Alexander Bertoloni Meli: Comparison of local Langlands correspondences for odd unitary groups

I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a $p$-adic field, there is a Langlands correspondence using trace formula techniques due to Mok and Kaletha--Minguez--Shin--White. Another correspondence was constructed recently by Fargues--Scholze using $p$-adic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.

Evangelia Gazaki: Weak Approximation for zero-cycles

Let $X$ be a smooth projective variety over the rational numbers. The set $X(\mathbb{Q})$ of all rational points of $X$ embeds to the set of all real points, as well as the set of all $p$-adic points, for every prime number $p$. The classical local-to-global principles for $X$ refer to the diagonal embedding of $X(\mathbb{Q})$ to the direct product of the above sets. We say that $X$ satisfies Weak Approximation, if the image of this embedding is dense. When this is not the case, one looks for possible obstructions. A classical obstruction is given by the Brauer group of $X$. This obstruction is expected to explain all phenomena for certain classes of varieties, but this is generally not true in general. In this talk we will discuss analogs of these questions for the Chow group $C{H}_0(X)$ of zero-cycles on $X$. In this case, there is a very open conjecture due to Colliot-Thélène and Sansuc and separately due to Kato and Saito that the obstruction induced by the Brauer group is the only one. We will give evidence for this conjecture for a product of elliptic curves. This is based on two papers, the first joint work with T. Hiranouchi, and the second with an Appendix by A. Koutsianas.

Joseph Hundley: Functorial Descent in the Exceptional Groups

In this talk, I will discuss the method of functorial descent. This method, which was discovered by Ginzbug, Rallis and Soudry, uses Fourier coefficients of residues of Eisenstein series to attack the problem of characterizing the image of Langlands functorial liftings. We'll discuss the general structure, the results of Ginzburg, Rallis and Soudry in the classical groups, some recent attempts to extend the same ideas to the exceptional group, and challenges and new phenomena which emerge in these attempts. The new work discussed will mainly be joint with Baiying Liu. Time permitting I may also comment on unpublished work of Ginzburg and joint work with Ginzburg.

Karol Kozioł: Derived K-invariants and the derived Satake transform

The classical Satake transform gives an isomorphism between the complex spherical Hecke algebra of a $p$-adic reductive group $G$, and the Weyl-invariants of the complex spherical Hecke algebra of a maximal torus of $G$. This provides a way for understanding the $K$-invariant vectors in smooth irreducible complex representations of $G$ (where $K$ is a maximal compact subgroup of $G$), and allows one to construct instances of unramified Langlands correspondences. In this talk, I'll present work in progress with Cédric Pépin in which we attempt to understand the analogous situation with mod $p$ coefficients, and working at the level of the derived category of smooth $G$-representations.

Jaclyn Lang: Counting vexing congruences

Fix a weight 2 CM modular form with trivial nebentypus and level ${\ell }^2$ for some prime $\ell$, say $f$. How many forms of the same weight and level are congruent to $f$ modulo a prime $p$? We will sketch a proof that when $\ell =-1\mathrm{mod}p$, this number is always divisible by $p$. Such an $f$ is an example of a modular form that we call “vexing at $\ell$ modulo $p$” (following Diamond), and the $p$-divisibility phenomenon is true for all such vexing forms (at least if $p$ is greater than $3$). Our methods combine modular representation theory with geometry and cohomology of modular curves. This work in progress is joint with Robert Pollack and Preston Wake.

Gilbert Moss: The local theta correspondence in families for type II dual pairs

Let $F$ be a nonarchimedean local field of residue characteristic $p$. A type II reductive dual pair over $F$ is a pair $({\mathrm{G}\mathrm{L}}_n(F),{\mathrm{G}\mathrm{L}}_m(F))$ for integers $n,m$. The local theta correspondence provides a bijective map between certain subsets of complex irreducible representations of ${\mathrm{G}\mathrm{L}}_m(F)$ into those of ${\mathrm{G}\mathrm{L}}_n(F)$, mediated by the commuting actions of these groups on the Weil representation. If instead of complex representations we work over an algebraically closed field of characteristic $\ell$ different from $p$, such a map no longer exists for bad primes $\ell$. In this talk we will present a new perspective on the type II theta correspondence in terms of the Bernstein center and cuspidal support, which works nicely mod-$\ell$, interpolates within $\ell$-adic integral families, and descends to $\mathbb{Z}[1/p]$.

Stefan Patrikis: Compatibility of the canonical $\ell$-adic local systems on non-abelian type Shimura varieties

Let $(G,X)$ be a Shimura datum, and let $K$ be a compact open subgroup of $G({\mathbb{A}}_f)$. One hopes that under mild assumptions on $G$ and $K$, the points of the Shimura variety $S{h}_K(G,X)$ form a family of motives; in abelian type this is well-understood, but in non-abelian type it is completely mysterious. I will discuss joint work with Christian Klevdal showing that for many non-abelian type Shimura varieties the points (over number fields, say) at least yield compatible systems of $\ell$-adic representations (that should be the $\ell$-adic realizations of the conjectural motives).

Naser Talebizadeh Sardari: Intersections of geodesics on the modular surface

Let $\alpha$ be a compact geodesic segment in the full modular surface, and let $C_D$ be the union of closed geodesics of discriminant $D$. I’m going to present a proof that the intersection points $\alpha \cap C_D$ become equidistributed along $\alpha$ as $D\to \mathrm{\infty }$. I will then discuss how to make the theorem effective. This is a joint work with Junehyuk Jung.

Yujie Xu: From affine Deligne--Lusztig varieties to geometry of integral models of Shimura varieties of abelian type

Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, etc.) have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic and deformation-theoretic aspects of integral models of Hodge type (or more generally abelian type) constructed by Kisin and Kisin-Pappas. I will talk about my recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models. I will also mention an application to toroidal compactifications of such integral models.

A key ingredient in the parahoric version of the main theorem is a new CM lifting result on parahoric level integral models of Shimura varieties, which uses as input a new result on connected components of affine Deligne–Lusztig varieties (joint with Gleason and Lim). This resolves a long-standing conjecture on connected components of affine Deligne-Lusztig varieties, using recently developed diamond- and v-sheaf-theoretic techniques from p-adic geometry. If time permits, I will give a sketch on how to deduce the new CM lifting result, as well as various other applications to the geometry of Shimura varieties.