L-functions and regulators of modular forms
The Beilinson conjectures express special values of L-functions at integers in terms of so-called regulator integrals. After presenting the case of the Riemann zeta function, I will discuss the case of elliptic curves and modular forms. I will explain a new construction of elements in the K4 group of modular curves, and their applications to L-functions of modular forms. If time permits, I will also speculate p-adically. This is partly joint work with Wadim Zudilin.
Duality and flip-flop in local Shimura varieties
We will discuss applications of p-adic comparison theorems and of the p-adic six functor formalism to various arithmetic and representation-theoretic problems related to dual local Shimura varieties. This is based on joint work with Guido Bosco, Wieslawa Niziol and also with Juan Esteban Rodriguez Camargo.
Conjectures on L-functions for varieties over function fields and their relations
I will report on joint work with Timo Keller and Yanshuai Qin. We consider versions for smooth varieties over a function field in positive characteristic p of several conjectures that can be traced back to Tate, and study their interdependence. In particular, for an abelian variety A/K over a function field, I will explain how to relate the BSD-rank conjecture for A to the finiteness of the p-primary part of the Tate-Shafarevich group of A using p-adic methods.
Local Langlands correspondence for p-adic covering groups
Recently, Fargues-Scholze constructed the local Langlands correspondence for p-adic reductive groups and formulated the categorical conjecture. In this talk, we discuss its generalization to covering groups of p-adic reductive groups. This is based on joint work in progress with Tony Feng, Ildar Gaisin, Teruhisa Koshikawa and Yifei Zhao.
Motivic cohomology for singular schemes
We discuss the procdh topology as a coarsening of the cdh topology. This produces a motivic cohomology which is closer to algebraic K-theory. Specifically, there is an AtiyahâHirzebruch style spectral sequence from procdh motivic cohomology to algebraic K-theory for possibly singular schemes. At the end we will make some connections to the motivic cohomology of Elmanto-Morrow and Tess Bouis. This is joint work with Shuji Saito.
Classicality theorems and its applications to the automorphy lifting theorem and the Breuil-Mezard conjecture in some GL_2(Q_p^2) cases
In this talk, we will study locally analytic vectors in a 'partial' completed cohomology of Shimura varieties associated to some rank 2 unitary group over a totally real field F satisfying that F_v is Q_p^2 or Q_p for any p-adic place v and prove a certain classicality theorem. This is a partial generalization of Lue Pan's work in the modular curve case. As applications, we will prove the Breuil-Mezard conjecture of GL_2(Q_p^2) and the automorphy lifting theorem of GL_2(F) in some cases without any technical conditions on the properties at p-adic places of liftings of the residual representation except Hodge-Tate regularity.
On vanishing of the supercuspidal part of the l-adic cohomology of local Shimura varieties
A local Shimura variety is a p-adic analogue of a Shimura variety. Its l-adic cohomology is expected to be related to the local Langlands correspondence. In the Lubin-Tate case, a supercuspidal representation appears only in the middle degree. In the GSp(4) case, a supercuspidal representation appears only in the degrees 2,3,4. In this talk, I will talk about a generalization of these results to the Hodge-Newton reducible case.
Cohomology of (phi,Gamma)-modules and duality
Cohomology of (phi,Gamma)-modules was studied by Herr, Liu, and Kedlaya-Pottharst-Xiao. Kedlaya-Pottharst-Xiao proved finiteness, duality, and Euler-characteristic formula for cohomology of families of (phi,Gamma)-modules. In this talk, we will present an alternative proof of finiteness and duality by using analytic geometry introduced by Clausen-Scholze and six-functor formalism refined by Heyer-Mann. One advantage of this proof is that it can handle families over Banach Qp-algebras that are not topologically of finite type over Qp. If time permits, we will also discuss potential future applications to the representability of the analytic Emerton-Gee stack.
On approximation to a real number by algebraic numbers of bounded degree
In his seminal 1961 paper, Wirsing studied how well a given transcendental real number xi can be approximated by algebraic numbers alpha of degree at most n for a given positive integer n, in terms of the so-called naive height H(alpha) of alpha. He showed that the exponent w_n^*(xi) which measures this quality of approximation is at least (n + 1)/2. He also asked if we could even have w_n^*(xi) >= n as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form n/2 + O(1) until Badziahin and Schleischitz showed in 2021 that w_n^*(xi) >= an for each n >= 4, with a = 1/sqrt(3) = 0.577. In this talk, I will first present the background and ideas behind the proof of Wirsing's lower bound. Secondly, using a new approach that is partly inspired by parametric geometry of numbers, I will explain how we can obtain w_n*(xi)>= an for each n >= 2, with a = 1/(2 - log 2) = 0.765.
Potentially crystalline deformation rings and the Breuil-Mézard conjecture for PGL_n
In this talk, I will explain how we can study the potentially crystalline deformation rings for representations with values in PGL_n by comparing them to the analogous rings for representations with values in GL_n. As a consequence, we get enough information on these rings to be able to state a formula for their multiplicities, generalizing to this situation the Breuil-Mézard conjecture. This is joint work with Agnès David.
Log monogenic extensions and ramification theory
In classical ramification theory, many theorems are proved under the assumption that the extension of residue fields is separable. Log monogenic extensions give an optimal class of extensions for which classical theorems are extended. I discuss the definition, various characterizations and generalizations of classical theorems including the Hasse--Arf theorem.
On mod p representations of GL2 in the cohomology of Shimura curves
Let F be an unramified extension of Qp. Let pi be a mod p representation of GL2(F) which can be realized as an Hecke eigenspace in the mod p cohomology of a tower of Shimura curves. It is expected that such a representation is in the image of the (hypothetical) mod p Langlands correspondence for GL2(F). We prove, under some assumptions of type 'Taylor--Wiles', that such a representation of GL2(F) is of finite length. This is a joint work with Christophe Breuil, Florian Herzig, Yongquan Hu and Stefano Morra.
Blow-up invariance for Hodge-Witt sheaves with modulus
For a field k, a Q-modulus pair over k is a pair (X,D) consisting of a k-variety X and an effective Cartier divisor with Q-coefficient D on X such that the complement of the support of D is smooth. When k is perfect of characteristic p>0, we define the Hodge-Witt sheaves for Q-modulus pairs over k and prove that their cohomologies are invariant under blow-ups of Q-modulus pairs with certain centers in boundary divisors. This is a generalization of a result of Koizumi for Witt sheaves and that of Kelly-Miyazaki and Koizumi for Hodge sheaves.
On the higher analytic vectors of B_e
The category of p-adic representations of G=G_{Q_p} can be viewed as subcategory of the category of equivariant vector bundles on the Fargues-Fontaine curve, obtained by "glueing" the period rings B_e and B_{dR}^+ from p-adic Hodge theory. The subrings stable under the action of the kernel H of the cyclotomic character are well-understood, which leaves us with the action of the p-adic Lie group Gamma=G/H on (modules over) these rings. In many context, passage to locally analytic vectors can serve as a 'decompletion' functor and it was observed by Berger and Colmez that, contrary to the case of admissible representations, locally analytic vectors can fail to be exact in this context. Using condensed mathematics, we show that the higher derived analytic vectors of B_e are non-zero and compute their analytic cohomology. We also give a description of the co-kernel of a 'decompleted' variant of the Bloch-Kato exponential map for Q_p(n) in terms of derived analytic vectors.
On depth-zero integral models of local Shimura varieties
An integral model is a tool to calculate the cohomology of the generic fiber via the nearby cycle. Integral models of local Shimura varieties are usually considered only at parahoric levels. Still, those of Lubin-Tate spaces are constructed at depth-zero levels by T. Yoshida in an ad-hoc way. In this talk, we will give another construction of depth-zero integral models using local shtukas introduced by Scholze-Weinstein. It can be applied to any hyperspecial local Shimura datum and recovers Yoshida's models for Lubin-Tate spaces.
On the (phi, Gamma)-modules Corresponding to Crystalline Representations and Semi-stable Representations
From the 1980s to the 1990s, Jean-Marc Fontaine introduced the theory of (phi, Gamma)-modules to study p-adic Galois representations. They are simpler than p-adic Galois representations, but he showed an equivalence between them. Among p-adic Galois representations, some classes are particularly important in number theory. Main examples are crystalline representations, semi-stable representations and de Rham representations. In this talk, I will explain how we can determine the (phi, Gamma)-modules corresponding to crystalline representations and semi-stable representations. These results can be seen, in a sense, as generalizations of Wach modules. If time permits, I will explain my result on the (phi, Gamma)-modules corresponding to de Rham representations.
Geometric translations of (phi, Gamma)-modules for GL2(Qp)
Translation functors change infinitesimal characters of representations of Lie algebras. Under the p-adic Langlands correspondence, infinitesimal characters of p-adic automorphic forms correspond to the Hodge-Tate-Sen weights of the p-adic Galois representations. In the GL2(Qp)-case, translation functors can be applied directly to (phi, Gamma)-modules over the Robba ring attached to Galois representations, as studied in the works of Colmez, Ding. I will explain that these translations of (phi, Gamma)-modules admit a geometric realization via explicit morphisms between loci of the moduli stack of (phi, Gamma)-modules corresponding to changes in Hodge-Tate-Sen weights.