Jan 27  A. Raghuram (Fordham)

Special values of RankinSelberg Lfunctions over a totally imaginary field.
I will talk about my recent rationality results on the ratios of critical values for RankinSelberg Lfunctions for GL(n) x GL(m) over a totally imaginary base field. In contrast to a totally real base field, when the base field is totally imaginary, some delicate signatures enter the reciprocity laws for these special values. These signatures depend on whether or not the totally imaginary base field contains a CM subfield. This is a generalization of my work with Günter Harder on rankone Eisenstein cohomology for GL(N), where N = n + m. The rationality result comes from interpreting Langlands's constant term theorem in terms of an arithmetically defined intertwining operator between Hecke summands in the cohomology of the BorelSerre boundary of a locally symmetric space for GL(N). The signatures arise from Galois action on certain local systems that intervene in boundary cohomology.

Feb 03  Jared Weinstein (Boston)

Higher Modularity of Elliptic Curves
Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), but one can also extend the idea further and ask whether E is 'rmodular' for r=2,3.... To define this generalization, the modular curve gets replaced with Drinfeld's concept of a 'shtuka space'. The rmodularity of E is predicted by Tate's conjecture. In joint work with Adam Logan, we give some classes of elliptic curves E which are 2 and 3modular.

Feb 10 (Online)  J. E. Rodríguez Camargo (Bonn)

Solid locally analytic representations, Dmodules and applications to padic automorphic forms
In this talk I will present a project in progress with Joaquín Rodrigues Jacinto concerning the study of locally analytic
representations of padic Lie groups and its relation with padic Dmodules of rigid spaces à la Ardakov. I will sketch
how both theories are essentially two different looks of the same kind of objects and how they can be interpreted in
terms of sheaves in suitable stacks on analytic rings. If time permits I will mention two possible applications in the
cohomology of Shimura varieties.

Feb 17  Robert Cass (Michigan)

Geometrization of the mod p Satake transform
The classical Satake isomorphism relates the spherical Hecke algebra of a reductive group $G$ over a local field $F$ to representations of the Langlands dual group.
When $F$ is of mixed characteristic $(0,p)$ and the Hecke algebra has characteristic prime to $p$, the Satake isomorphism has been geometrized by X. Zhu, J. Yu, FarguesScholze, and RicharzScholbach using techniques from padic geometry.
In this talk, we consider the case where the Hecke algebra has characteristic p. I will speak on my recent joint work with Yujie Xu, where we geometrize and obtain explicit formulas for the mod p Satake isomorphism of Herzig and HenniartVignéras using mod p étale sheaves on Witt vector affine flag varieties.
Our methods involve the constant term functors inspired from the geometric Langlands program, especially the geometry of certain generalized MirkovićVilonen cycles. The situation is quite different from ladic sheaves ($l \neq p$) because only three of the six functors preserve constructible sheaves.

Feb 24  Yujie Xu (MIT)

Hecke algebras for $p$adic groups and the explicit Local Langlands Correspondence for $G_2$
I will talk about my recent joint work with Aubert where we prove the Local Langlands Conjecture for $G_2$ (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of reductive $p$adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties. In particular, we obtain "mixed" Lpackets containing Fsingular supercuspidals and nonsupercuspidals.

March 03  Mathilde GerbelliGauthier (McGill)

Counting nontempered automorphic forms using endoscopy
How many automorphic representations of level n have a specified local factor at the infinite places? When this local factor is a discrete series representation, this questions is asymptotically wellunderstood as n grows. Nontempered local factors, on the other hand, violate the Ramanujan conjecture and should be very rare. We use the endoscopic classification for representations to quantify this rarity in the case of cohomological representations of unitary groups, and discuss some applications to the growth of cohomology of Shimura varieties.

March 10  Stefan Patrikis (Ohio)

Compatibility of canonical ladic local systems on some Shimura varieties of nonabelian type
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 $Sh_K(G, X)$ form a family of motives; in abelian type this is wellunderstood, but in nonabelian type it is completely mysterious. I will discuss joint work with Christian Klevdal showing that for nonabelian type Shimura varieties the points (over number fields, say) at least yield compatible systems of ladic representations (to be precise, after projection to the adjoint group of G). These should be the ladic realizations of the conjectural motives.

March 24  Tasho Kaletha (Michigan)

Covers of reductive groups and functoriality
To a connected reductive group $G$ over a local field $F$ we
define a compact topological group $\tilde\pi_1(G)$ and an extension
$G(F)_\infty$ of $G(F)$ by $\tilde\pi_1(G)$. From any character $x$ of
$\tilde\pi_1(G)$ of order n we obtain an nfold cover $G(F)_x$ of the
topological group $G(F)$. We also define an Lgroup for $G(F)_x$, which is a
usually nonsplit extension of the Galois group by the dual group of $G$,
and deduce from the linear case a refined local Langlands correspondence
between genuine representations of $G(F)_x$ and Lparameters valued in
this Lgroup.
This construction is motivated by Langlands functoriality. We show that
a subgroup of the Lgroup of $G$ of a certain kind naturally lead to a
smaller quasisplit group $H$ and a double cover of $H(F)$. Genuine
representations of this double cover are expected to be in functorial
relationship with representations of $G(F)$. We will present two concrete
applications of this, one that gives a characterization of the local
Langlands correspondence for supercuspidal Lparameters when p is
sufficiently large, and one to the theory of endoscopy.

March 31  Gregorio Baldi (IHES)

The Hodge locus
I will report on a joint work with Klingler and Ullmo. Given a polarizable variation of Hodge structures on a smooth complex quasiprojective variety S (e.g. the one associated to a family of pure motives over S), Cattani, Deligne and Kaplan proved that its Hodge locus (the locus of closed points of S where exceptional Hodge tensors appear) is a *countable* union of closed algebraic subvarieties of S. In this talk I will discuss when this Hodge locus is actually algebraic.
The first part of the talk will introduce the Hodge theoretic formalism and highlight differences and similarities with the world of Shimura varieties. If time permits I will present some applications of such a viewpoint to either the LawrenceVenkatesh method or to the existence of genus four curves of "Mumford's type".

April 07  Yihang Zhu (Maryland)

Zeta Functions of Shimura Varieties
I will first recall the general expectations of Shimura, Langlands, and Kottwtiz on the shape of the zeta function of a Shimura variety, or more generally its étale cohomology. I will then report on some recent progress which partially fulfills these expectations, for Shimura varieties of unitary groups and special orthogonal groups. Finally, I will give a preview of some foreseeable developments in the near future.

April 14  Haruzo Hida (UCLA)

Adjoint Lvalue formula and Tate conjecture
For a Hecke eigenform $f$, we state an adjoint Lvalue formula relative to each quaternion algebra $D$ over $\mathbb{Q}$ with discriminant $\partial$ and reduced norm $N$.
A key to prove the formula is the theta correspondence for the quadratic $\mathbb{Q}$space $(D,N)$. Under the $R=\mathbb{T}$theorem, the $p$part of the BlochKato conjecture is known; so, the formula is
an adjoint Selmer class number formula. We also describe how to relate the formula to a consequence of the Tate conjecture for quaternionic Shimura varieties.

April 21  Michael Groechenig (Toronto)

padic points of stacks and applications
The first half of this talk will be devoted to describing the structural properties of the set of local field valued points of a certain class of algebraic stacks. I will then describe two applications in the second half, one joint with Wyss and Ziegler, and the other one with Esnault. The first application relates the padic volume of certain moduli spaces to BPS invariants and the second application is an elementary proof of the existence of a FontaineLaffaille structure for rigid flat connections.
