Résumés

Vous trouvez ici le programme et la liste des résumés au format .pdf.

 

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

I will discuss a way to construct anticyclotomic 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. 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-Prasanna (for cuspidal eigenforms). This is joint work with Adrian Iovita.

 

Anne-Marie Aubert: Profondeurs et correspondance de Langlands

Il est possible d'attacher des notions de "profondeurs" d'une part aux représentations irréductibles d'un groupe réductif p-adique G et d'autre part aux paramètres de Langlands de G. Nous décrirons comment la profondeur se comporte via la correspondance de Langlands, notamment dans le cas où G est une forme intérieure d'un groupe général linéaire ou spécial linéaire, ou le groupe des points rationnels d'un tore algébrique induit.
 
 

John Coates: A non-vanishing theorem for central L-values

I will report on recent work with Yongxiong Li proving the following non-vanishing result. Let q be any prime ≡ 7 mod 16, K = Q(√−q), and H the Hilbert class field of K. Let A/H be Gross's Q-curve with complex multiplication by the maximal order of K, and whose associated Hecke character has conductor (√−q). Let k be any non-negative integer, and let R = r1,...,rk, where the ri are distinct rational primes such that ri ≡ 1 mod 4 and ri is inert in K for 1 ≤ i ≤ k. Let A(R)/H be the quadratic twist of A by H(√R)/H. Writing L(A(R)/H,s) for the complex L-series of A(R)/H), we prove by arguments from Iwasawa theory that L(A(R)/H,1) 6= 0 always. When k = 0 this is an old result of D. Rohrlich, which he proved using complex methiods, but there seems little hope of proving the more general statement by such complex methods. It can be shown that this non-vanishing result proves the finiteness of the Tate-Shafarevich group of A(R)/H, and in subsequent joint work with Y. Kezuka and Y. Tian we hope to prove the exact Birch-Swinnerton- Dyer formula for the order of this group.

 

Mladen Dimitrov: Geometry of the eigencurve at CM points and trivial zeroes of Katz-Hida-Tilouine's p-adic L-functions

While tackling the anticyclotomic Main Conjecture, Hida and Tilouine showed that the p-adic L-function divides a certain congruence power series, while strongly suggesting that the two should in fact be equal. Beyond the proof, introducing congruences brings a whole new geometric perspective on the original conjecture (for example, the zeros of the p-adic L-function can be visualised as singular points on an eigenvariety).
In a joint work with Adel Betina, we focus on the intriguing case of a trivial zero which, according to Katz' p-adic Kronecker limit formula, occurs when the anti-cyclotomic character \psi (over an imaginary quadratic field) is locally trivial at p. We precisely determine the local structure of the eigencurve at the corresponding weight one theta series, and we prove in passing an R=T theorem for non-Gorenstein algebras. As an application, we show that the prediction of Hida and Tilouine is correct in our case and deduce an anti-cyclotomic version of the famous result of Ferrero and Greenberg: Katz' anti-cyclotomic p-adic L-function of either \psi, or of its inverse, has a simple trivial zero.

 

Matthew Emerton: Crystalline lifts of mod p representations of p-adic Galois groups

I will discuss the following theorem (joint with Toby Gee):
If rhobar: G_K --> GL_d(F_p-bar) is a continuous representation (with G_K being the absolute Galois group of a finite extension K of Q_p) then rhobar admits a lift rho: G_K --> GL_d(Z_p-bar) which is a lattice in a crystalline representation.
In fact, we can furthermore ensure that \rho is potentially diagonalizable, and gain essentially optimal control over the possible HT weights of rho.
As a consequence, we can also show (under the additional assumption that p is coprime to 2d) that rhobar admits an automorphic globalization.
Previously, such results were only known for small values of d, and in general, it didn't seem to be known whether rhobar admitted any p-adic lift at all. Our argument combines previous approaches to this problem (lifting extension classes from characteristic p to characteristic 0) - interpreted through a combination of geometry and homological algebra - with a computation of the structure of the coherent sheaf H^2(G_K,rho^u), where rho^u denotes the universal deformation of rho over a certain crystalline deformation ring. This latter computation depends on the construction of moduli stacks of (phi,Gamma)-modules, which I will also discuss in the talk (if time permits).
 
 
 
Wushi Goldring: Propagating algebraicity via functoriality

Our general motivation is to understand when an automorphic representation which does not admit some particular geometric realization admits a transfer via Langlands functoriality to one that does. In particular, we aim is to classify the automorphic representations for which the algebraicity of the Satake parameters is reducible via functoriality to the well-known, Hecke-stable rational structure on the coherent cohomology of Shimura varieties. In the
positive direction, the above strategy is used to illustrate the first examples of algebraicity and associated Galois representations for automorphic representations with a singular, tempered, non-limit of discrete series archimedean component (and which do not arise from a known case on a smaller group e.g. do not descend via base change to a
known case). In the negative direction, we prove two general results on the serious limitations of the strategy above; the second gives a conceptual explanation for why the algebraicity of Maass forms of eigenvalue 1/4 on GL(2) is not
reducible via functoriality to the coherent cohomology of Shimura varieties. We will also mention the remaining open cases not covered by either the positive or negative results.
 
 
 
Michael Harris: Square root p-adic L-functions
 
About 20 years ago, Jacques Tilouine and I were completing our paper on the construction of a p-adic analytic function whose square interpolates the normalized central critical values of the L-function of a triple of holomorphic modular forms of weights k, l, and m, in the unbalanced case (k \geq l + m), where l and m are fixed and the forms of weight k vary in an ordinary Hida family. The first such construction goes back to Andrea Mori's 1989 thesis (although it wasn't published for many years), and when we wrote our paper the phenomenon was still rather rare.  Since then 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,BC(\pi)xBC(\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 BC denotes the twisted base change to GL(n) x GL(n-1). 
The talk will be a report on the first steps toward generalizing the construction of my paper with Jacques to this situation. It will be assumed that \pi is a holomorphic representation and \pi' varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to \pi uses recent work of Eischen, Fintzen, Mantovan, and Varma.

 

Adrian Iovita: p-Adic Gross-Zagier formulae

Together with Fabrizio Andreatta we have defined p-adic L-functions (à la Katz and Bertolini-Darmon-Prasanna) associated to a classical, elliptic eigenform and a quadratic imaginary field K, satisfying certain conditions of which the most important is that the prime p is non-split in K.
In this talk I will discuss the special values of these p-adic L-functions at algebraic Hecke characters of K which are not in the interpolation set and which involve Abel-Jacobi images of certain generalized Heegner cycles, defined by Bertolini-Darmon-Prasanna.

 

Payman Kassaei: Minimal Weights of Mod p Hilbert Modular Forms

We prove that any mod p Hilbert modular form arises by multiplication by partial Hasse invariants from one whose weight vector lies in a certain 'minimal' cone contained in the set of non-negative weights. In an earlier work, we proved this result in the case the prime p is unramified in the relevant totally real field using properties of the Goren-Oort stratification on tame level Hilbert modular varieties established by Goren and Oort, and Tian and Xiao. In the general case, we use instead results on the global geometry of a stratification of the Iwahori-level Hilbert modular variety studied by Goren-Kassaei for unramified p, and generalized for all p by Emerton-Reduzzi-Xiao. This is joint work with Fred Diamond.

 

Chandrashekhar Khare: Relative deformation theory and lifting Galois representations

I will talk about lifting "odd"  Galois representations with values in finite groups of Lie type to geometric characteristic 0 representations. The basic method is that of Ravi Ramakrishna, but to make the method maximally flexible requires some new techniques which includes defining (and annihilating) relative Selmer groups, liftings pairs of representations (relative deformation theory), and using generic rather than formal smoothness of local deformation rings. I will focus on these novelties in the joint work with N. Fakhruddin and S. Patrikis.

 

Jaclyn Lang: Images of 2-dimensional pseudorepresentations

There is a general philosophy that the image of a Galois representation should be as large as possible, subject to the symmetries of the geometric object from which it arose.  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 Hida and Coleman families of modular forms.  Recently, Bellaiche developed a way to measure the image of an arbitrary Galois representation taking values in GL(2) of 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 representation are reflected in its image.  This is joint work with Andrea Conti and Anna Medvedovsky.

 

Zheng Liu: p-adic L-functions and doubling archimedean zeta integrals for symplectic groups
 
In order to prove the desired interpolation properties of the p-adic standard L-functions for Siegel modular forms, one needs to calculate a doubling archimedean zeta integral for holomorphic discrete series on Sp(2n,R). When the holomorphic discrete series is of scalar weight, it has been computed by Bocherer--Schmidt and Shimura. I will explain a way to compute this archimedean zeta integral for general vector weight by using the theory of theta correspondence, and verify that the results are compatible with the conjecture of Coates--Perrin-Riou.
 
 
 
David Loeffler: p-adic L-functions and Euler systems for GSp4
 
I will explain how the higher Hida theory recently introduced by Pilloni can be used to construct p-adic L-functions interpolating the critical values of the degree 4 (spin) L-functions of automorphic forms on GSp4, and the degree 8 L-functions of cusp forms on GSp4 x GL2. This is joint work with Vincent Pilloni, Chris Skinner and Sarah Zerbes. I will conclude by describing work in progress to relate the GSp4 p-adic L-function to the images of Euler system classes under the p-adic syntomic regulator map.

 

Jan Nekovar: The plectic polylogarithm

We are going to describe the Hodge realisation of the plectic polylogarithm and its relation to special values of L-functions. This is a joint work with A.J. Scholl.

 

Vincent Pilloni: Théorie de Hida supérieure en poids réguliers pour GL_2

Nous allons parler de la partie ordinaire de la cohomologie cohérente des fibrés vectoriels automorphes de poids régulier pour les courbes modulaires et les variétés de Hilbert. Travail en commun avec G. Boxer.



Dipendra Prasad: Cohomological representations of real reductive groups

We discuss cohomological representations of real reductive groups - especially for classical groups - from the point of view of parameters. Joint work with A. Nair.

 

Giovanni Rosso: Families of Drinfeld modular forms

Seminal work of Hida tells us that for eigenforms that are ordinary at p we can always find other eigenforms, of different weights, that are congruent to our given form. Even better, it also says that we can find q-expansions whose coefficients are analytic functions of the weight variable k, that when evaluated at positive integers give the q-expansion of classical ordinary eigenforms.
This talk will explain how similar results can be obtained for Drinfeld modular forms. We shall explain how to construct families for Drinfeld modular forms, both ordinary and of positive slope, and how to decide if an overconvergent form of small slope is classical. Joint work with Marc-Hubert Nicole.
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