We study arithmetic analogues of theta series. Given a symplectic vector space \(V\) and a Schwartz function \(f\) on \(V\), there is a collection of cycles \(Z(n,f)\), consisting of CM points, on the Siegel modular variety. Assuming that \(f\) satisfies a strong regular semisimple condition at some prime \(p\), we show that the generating series of the degrees of the cycles \(Z(n,f)\) is a modular form. We identify it explicitly with a classical theta series for a quaternion unitary similitude group. The proof relies on the \(p\)-adic uniformization of the supsersingular locus on the Siegel modular variety.

We compute some examples of gluing sheaves on the moduli stack of rank \(2\) vector bundles on the Fargues-Fontaine curve. In the case of prime-to-\(p\) torsion coefficients, the category \(D_{\mathrm{\acute{e}t}}(\mathrm{Bun}_G)\) can be thought of as an approximation of the automorphic data appearing in the geometrization of the local Langlands correspondence due to Fargues-Scholze. The stratification of \(\mathrm{Bun}_G)\) arising from the Harder-Narasimhan slope formalism on G-isocrystals yields a semi-orthogonal decomposition of \(D_{\mathrm{\acute{e}t}}(\mathrm{Bun}_G)\) into the derived categories of smooth representations of inner forms of Levi subgroups of \(G\). Between such categories there is a full six functor formalism that can be used to compute how sheaves arising on a quasi-compact open substack interact with sheaves on higher strata via nearby cycles functors, which can be interpreted as some derived analogue of Jacquet restriction functors for parabolic subgroups of \(G\) up to inner twisting. We restrict to \(G=\mathrm{GL}_2\) and to sufficiently nice coefficients (notably this includes an algebraic closure of \(\mathbb F_\ell\) and \(\mathbb Z/\ell^n \mathbb Z\) for almost all \(\ell\) prime to \(p\)), and we will explain how these computations fundamentally reduce to the étale cohomology of local Shimura varieties (more generally local shtuka spaces)

Given a regular scheme \(X\) and a normal crossing divisor \(D\) one may consider two different cohomology groups. The first one is the log étale cohomology developed by Illusie, K. Kato and many others: We associate a logarithmic structure \(M\) to \(X\) and define the log étale site over \((X,M)\). The second one is the tame cohomology developed by Hübner and Schmidt. Here we consider the tame site over the discretely ringed adic space \(\mathrm{Spa}(X\backslash D,X)\). Tame morphisms are those which are étale and induce at most tamely ramified extension on the valuations. We construct a comparison morphism between these cohomology groups and prove that they are equal for schemes over \(\mathbb{F}_p\) and locally constant finite sheaves once we assume resolution of singularities.

The talk will be about the relationship between two different categories associated with the category of lattices in crystalline representations with small Hodge-Tate weights. In particular, I will explain how to attach a prismatic Frobenius crystal to a (crystalline) strongly divisible module. Time permitting, I will also sketch how this can be extended to higher dimensions.

Emerton has given a factorization of the completed cohomology of the tower of modular curves, separating the contributions of all the groups that act (i.e., the absolute Galois group of \({\mathbb Q}\) and the \({\mathrm {GL}}_2({\mathbb Q}_\ell)\) for all primes \(\ell\). I will explain how one can use \(p\)-adic Hodge theory to construct a Kirillov model for the completed cohomology and obtain a more direct construction of this factorization.

Berkovich spaces over \(\mathbb Z\) may be seen as fibrations containing complex analytic spaces as well as \(p\)-adic analytic spaces, for every prime number \(p\). We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on \(\mathbb P^1\) of torsion points of two elliptic curves.

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 \(\mathbb Q_p\)-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.

The coefficient theory of prismatic cohomology has been investigated a lot recently. For example, Morrow--Tsuji and Tian have discovered the relationship between Hodge--Tate crystals and Higgs bundles (at least in the local case). In this talk, I will briefly talk about how to get a global correspondence between Hodge--Tate crystals and Higgs bundles after inverting \(p\). Then I will discuss what we can say without inverting \(p\) in the geometric case (over \(\mathcal O_C\)) (if time permits, also in the arithmetic case (over \(W(k)\))). This is joint work with Yupeng Wang.

We introduce a new category of covering spaces in rigid geometry, called geometric coverings, and show it is classified by a certain topological fundamental group. Geometric coverings generalize the class of étale coverings, introduced by de Jong, and its various natural modifications, and have certain desirable properties that were missing from those older notions: they are étale local and closed under taking infinite disjoint unions. The definition is based on the property of unique lifting of “geometric arcs.” On the way, we answer some questions from the foundational paper of de Jong. This is joint work with Piotr Achinger and Alex Youcis.

Let \(X\) be a smooth and proper variety over a local field \(K\). Then the weight-monodromy conjecture predicts that the monodromy and weight filtrations coincide up to a shift. Recently, P. Scholze proved this conjecture for set-theoretic complete intersections inside the projective space using the theory of perfectoid spaces. Alternatively, one can formulate a (local) version of the weight-monodromy conjecture for the nearby cycles. We will give a precise formulation of this conjecture and prove it in some cases following the strategy of Scholze in the global case. This is joint work with David Hansen.