starting point of the mod p-local Langlands program, and beyond GL_2(Q_p) this conjecture was known in the few cases when the local Galois deformation rings were directly related to moduli of finite flat group schemes. In general, the theory of Breuil and Kisin shows that potentially crystalline deformation rings can be obtained from localizations of an affine flag variety over Z_p by imposing a transcendental condition. In ongoing work joint with Daniel Le, Bao Viet Le Hung and Brandon Levin we algebraize this condition to construct a local model whose localizations are equisingular to a large class of potentially crystalline deformation rings. We furthermore show that in the special fiber of this local model there exist cycles which have a natural interpretation in terms of modular representation theory, leading us to the proof of many cases of the Breuil-Mézard conjecture, and of the Serre weight conjecture when the Galois representation is semisimple.

Tadashi Ochiai (Osaka University)

Title: Endoscopic congruences and adjoint L-values for GSp(4)

Abstract: In his paper published in 1981, Hida related the existence of a mod p congruence between two elliptic normalized eigen cuspforms f and g of the same weight and level and the divisibility by p of a special value of the adjoint L-function of the form f. After reviewing this classical result, we explain the existence of congruences between a fixed endoscopic cuspidal automorphic representation of GSp(4) of square-free conductor and non-endoscopic cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at 1 of the adjoint L-function normalized by a suitable period. This is a joint work with Francesco Lemma.

Takamichi Sano (Osaka City University)

Title: On a generalization of Perrin-Riou's conjecture on Kato's zeta elements

Abstract: In this talk, I will formulate a new conjecture on Kato's zeta elements for elliptic curves, which is a generalization of Perrin-Riou's conjecture. I will show, by generalizing Rubin's formula concerning p-adic heights, that our conjecture implies both the Mazur-Tate conjecture and the Mazur-Tate-Teitelbaum (p-adic Birch and Swinnerton-Dyer) conjecture. Lastly, I will apply our conjecture to the descent argument in Iwasawa theory to give a general strategy for proving the conjectural Birch-Swinnerton-Dyer formula. This is joint work with D. Burns and M. Kurihara.

Shu Sasaki (Universität Duisburg-Essen & Queen Mary University of London)

Title: Serre's conjecture about weight of mod p modular forms: old conjectures, not so old theorems and new conjectures

Abstract: In 1987, J.-P. Serre made some remarkable conjectures about weights and levels of two-dimensional (modular) mod p Galois representations of the absolute Galois group of Q. They have been completely proved by C. Khare and J.-P. Wintenberger (2009) building on the work of many mathematicians (R. Taylor and M. Kisin to name a few), but they have also inspired a good deal of new mathematics. One strand of research spurred on by the development is about generalising Serre's conjectures over to totally real number fields. And this was initiated by the work (2009) of K. Buzzard, F. Diamond and F. Jarvis, while focusing exclusively on regular weights of mod p Hilbert modular forms. In my joint work with F. Diamond, we have improved on the Buzzard-Diamond-Jarvis conjectures and formulated new conjectures about general weights of (geometric) mod p Hilbert modular forms (analogous to what B. Edixhoven did in 1992). I will explain what our conjectures say and demonstrate some evidence that we are on the right track. My talk will be a warm-up for Diamond's talk later in the conference.

Marco Adamo Seveso (Università di Milano)

Title: p-adic interpolation of algebraic cycles and reciprocity laws

Abstract: The aim of this talk is to explain a method for interpolating the Abel-Jacobi image of suitable algebraic cycles, illustrate it in a couple of examples, then explain the resulting reciprocity laws and some arithmetic applications. This is joint work with M. Bertolini and R. Venerucci.

Eric Urban (Columbia University)

Title: An application of the p-adic Langlands correspondence to Euler system constructions

Abstract: The study of Eisenstein congruences leads to the construction of norm compatible Galois cohomology classes. The proof of the integrality of these classes relies on understanding a certain module over some universal locally Galois reducible quotient of the Hecke algebra. The goal of this talk is to explain how the p-adic Langlands correspondence for GL_2(Q_p) can be a tool to reach this understanding.

Rodolfo Venerucci (Università di Milano)

Title: Endoscopy for GSp(4) and rational points on elliptic curves

Abstract: I will report on joint work with M. Bertolini and M. A. Seveso, aimed at studying the BSD conjecture for rational elliptic curves twisted by certain 4-dimensional self-dual Artin representations in situations of analytic rank one. The endoscopy for GSp(4) plays a central role in our approach.