
Sep 6  Spencer Leslie (BC)  Relative Langlands and endoscopy
Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by BenZvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre)stabilization of relative trace formulas.

Sep 13  Ryan Chen (MIT)  Fourier coefficients, orbital integrals, and arithmetic 1cycles
Arithmetic SiegelWeil formulas predict that derivatives of Eisenstein series Fourier coefficients should encode arithmetic volumes of special cycles on Shimura varieties.
Such formulas are arithmetic versions of formulas for complex volumes of moduli spaces in terms of Lfunctions and Eisenstein series.
These "geometrize" the classical Siegel mass and SiegelWeil formulas, on lattice and lattice vector counting.
I will introduce these ideas and overview my proof for arithmetic 1cycles on unitary Shimura varieties of arbitrarily high dimension.
The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic.
I will also mention a closely related "nearcenter arithmetic fundamental lemma (AFL)" relating arithmetic 1cycles and orbital integrals.
This "AFL" is from joint work in progress with Weixiao Lu and Wei Zhang.

Sep 20  Aaron Pollack (UCSD)  Automatic convergence of modular forms
Quaternionic modular forms (QMFs) are a class of nonholomorphic automorphic functions that behave similarly to classical Siegel modular forms. They exist on certain real forms of the exceptional groups, and also groups of type B and D with real rank 4, e.g. SO(4,n) with n at least 4. QMFs have semiclassical Fourier expansions and Fourier coefficients. I will explain the proof that, on the exceptional groups, there is a basis of the space of QMFs such that every Fourier coefficient of every element of the basis lies in Qbar. The main technique is an "automatic convergence" theorem, which says that certain infinite series, which "look like" the Fourier expansion of a QMF, actually converge absolutely and thus are the Fourier expansion of a QMF.

Sep 27  Florian Herzig (Toronto)  Finite length results for the mod p cohomology of $\mathrm{GL}_2$
Suppose that $F/\mathbb{Q}_p$ is a finite unramified extension. When $F = \mathbb{Q}_p$, Emerton
showed that the modp local Langlands correspondence is realized globally
in the modp cohomology of modular curves. For larger $F$, the modp local
Langlands correspondence of $\mathrm{GL}_2(F)$ is unknown, but it is interesting to
study the subrepresentations of the modp cohomology of Shimura curves cut
out by a global Galois representation, in analogy with $F = \mathbb{Q}_p$. Under some
reasonable hypotheses we prove that these representations of $\mathrm{GL}_2(F)$ are
of finite length, as was expected since the work of BreuilPaskunas from
around 2007. This is joint work with C. Breuil, Y. Hu, S. Morra, and B.
Schraen.

Oct 4  Adrian Diaconu (UMN)  Moments of quadratic $L$functions over function fields
In 2001, Conrey, Farmer, Keating, Rubinstein, and Snaith developed a "recipe" utilizing heuristic arguments to predict the asymptotics of moments of various families of Lfunctions. This heuristic was later extended by Andrade and Keating to include moments and ratios of the family of Lfunctions associated to hyperelliptic curves of genus g over a fixed finite field. In joint work with Bergström, Petersen, and Westerland, we related the moment conjecture of Andrade and Keating to the problem of understanding the homology of the braid group with symplectic coefficients. We computed the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations, and showed that the answer matches the numbertheoretic predictions. Our results, combined with a recent homological stability theorem of Miller, Patzt, Petersen, and RandalWilliams, imply the conjectured asymptotics for all moments in the function field case, for all large enough odd prime powers $q$. 
Oct 11 (Rm 312)  AnneMarie Aubert (CNRS)  Bernstein blocks of enhanced $L$parameters and applications
The decomposition into Bernstein blocks of the category of smooth representations of a $p$adic group $G$ admits a Galois analogue in terms of enhanced $L$parameters, in which the role of supercuspidal representations is played by « cuspidal » enhanced $L$parameters, a notion that originates in the construction by Lusztig of the generalized Springer correspondence. We will formulate sufficient conditions to be satisfied by the local Langlands correspondence for the supercuspidal representations of the Levi subgroups of G in order to allow to construct a canonical bijection between the associated blocks on both sides. Next, we will prove that, when these conditions are satisfied, every compound $L$packet of G contains at least one irreducible representation with nonsingular supercuspidal support. The conditions evacuated above are notably satisfied when $G$ is the exceptional group $G_2$, and when G is a classical group. We will show that, for every irreducible $p$adic orthosymplectic dual pair $(G, G')$, the Howe correspondence can be described as a collection of correspondences between simple modules of affine Hecke algebras attached to the Bernstein blocks of enhanced $L$parameters of $G$ and $G^′$. Parts of this talk are based on joint works with Ahmed Moussaoui and Maarten Solleveld, and with Yujie Xu. 
Oct 18  Jared Weinstein (BU)  Excursion functions for padic reductive groups
The local Langlands correspondence (LLC) is a conjectural finitetoone map from representations of a padic reductive group G
to the set of Lparameters for G. Recently there have been two major advances in this area: Kaletha's characterization of the LLC in terms of explicit character formulas and harmonic analysis, and
then a more geometric construction, due to FarguesScholze, involving excursion operators. This talk represents work in progress on what happens when you marry these approaches: you can write down an expected formula for the excursion operator, assuming Kaletha's formulas. Interestingly, the excursion operator is convolution with a function, the "excursion function", which comes entirely out of harmonic analysis.

Oct 25  Chandrashekhar Khare (UCLA)  Congruence modules in higher codimension, Wiles defect and the BlochKato conjecture
I will talk about recent work (joint with Boeckle, Iyengar, Manning, Urban) that studies situations when one can prove isomorphism of deformation rings R and Hecke rings T without them being complete intersections. Our work has led to new algebraic invariants which capture the failure of R=T to be isomorphisms of complete intersections. One can try and relate these new invariants to the numerical invariants like Tamagawa factors that appear in the formula conjectured by BlockKato for the special value L(1,Ad_f) of the adjoint motive associated to a newform f of weight at least 2. One is led to make new guesses about the relation of the higher congruence numbers (defined in work with Iyengar and Manning) associated to Kisin’s local deformation rings at at the augmentation induced by f and the Tamagawa numbers of the adjoint motive of f. The talk will summarize known results and conclude with some more speculative thoughts.

Nov 1  Lie Qian (UChicago)  Another Formulation of Trianguline Deformation Ring
Trianguline representations are a big class of $p$adic representations that contain all nice enough (cristalline) ones but allow a continuous variation of weights. The trianguline deformation ring is defined as the Zariski closure of such representations in certain ambient spaces. We give a new definition of trianguline deformation rings in 2 dimensional cases as the support of certain modules, so that one does not need to take Zariski closure. It is also integral in nature and can be seen to be compatible with Emerton's Jacquet functor.

Nov 8  Xinwen Zhu (Stanford)  Etale cohomology of Shimura varieties via coherent sheaves
Studying the étale cohomology of Shimura varieties with Hecke and Galois actions provides an avenue toward understanding the Langlands correspondence.
While the structure of the rational cohomology groups is predicted by conjectures of Kottwitz and Arthur,
the integral or mod $\ell$ cohomology groups are more intricate. Although a conjectural formula exists for computing these groups, it relies on an additional conjecture on the existence of moduli spaces of global Galois representations.
However, much more can be said if we focus on these cohomology groups with Hecke and local Galois actions. Specifically,
I will present an unconditional formula for computing the mod $\ell$ cohomology groups of many Shimura varieties with Iwahori level at $p(\neq \ell$), expressed in terms of coherent sheaves on the moduli space of local Galois representations. This result not only provides evidence supporting the aforementioned conjectures but also has surprising arithmetic applications.

Nov 15  Mahesh Kakde (IISc) 

Nov 22  Jacob Tsimerman (Toronto) 

Dec 6  KaiWen Lan (UMN) 

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Organizers: Qiao He, Eric Urban, and Yujie Xu