Automorphic Forms and Arithmetic

The work of Bertolini, Darmon and their schools has shown that p-adic 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 Stark-Heegner points in great generality. Their results strongly support the expectation that Stark-Heegner points completely control the Mordell-Weil 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 p-adic construction of Heegner points via Cerednik-Drinfeld uniformization. Inspired by Nekovar and Scholl's plectic conjectures, we expect the non-triviality of these plectic Heegner points to control the Morderll-Weil group of higher rank elliptic curves. We provide some evidence for our conjectures by showing that higher derivatives of anticyclotomic p-adic L-functions compute plectic Heegner points.

The goal of this talk is two explain to closely related phenomena: the existence of the categorical geometric Langlands theory for l-adic 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.

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 p-adic 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.

Almost a decade ago, Sakellaridis conjectured a vast generalization of the Rankin-Selberg method to produce integral representations of L-functions using affine spherical varieties. The conjecture is still very much unknown, but generalized Ichino-Ikeda formulas of Sakellaridis-Venkatesh 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 L-functions 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 L-functions (in a right half plane) over global function fields when the integral "unfolds".

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 p-adically. This is joint work with Vincent Pilloni.

In my talk, I will sketch a proof of new cases of the Bloch-Kato conjecture for the spin Galois representation attached to genus 2 Siegel modular forms. More precisely, I will show that if the L-function is non-vanishing 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 Birch-Swinnerton-Dyer conjecture for abelian surfaces. This is joint work with David Loeffler.

The Gross--Prasad conjecture for orthogonal groups relates special values of L-functions 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 p-adic 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 p-adic L-functions interpolating the Gross--Prasad periods in Hida families, and an 'explicit reciprocity law' relating these p-adic L-functions 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.

Following a strategy similar to the work of Loeffler-Skinner-Zerbes, 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 L-functions, due to Novodvorsky in our case, plays an important role. This is a joint work with Zhaorong Jin and Ryotaro Sakamoto.

The uniqueness of Whittaker models is an important ingredient in the study of certain Langlands L-functions. 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.

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 $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5\psi X_0X_1X_2X_3X_4=0,\ \psi\in K$$ defined over a number field $K$ under the irreducibility condition of the quintic trinomial $f_\psi(x)=4x^5-5\psi x^4+1$. 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 5-th symmetric group $S_5$. When K=F is totally real field, we apply the modularity of 2-dimensional, 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.

Automorphy lifting theorems aim to show that a p-adic 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 semi-stable case much less is known (beyond the ordinary case and the 2-dimensional case).
I will explain recent progress on classicality theorems for p-adic automorphic forms whose associated Galois representation is semi-stable at places dividing p. In the context of automorphy lifting problems, these results can be used to deduce the semi-stable case from the crystalline case.

The local Gan-Gross-Prasad conjecture studies the restriction and branching problems for representations of classical and metaplectic groups. In this talk, I will talk about my proof for the tempered part of the local Gan-Gross-Prasad conjecture (multiplicity one in Vogan packets) for special orthogonal groups over any local fields of characteristic zero, which combines the work of Waldspurger (for the tempered part of the conjecture for special orthogonal groups over $p$-adic fields) and Beuzart-Plessis (for the tempered part of the conjecture for unitary groups over real field) in a non-trivial way. In the proof, an indispensable result which is also of independent interest is a formula expressing the regular nilpotent germs of quasi-split reductive Lie algebras over any local fields of characteristic zero via endoscopic invariants, which was previously proved by Shelstad over $p$-adic fields. We also relate the formula with the Kostant's sections.

Let $K$ be a complete discrete valuation field of residue characteristic $p>0$ and $\ell\neq p$ a prime number. To a finite dimensional $\mathbb{F}_{\ell}$-representation $M$ of the absolute Galois group $G_K$, the ramification theory of Abbes and Saito attaches a Swan conductor ${\rm sw}(M)$ and a characteristic cycle ${\rm CC}(M)$. Let $D$ be the rigid unit disc over $K$ and $\mathcal{F}$ a lisse sheaf of $\mathbb{F}_{\ell}$-modules on $D$. For $t\in \mathbb{Q}_{\geq 0}$, the normalized integral model $\mathcal{D}^{(t)}$ of the subdisc $D^{(t)}$ of radius $t$ is defined over some finite extension of $K$. The restriction $\mathcal{F}_{\lvert D^{(t)}}$ defines, at the generic point $\mathfrak{p}^{(t)}$ of the special fiber of $\mathcal{D}^{(t)}$, a Galois representation $M_t$ over a complete discrete valuation field, thus yielding a Swan conductor ${\rm sw}(M_t)$ and a characteristic cycle ${\rm CC}(M_t)$. The goal of the talk is to explain how we connect earlier works, of Lütkebohmert on a discriminant function attached to a cover of $D$, and of Kato on the ramification of valuation rings of height $2$, and prove that the function $t\mapsto {\rm sw}(M_t)$ is continuous and piecewise linear with finitely many slopes which are all integers, and that its right derivative is $t\mapsto -{\rm ord}_{\mathfrak{p}^{(t)}}({\rm CC}(M_t)) + \dim_{\mathbb{F}_{\ell}}(M_t/M_t^{(0)})$, where ${\rm ord}_{\mathfrak{p}^{(t)}}$ is a normalized discrete valuation at $\mathfrak{p}^{(t)}$ extended to differentials and $M_t^{(0)}$ is the tame part of $M_t$.