Anne-Marie Aubert (IMJ-PRG)

The Langlands correspondence for p-adic classical groups via isomorphisms of Hecke algebras 

Abstract: I will report on a joint project with Ahmed Moussaoui and Maarten Solleveld. We introduced notions of cuspidality and cuspidal support for enhanced L-parameters of reductive p-adic groups. We obtained a partition of the set of enhanced parameters that is modeled upon the Bernstein decomposition of the smooth dual of the p-adic group.

On the other hand, in the case of classical groups, works of Arthur and Moeglin allow to attach, via twisted endoscopy, a cuspidal enhanced L-parameter to each irreducible supercuspidal representation. Focusing on examples, we will show that there exists an isomorphism between the Hecke algebras associated to the corresponding Bernstein blocks which coincides with the Langlands correspondence constructed by Arthur. 

Michael Harris (Columbia University)

Square root p-adic L-functions 

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, gives explicit formulas for central critical values of a large class of Rankin-Selberg tensor products.  The version for unitary groups is now a theorem, and expresses the central critical value of $L$-functions of the form 

$L(s,\Pi \times \Pi')$ in terms of squares of automorphic periods on unitary groups.  Here $\Pi \times \Pi'$ is an automorphic representation of 

$GL(n,F)\times GL(n-1,F)$ that descends to an automorphic representation of $U(V) \times U(V')$, where $V$ and $V'$ are hermitian spaces over $F$, with respect to a Galois involution $c$ of $F$, of dimension $n$ and $n-1$, respectively.  

I will report on the construction of a $p$-adic interpolation of the automorphic period — in other words, of the square root of the central values of the $L$-functions — when $\Pi'$ varies in a Hida family.  The construction is based on the theory of $p$-adic differential operators due to Eischen, Fintzen, Mantovan, and Varma.

Lea Beneish (University of Berkley, California) 

The Gross--Kohnen--Zagier formula via $p$-adic uniformisation 

Abstract: The Gross-Kohnen-Zagier theorem says that certain generating series of CM points are modular forms of weight 3/2 in the Jacobian of the modular curve $X_0(N)$. In this talk, I will discuss a new proof of the Gross--Kohnen--Zagier formula for Shimura curves which uses the $p$-adic uniformisation of Cerednik--Drinfeld. The explicit description of CM points via this uniformisation leads to an expression for the Gross--Kohnen--Zagier generating series as the ordinary projection of the first derivative of a $p$-adic family of positive definite ternary theta series. This is joint work with Henri Darmon, Lennart Gehrmann and Marti Roset Julia.

Kazim Büyükboduk (University College Dublin)

Artin formalism for non-genuine p-adic Garrett-Rankin L-functions 

I will report joint work with D. Casazza and R. Sakamoto, where we formulate a conjecture (and prove it in many cases) on the factorization of a certain triple product p-adic L-function whose range of interpolation is empty. The relevant factorization statement reflects not only the Artin formalism for the underlying family of motives (which decompose as the sum of 2 motives of respective degrees 2 and 6) but also dwell on the interplay between various Gross--Zagier formulae for the relevant complex L-series, and the subtle relationship between the derivatives of complex L-series at their central critical point and p-adic L-functions. 

Aprameyo Pal  (Harish-Chandra Reasearch Institute) 

TBA

Anna Szumowicz (Caltech)

Uniform bounds on the Harish-Chandra characters

Abstract: Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}$ tends to $0$ when $\pi$ runs over equivalence classes of irreducible supercuspidal representations of $G(F)$ whose central character is unitary and the formal degree of $\pi$ tends to infinity. In fact something stronger holds under some additional conditions. I give the sketch of the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open compact modulo center subgroup.

Kim Tuan Do (UCLA)

Construction of an anticyclotomic Euler System 

We construct a new anticyclotomic Euler system for the Galois representation attached to a modular form of weight 2k, twisted by an anticyclotomic Hecke character \chi of infinity type (m,-m), denoted V_{f,\chi}, when the Heegner hypothesis is not satisfied and m\ge k. If time permits, we will then show some arithmetic applications (in a joint work with F. Castella) of the constructed Euler system, including evidence of the Bloch-Kato Conjecture and the Iwasawa Main Conjecture for V{f,\chi}. 

Yuxin Lin (Caltech)

Abelian covers of P^1 of p-ordinary Ekedahl-Oort type.

Given a family of abelian covers of P1 and a prime p of good reduction, by considering the associated Deligne–Mostow Shimura variety, we obtain new lower bounds for the Ekedahl-Oort type, and the Newton polygon, at p of the curves in the family. In this paper, we investigate whether such lower bounds are sharp. In particular, we prove sharpeness when the number of branching points is at most five and p sufficiently large. Our result is a generalization of the result by Bouw, which proves the analogous statement for the p-rank, and it relies on the notion of Hasse-Witt triple introduced by Moonen.

Philippe Michel (EPFL)

TBA

Wanlin Li (CRM Montreal)

Somnath Jha (IIT Kanpur) 

Liyang Yang