Automorphic Forms and Arithmetic

The étale cohomology of varieties over $\mathbb{Q}$ enjoys a Galois action. In the case of Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higher-rank 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 mixed-characteristic version of fusion due to Fargues-Scholze.

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 Schlickewei-Schmidt 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 semi-simple (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.

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.

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 Wiles-Lenstra-Diamond 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.

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 Gross-Zagier, 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.

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, hands-on introduction to inductive valuations. We will then discuss an application to explicit resolutions of singularities on arithmetic surfaces, ultimately giving a generalization of a conductor-discriminant inequality of Qing Liu in genus 2 to arbitrary genus.
This is joint work with Padmavathi Srinivasan

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

We discuss the integrality of theta liftings of anti-cyclotomic 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 anti-cyclotomic $p$-adic L-function. The result is an input to Urban's construction of Euler systems.

We show that the Betti moduli space of a smooth complex quasi-projective 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

In the recent framework proposed by Ben-Zvi--Sakellaridis--Venkatesh, 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 L-functions.

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.

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 Gouvea-Mazur and then generalized by Chenevier in dimension 3. Both strategies uses the Infinite fern, a fractal-like object which is the image of an Eigenvariety. More recently Hellmann-Margerin-Schraen extended Chenevier's result under strong Taylor-Wiles hypothesis, with main input the local model of the trianguline variety and the patched Eigenvarieties of Breuil-Hellmann-Schraen. Our strategy is to study further the geometry of the trianguline variety, and to use the geometry of classical points on the (non-patched) Eigenvariety to remove the Taylor-Wiles hypothesis.
This is a joint work with B. Schraen.

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