Abstracts
The following are the details of the talks scheduled for TORA XIV at Oklahoma State University:
- Lea Beneish, University of North Texas, Lea.Beneish@unt.edu
Title: Replicable functions arising from code-lattice VOAs fixed by automorphisms
Abstract: We ascertain properties of the algebraic structures in towers of codes, lattices, and vertex operator algebras (VOAs) by studying the associated subobjects fixed by lifts of code automorphisms. In the case of sublattices fixed by subgroups of code automorphisms, we identify replicable functions that occur as quotients of the associated theta functions by suitable eta products. We show that these lattice theta quotients can produce replicable functions not associated to any individual automorphisms. Moreover, we show that the structure of the fixed subcode can induce certain replicable lattice theta quotients and we provide a general code theoretic characterization of order doubling for lifts of code automorphisms to the lattice-VOA. Finally, we prove results on the decompositions of characters of fixed subVOAs. This talk is based on joint work with Jennifer Berg, Eva Goedhart, Hussain M. Kadhem, Allechar Serrano L´opez, and Stephanie Treneer.
- Alex Hazeltine, University of Michigan, ahazelti@umich.edu
Title: Computing local Arthur packets for classical groups
Abstract: Local Arthur packets play a fundamental role within the Langlands program. In this talk, we discuss a new algorithm which computes local Arthur packets for quasi-split symplectic or orthogonal groups with only minimal prerequisite knowledge. In a sense, we compute local Arthur packets using pictures. This is based on joint work with Liu and Lo for even orthogonal groups and Jiang, Liu, Lo, and Zhang for the other groups.
- Marti Roset Julia, McGill University, marti.rosetjulia@mail.mcgill.ca
Title: Rigid cocycles for SL(n) and explicit class field theory for totally real fields
Abstract: The theory of complex multiplication implies that the values of modular functions at CM points belong to abelian extensions of imaginary quadratic fields. In this talk, we propose a conjectural approach to extending this phenomenon to the setting of totally real fields. Generalizing the work of Darmon, Pozzi, and Vonk, we construct rigid cocycles for SL(n), which play the role of modular functions, and define their values at points associated with totally real fields. The construction of these cocycles originates from a topological source: the Eisenstein class of a torus bundle. This is ongoing joint work with Peter Xu.
- Ju-lee Kim, MIT, juleekim@mit.edu
Title: Γ-asymptotic expansions and a Wavefront set conjecture
Abstract: We are interested in the interplay between the Langlands parameter of a smooth irreducible representation of the reductive p-adic group and the wavefront set of the representation, an invariant that appears naturally in the local character expansion of Howe and Harish-Chandra. Let G be a connected reductive p-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the Γ-asymptotic wavefront sets generalizing the notion of wavefront sets via the Γ-asymptotic expansions (in the sense of Kim-Murnaghan), and then study their relation with the Langlands parameters. When G= GLn, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. This is a joint work with Dan Ciubotaru.
- Stephen Kudla, University of Toronto, skudla@math.toronto.edu
Title: On the subring of special cycles on orthogonal Shimura varieties
Abstract: Over a totally real field F of degree d over Q, quadratic spaces V with signature (m, 2), ..., (m, 2), (m+2, 0), ..., (m + 2, 0) define Shimura varieties of dimension md+ where d+ is the number of indefinite places. The structure of the subring of cohomology generated by special cycles up to numerical equivalence can be described using the Siegel-Weil formula. The structure constants for this ring can be expressed in term of the Fourier coefficients of triple pullbacks of Siegel Eisenstein series. The matching principle for such series implies some identities which have no evident geometric proof.
- Eun Hye Lee, TCU, EUN.HYE.LEE@tcu.edu
Title: Subconvexity of Shintani zeta functions
Abstract: Subconvexity problem has been a central interest in analytic number theory for over a century. The strongest possible form of the subconvexity problem is the Lindelof hypothesis, which is a consequence of the RH, in the Riemann zeta function case. There have been many attempts to break convexity for diverse zeta and L functions, usually using the moments method. In this talk, I will introduce the Shintani zeta functions, and present another way to prove a subconvex bound.
- Ameya Pitale, University of Oklahoma, apitale@ou.edu
Title: Sup-norm bounds for Maass forms on O(1,8n+1)
Abstract: Lower and upper bounds for sup norms of eigenfunctions of Laplacian on Riemannian manifolds is an important area of research in analytic number theory. Motivated by the 1995 paper by Iwaniec and Sarnak on sup norm bounds for Maass forms on arithmetic quotients of the complex upper half plane, there has been a lot of progress made in the last 2-3 decades in the context of Maass forms on GL(2) and other groups. We will present recent joint work with Hiroaki Narita and Simon Marshall concerning sup norm bounds for Maass forms on higher dimensional hyperbolic spaces. We restrict ourselves to Maass forms that are obtained as lifts from Maass forms on the complex upper half plane. The reason is two-fold—we obtain explicit formulas for the Fourier coefficients and Petersson norms of the lifts in terms of the Maass form we start with. These are the two main ingredients of the Fourier expansion method to obtain bounds for sup norms, leading to new results for the group O(1,8n+1).
- Loren Spice, TCU, l.spice@tcu.edu
Title: Fixed points under quasisemisimple and locally quasisemisimple actions
Abstract: The notion of quasisemisimplicity is a generalization of semisimplicity, due to Steinberg, that allows us to describe nicely behaved outer automorphisms. The geometric behaviour of fixed-point groups under a single quasisemisimple automorphism was first investigated by Steinberg. In joint work with Adler and Lansky, we investigated generalisations in several directions: first, dealing with the rational theory (over a non-algebraically closed base field); second, dealing with the jointly quasisemisimple actions; and, third, dealing with the more general class of locally quasisemisimple actions. In this talk, we will try to give the flavour of this third generalisation, including how it turns out to involve surprisingly deep results in the theory of abstract finite groups.
- Shuichiro Takeda, Osaka University, takedas@math.sci.osaka-u.ac.jp
Title: The local theta correspondence - infomercial
Abstract: In this talk, I will talk about the theory of local theta correspondences, especially the two main theorems: conservation relation and Howe duality. The talk is based on the forthcoming book jointly written by W.T. Gan and S. Kudla.