Automorphic Forms and Arithmetic

Euler Systems (ESs) are collections of Galois cohomology classes that verify some co-restriction compatibilities. The key feature of ESs is that they provide a way to bound Selmer groups, thanks to the machinery developed by Rubin, inspired by earlier work of Thaine, Kolyvagin, and Kato. In this talk, I will present joint work with C. Skinner, in which we develop a new method for constructing Euler Systems and apply it to build an ES for the Galois representation attached to the symmetric square of an elliptic modular form. I will stress how this method gives a unifying approach to constructing ESs, in that it can be successfully applied to retrieve most classical ESs (the cyclotomic units ES, the elliptic units ES, Kato's ES, Lei-Loeffler-Zerbes ES for the Rankin-Selberg convolution of two elliptic modular forms...).

Any finite-dimensional p-adic representation of the absolute Galois group of a p-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a p-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. When the representation comes from a Q_p-representation of the fundamental group, we relate the infinitesimal action of inertia subgroups with Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan.

For a prime number N, Ogg's conjecture states that the torsion in the Jacobian of the modular curve X_0(N) is generated by the cusps. Mazur proved Ogg's conjecture as one of the main theorems in his "Eisenstein ideal" paper. I'll talk about a generalization of Ogg's conjecture for squarefree N and a proof using the Eisenstein ideal. This is joint work with Ken Ribet.

Geometric Iwasawa theory studies the behavior of p-adic towers of curves. Classically, the focus has been on the p-part of class groups, mirroring Iwasawa theory for number fields. However, there are many interesting features of Iwasawa theory for curves that have no number field analogy. The p-part of the class group is only a small part of the p-divisible group, a much more intricate object with no number field analogy. In this talk I will survey various results and conjectures about the behavior of p-divisible groups along towers of curves. I will also discuss what geometric Iwasawa theory for motives should look like and explain new results in this direction.

The $p$-adic Simpson correspondence aims to establish an equivalence between generalized representations and Higgs bundles over a $p$-adic variety. In this talk, we will explain how to upgrade such an equivalence to a twisted isomorphism of moduli stacks in the curve case. This is based on a joint work in progress with Heuer.

Scholze's category of diamonds gives a robust framework for p-adic geometry that bridges the gap between Tate's classical theory of rigid analytic varieties and the modern theory of perfectoid spaces. This flexibility is crucial, for example, if one wants to study Hodge-Tate period maps or other natural period maps that arise in the study of p-adic cohomology. However, this flexibility also comes at a price: many diamonds that appear naturally, including perfectoid spaces, are of a fundamentally topological rather than analytic nature. This topological nature eliminates some basic tools that we might expect to have available to us based on our experience in complex analytic geometry: for example, the existence of approximate p-power roots in a perfectoid algebra guarantees that it will admit no continuous derivations and thus no tangent space via the classical Kahler theory, and because of this one cannot naively differentiate Hodge-Tate period maps. In this talk, we will explain why many diamonds are nonetheless secretly equipped with the extra data of a Tangent Space. An important motivating example comes from the Fargues-Scholze Jacobian Criterion, but we will go well beyond this case. In particular, we construct Tangent Spaces for p-adic Lie torsors over rigid analytic varieties and differentiate the Hodge-Tate period map in the de Rham case. The key tools in the construction and computations are the theory of coherent sheaves on the Fargues-Fontaine curve and its relation to the theory of Banach-Colmez spaces due to le Bras, the geometric Sen theory of Pan and Camargo, and the p-adic Simpson/Riemann-Hilbert correspondence of Liu and Zhu. Motivated by the success of the Fargues-Scholze criterion, it is natural to ask: after these Tangent Spaces and derivatives are constructed, what can they tell us about the topological properties of diamonds and morphisms between them? We will address this by formulating two general conjectures in the spirit of a "differential topology for diamonds" and then conclude by exploring some examples.

I will talk about two results. The first is a new case of the BSD conjecture, contained in a joint work with Hamacher and Zhao. Namely, we prove the conjecture for elliptic curves of height 1 over a global function field of genus 1 under a mild assumption. This is obtained by specializing a more general theorem on the Tate conjecture. The key geometric idea is an application of rigidity properties of the variations of Hodge structures to study deformation of line bundles in positive and mixed characteristic. Then I will talk about a generalization of such deformation results recently obtained with Urbanik. Namely, we show that for a sufficiently big arithmetic family of smooth projective varieties, there is an open dense subscheme of the base over which all line bundles in positive characteristics can be obtained by specializing those in characteristic 0.

Faltings has initiated in 2005 a p-adic analogue of the (complex) Simpson's correspondence whose construction has been taken up by different authors, according to several approaches. After a presentation of the one Ahmed Abbes and I have developed, I will explain how we establish the functoriality of the p-adic Simpson correspondence by proper direct image, which leads to a generalization of the relative Hodge-Tate spectral sequence.

Joint work with Ahmed Abbes.

Motivated by the Langlands functoriality conjecture and the work of Godement-Jacquet on automorphic L-functions, Braverman and Kazhdan introduced a non-linear version of the Fourier-Deligne transform on reductive groups over finite fields and they conjecture that this new type of non-linear Fourier-transform satisfies several remarkable properties similar to the linear case. It was shown that their conjecture follows from a certain vanishing conjecture (a generalization of the well-known acyclicity of Artin-Schreier sheaf on affine line to reductive groups). I will give an introduction to the work of Braverman and Kazhdan on non-linear version Fourier transforms and explain a proof of their vanishing conjecture. Time permitting I will also discuss applications to stable Bernstein center conjecture.

I will discuss a conjectural categorical form of the local Langlands correspondence for p-adic groups and establish the unipotent part of such correspondence (for characteristic zero coefficient field).

This will be a survey of some joint work with Madhav Nori on the theory of Gamma Motives. Classical Alexander modules are examples, and we will explain the analogs of the classical monodromy theorem and period isomorphisms in this context. If time permits, we will discuss some motivation coming from Beilinson's conjectures on special values of L-functions.

We explain how one can define special cocycles for arithmetic groups via explicit maps of complexes parameterizing linear algebraic data, in a framework simultaneously generalizing work of Bergeron-Charollois-Garcia and Sharifi-Venkatesh. We explain some representation-theoretic aspects of these cocycles, and point towards some ongoing and future arithmetic applications.