Sep 18  Michele Fornea (Columbia)

Points on elliptic curves via padic integration
The work of Bertolini, Darmon and their schools has shown that padic multiplicative integrals can be successfully employed to study the global arithmetic of elliptic curves. Notably, Guitart, Masdeu and Sengun have recently constructed and numerically computed StarkHeegner points in great generality. Their results strongly support the expectation that StarkHeegner points completely control the MordellWeil group of elliptic curves of rank 1.
In our talk, we will report on work in progress about a conjectural construction of global points on modular elliptic curves, generalizing the padic construction of Heegner points via CerednikDrinfeld uniformization. Inspired by Nekovar and Scholl's plectic conjectures, we expect the nontriviality of these plectic Heegner points to control the MorderllWeil group of higher rank elliptic curves. We provide some evidence for our conjectures by showing that higher derivatives of anticyclotomic padic Lfunctions compute plectic Heegner points.

Sep 25  Dennis Gaitsgory (Harvard)

The stack of local systems with restricted variation and the passage from
geometric to classical Langlands theory
The goal of this talk is two explain to closely related phenomena: the
existence of the categorical geometric Langlands theory for ladic sheaves
and the link between geometric to classical Langlands via the operation of
categorical trace. A key ingredient is played by a new geometric object:
the stack of local systems with restricted variation.
Video  Notes

Oct 2  Hector Pasten (PUC Chile)

A ChabautyColeman estimate for surfaces in abelian threefolds
Coleman's explicit version of Chabauty's theorem gives a remarkable upper bound for the number of rational points in hyperbolic curves over number fields, under a certain rank condition. This result is obtained by padic methods. Despite considerable efforts in this topic, higher dimensional extensions of such a bound have remained elusive. In this talk I will sketch the proof for hyperbolic surfaces contained in abelian threefolds, which provides the first case beyond the scope of curves. This is joint work with Jerson Caro.
Video  Slides

Oct 9  Jonathan Wang (MIT)

Local Lvalues and geometric harmonic analysis on spherical varieties
Almost a decade ago, Sakellaridis conjectured a vast generalization of the RankinSelberg method to produce integral representations of Lfunctions using affine spherical varieties. The conjecture is still very much unknown, but generalized IchinoIkeda formulas of SakellaridisVenkatesh relate the global problem to certain problems in local harmonic analysis. I will explain how we can use techniques from geometric Langlands to compute integrals which give special values of unramified local Lfunctions over a local function field, for a large class of spherical varieties. This is joint work with Yiannis Sakellaridis. Our results give new integral representations of Lfunctions (in a right half plane) over global function fields when the integral "unfolds".
Video  Slides

Oct 16  George Boxer (ENS de Lyon)

Higher Coleman Theory
We introduce a higher coherent cohomological analog of overconvergent modular forms on Shimura varieties and explain how to compute the finite slope part of the coherent cohomology of Shimura varieties in terms of them. We also discuss how they vary padically. This is joint work with Vincent Pilloni.
Video  Notes

Oct 23  Sarah Zerbes (UCL)

The BlochKato conjecture for GSp(4)
In my talk, I will sketch a proof of new cases of the BlochKato conjecture for the spin Galois representation attached to genus 2 Siegel modular forms. More precisely, I will show that if the Lfunction is nonvanishing at some critical value, then the corresponding Selmer group is zero, assuming a number of technical hypotheses. I will also mention work in progress on extending this result to Siegel modular forms of parallel weight 2, with potential applications to the BirchSwinnertonDyer conjecture for abelian surfaces. This is joint work with David Loeffler.
Video

Oct 30  David Loeffler (Warwick)

Padic interpolation of GrossPrasad periods and diagonal cycles
The GrossPrasad conjecture for orthogonal groups relates special values of Lfunctions for SO(n) x SO(n+1) to period integrals of automorphic forms. This conjecture is known for n = 3, in which case the group SO(3) x SO(4) is essentially GL2 x GL2 x GL2; and the study of these GL2 triple product periods, and in particular their variation in padic families, has had important arithmetic applications, such as the work of Darmon and Rotger on the equivariant BSD conjecture for elliptic curves.
I'll report on work in progress with Sarah Zerbes studying these periods in the n = 4 case, where the group concerned is isogenous to GSp4 x GL2 x GL2. I'll explain a construction of padic Lfunctions interpolating the GrossPrasad periods in Hida families, and an 'explicit reciprocity law' relating these padic Lfunctions to diagonal cycle classes in etale cohomology. These constructions are closely analogous to the Euler system for GSp(4) described in Sarah's talk, but with cusp forms in place of the GL2 Eisenstein series.
Video

Nov 6  ChiYun Hsu (UCLA)

Construction of Euler systems for GSp4×GL2
Following a strategy similar to the work of LoefflerSkinnerZerbes, we construct Euler systems for Galois representations coming from automorphic representations of GSp4×GL2. We will explain how the tame norm relations follow from a local calculation in smooth representation theory, in which the integral formula of Lfunctions, due to Novodvorsky in our case, plays an important role. This is a joint work with Zhaorong Jin and Ryotaro Sakamoto.
Video  Notes

Nov 13  Yuanqing Cai (Kyoto)

Certain representations with unique models
The uniqueness of Whittaker models is an important ingredient in the study of certain Langlands Lfunctions. However, this property fails for groups such as GL(n,D), where D is a central division algebra over a local field.
In this talk, we discuss a family of irreducible representations of GL(n,D) that admit unique models. We also discuss some related local and global questions.
Video  Slides

Nov 20  Takuya Yamauchi (Tohoku)

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials
In this talk, I will explain my recent work with Tsuzuki Nobuo on computing
mod $2$ Galois representations $\overline{\rho}_{\psi,2}:G_K:={\rm Gal}(\overline{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$
associated to the mirror motives of rank 4 with pure weight 3 coming from the
Dwork quintic family
$$X^5_0+X^5_1+X^5_2+X^5_3+X^5_45\psi X_0X_1X_2X_3X_4=0,\ \psi\in K$$
defined over a number field $K$ under the irreducibility condition of the quintic trinomial
$f_\psi(x)=4x^55\psi x^4+1$.
In the course of the computation, we observe that the image of such a mod $2$ representation is governed by reciprocity of
$f_\psi(x)$ whose decomposition field is generically of type
5th symmetric group $S_5$.
When K=F is totally real field, we apply the modularity of
2dimensional, totally odd Artin representations of ${\rm Gal}(\overline{F}/F)$ due to Shu Sasaki
to obtain automorphy of $\overline{\rho}_{\psi,2}$ after a suitable (at most) quadratic base extension.
Video

Dec 4  Eugen Hellmann (Münster)

Towards automorphy lifting for semistable Galois representations
Automorphy lifting theorems aim to show that a padic global Galois representation that is unramified almost everywhere and de Rham at places dividing p is associated to an automorphic representation, provided its reduction modulo p is. In the past years there has been a lot of progress in the case of polarizable representations that are crystalline at p. In the semistable case much less is known (beyond the ordinary case and the 2dimensional case).
I will explain recent progress on classicality theorems for padic automorphic forms whose associated Galois representation is semistable at places dividing p. In the context of automorphy lifting problems, these results can be used to deduce the semistable case from the crystalline case.

Dec 11  Zhilin Luo (Minnesota)


Dec 18 

