Sep 09  Daniel LiHuerta (Harvard)

The Plectic conjecture over local fields
The étale cohomology of varieties over $\mathbb{Q}$ enjoys a Galois action. In the case of Hilbert modular varieties, NekovářScholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higherrank Euler systems, they conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.
We present a proof of the analog of this conjecture for local Shimura varieties. Consequently, we obtain results for the basic locus of global Shimura varieties, after restricting to a decomposition group. The proof crucially uses a mixedcharacteristic version of fusion due to FarguesScholze.

Sep 16 (Online)  Jinbo Ren (Xiamen University)

Applications of the Subspace Theorem in Group Theory
An abstract group is said to have the bounded generation property (BG) if it can be written as a product of finitely many cyclic subgroups. Being a purely combinatorial notion, bounded generation has close relation with many group theoretical problems including semisimple rigidity, Kazhdan's property (T) and Serre's congruence subgroup problem.
In this talk, I will explain how to use the SchlickeweiSchmidt subspace theorem in Diophantine approximation to prove that the distribution of the points of a set of matrices over a number field $F$ with (BG) by certain fixed semisimple (diagonalizable) elements is of at most logarithmic size when ordered by height. Moreover, one obtains that a linear group $\Gamma \subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus.
This is joint work with Corvaja, Demeio, Rapinchuk and Zannier.

Sep 23  Oana Padurariu (Boston)

Quadratic Analogues of Kenku's Theorem
Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$, building on Mazur's 1978 work on prime degree isogenies. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has until now not been realized.
In this talk I will present a procedure to assist in establishing such a determination for a given quadratic field. Running this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $d<104$, we obtain, conditional on the GRH, the determination of cyclic isogenies of elliptic curves over 19 quadratic fields.
This is joint work with Barinder Banwait and Filip Najman.

Sep 30  Chandrashekhar Khare (UCLA)

The WilesLenstraDiamond numerical criterion in higher codimensions
I will report on recent joint work with Srikanth Iyengar and Jeff Manning. We give a development of numerical criterion that was used by Wiles as an essential ingredient in
his approach to modularity of elliptic curves over $\mathbb{Q}$. The patching method introduced by Wiles and Taylor has been developed considerably while the numerical criterion has lagged behind.
We prove new commutative algebra results that lead to a generalisation of the WilesLenstraDiamond numerical criterion in situations of positive defect (as arise when proving modularity of elliptic curves over number fields with a complex place). A key step in our work is the definition of congruence modules in higher codimensions which should be relevant to studying properties of eigenvarieties at classical points.

Oct 07  Naomi Sweeting (Harvard)

Kolyvagin's Conjecture and Higher Congruences of Modular Forms
Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he deduced remarkable consequences for the Selmer rank of E. For example, his results, combined with work of GrossZagier, implied that a curve with analytic rank one also has algebraic rank one; a partial converse follows from his conjecture.
In this talk, I will report on work proving several new cases of Kolyvagin's conjecture. The methods follow in the footsteps of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. By considering congruences modulo higher powers of p, we remove many of those hypotheses. The talk will provide an introduction to Kolyvagin's conjecture and its applications, explain an analog of the conjecture in an opposite parity regime, and give an overview of the proofs, including the difficulties associated with higher congruences of modular forms and how they can be overcome via deformation theory.

Oct 14  Andrew Obus (CUNYBaruch)

Mac Lane Valuations and applications to conductordiscriminant inequalities
Mac Lane's technique of "inductive valuations" is over 85 years old, but has only recently been used to attack problems in arithmetic geometry. We will give an explicit, handson introduction to inductive valuations. We will then discuss an application to explicit resolutions of singularities on arithmetic surfaces, ultimately giving a generalization of a conductordiscriminant inequality of Qing Liu in genus 2 to arbitrary genus.
This is joint work with Padmavathi Srinivasan

Oct 21  Jacques Tilouine (Sorbonne Paris Nord)

Iwasawa theory of classical and derived deformation rings
In a joint work with E. Urban, we define Iwasawatheoretic deformation
rings for the Galois representation associated to a pordinary cusp form on a connected reductive group
and study their relations to the Iwasawa theory of Selmer groups associated to its adjoint representation.

Oct 28  YuSheng Lee (Columbia)

Arithmetic of theta liftings
We discuss the integrality of theta liftings of anticyclotomic characters to a definite unitary group $\mathrm{U}(2)$ of two variables. This will allow us to construct a Hida family of the theta liftings and relate the congruence module of which to an anticyclotomic $p$adic Lfunction. The result is an input to Urban's construction of Euler systems.

Nov 04  Hélène Esnault (Universität Berlin & Columbia)

Integrality of the Betti moduli space
We show that the Betti moduli space of a smooth complex quasiprojective variety $X$ has a weak integrality property which in particular yields a new obstruction for a finitely presented group to be the topological fundamental group of $X$. We define weak arithmetic complex points of the Betti moduli space and prove density of those. Our method relies on the arithmetic (via companions) and the geometric (via de Jong's conjecture solved by Gaitsgory) Langlands correspondence. It also yields other properties of the Betti moduli space which we shall mention if time permits.
Joint work in progress with Johan de Jong

Nov 11  Eric Chen (Princeton)

Duality of singular automorphic periods
In the recent framework proposed by BenZviSakellaridisVenkatesh, automorphic periods ought to, very roughly speaking, come in Langlands dual pairs. I will give a short introduction of this prediction and motivate the need to consider certain singular automorphic periods. In particular, I will present an example in joint work with Akshay Venkatesh, where we establish duality using a generalization of Lfunctions.

Nov 18  Robert Pollack (Boston)

Predicting slopes of modular forms and reductions of crystalline representations
The ghost conjecture predicts slopes of modular forms whose residual representation is locally reducible. In this talk, we'll examine locally irreducible representations and discuss recent progress on formulating a conjecture in this case. It's a lot trickier and the story remains incomplete, but we will discuss how an irregular ghost conjecture is intimately related to reductions of crystalline representations.

Dec 02 (Online)  Valentin Hernandez (Université ParisSaclay)

The infinite fern in higher dimensions
In full generality deformation spaces of Galois representations are mysterious objects. A natural question to ask is if they contain at least enough modular points in their generic fiber. In this talk I will explain result about the Zariski density of such points for conjugate self dual deformations spaces. Such a result was obtained for GL_2 by GouveaMazur and then generalized by Chenevier in dimension 3. Both strategies uses the Infinite fern, a fractallike object which is the image of an Eigenvariety. More recently HellmannMargerinSchraen extended Chenevier's result under strong TaylorWiles hypothesis, with main input the local model of the trianguline variety and the patched Eigenvarieties of BreuilHellmannSchraen. Our strategy is to study further the geometry of the trianguline variety, and to use the geometry of classical points on the (nonpatched) Eigenvariety to remove the TaylorWiles hypothesis.
This is a joint work with B. Schraen.

Dec 09  Ari Shnidman (Jerusalem & Dartmouth)

Torsion points on abelian surfaces with potential quaternionic multiplication
I'll discuss work in progress with Laga, Schembri, and Voight, which aims to classify the finite subgroups that arise inside MordellWeil groups of abelian surfaces over $\mathbb{Q}$ with geometric endomorphism ring isomorphic to a maximal quaternion order ("potentially QM").
