The 24th Autumn Workshop on Number Theory

• Siegfried Böcherer: On denominators of critical values of standard-L-functions twisted by characters: the conductor part

Abstract: Katsurada started the investigation of denominators of critical values of standard L-functions for Siegel modular forms. Such denominators are related to congruence primes. Later it was shown that this remains true for L-values twisted by Dirichlet characters but one had to restrict to the denominator part “away from the conductor of the character”. In the talk, we describe a (weak) result concerning the conductor part. For degree one we get a more precise result. This is inspired by a similar result for triple product L-functions obtained in an ongoing Mannheim PhD thesis by Tobias Keller.

• Shih-Yu Chen: Algebracity of Ratios of Rankin-Selberg L-functions 

Abstract: For consecutive critical values of Rankin–Selberg L-functions for GLn × GLn′, we have the celebrated result of G. Harder and A. Raghuram on the algebraicity of the ratios when nn′ is even. As a different aspect of ratios of critical values, we consider ratios of products of different Rankin–Selberg L-functions at a fixed critical point. In this talk, we introduce our result on the algebraicity of the ratios under a regularity condition. As applications, we prove new cases of Blasius’ and Deligne’s conjectures on critical values of tensor product L-functions and symmetric power L-functions of modular forms.

• Sungmun Cho: Orbital integrals for classical Lie algebras and smoothening: Part I

Abstract: We will introduce a new method to study stable orbital integrals for regular semisimple elements and for the unit element of the Hecke algebras in classical Lie algebras. In Part I, I will explain it in the context of gl_n by suggesting the values for gl_2 and gl_3 and a lower bound for gl_n. I will also propose a conjecture about the second leading term for gl_n. Finally, I will suggest a uniform shape of its values covering other classical Lie algebras, which will be explained in Part II. 

• Solomon Friedberg: Towards a New Shimura Lift

Abstract: The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$, corresponding to classical half-integral weight forms, to automorphic representations on $PGL_2$. Though efforts have been made for many years to generalize this map to higher rank groups and higher degree covers, our knowledge is limited. In this talk, I present joint work with Omer Offen that points to a new Shimura lift for automorphic representations on the triple cover of $SL_3$.  We establish the Fundamental Lemma for a relative trace formula.  Moreover, this project will characterize the image of the lift by means of a period involving a theta function on $SO_8$, confirming a 2001 conjecture of Bump, Friedberg, and Ginzburg.

• Hirotaka Kakuhama: Langlands parameters for quaternionic unitary groups and Local theta correspondences

Abstract: The descriptions of local theta correspondences in terms of the Langlands parameters have been studied and established by many researchers for a large part of the reductive dual pairs. However, for quaternionic dual pairs, it has not been attained. In this talk, we analyze the Langlands parameters for quaternionic unitary groups defined in the theory of rigid inner twists and construct a correspondence of the parameter side for quaternionic dual pairs, which should commute with the local theta correspondences.

• Yuchan Lee: Orbital integrals for classical Lie algebras and smoothening: Part II

Abstract: In this talk, I will explain that arguments used in the case of gln are extended to other classical Lie algebras such as u_n, sp_{2n}, or so_{2n+1}, by suggesting the values when n=2 and lower bounds for a general n. I will also propose a conjecture of the first leading term of the value for any n. All of these values have uniform shapes.

• WenーWei Li: An intertwining relation via Takeda-Wood isomorphism

Abstract: For p-adic local fields of characteristic not equal to 2, but with arbitrary residual characteristic p, Takeda and Wood obtained an isomorphism between the Iwahori-Hecke algebra of SO(2n+1) and the Hecke algebra of Mp(2n) for the Bernstein block containing the even Weil representation; for the odd component one takes the non-split inner form of SO(2n+1) instead. I will try to explain how their isomorphism behaves under parabolic induction, whose proof is not entirely trivial. Then I will sketch a Gindikin-Karpelevich formula for Mp(2n) that applies to dyadic local fields as well. The motivation comes from Arthur's local intertwining relation for Mp(2n), which becomes "wild" when p=2. This is a joint work with Fei Chen.

• Kazuma Ohara: On progenerators of Bernstein blocks

Abstract: Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G(F), we have two kinds of progenerators: the compactly induced representation ind_K^{G(F)} (rho) of a type (K, rho), and the parabolically induced representation I_{P}^{G}(Pi^{M}) of a progenerator Pi^{M} of a Bernstein block for a Levi subgroup M of G. In this talk, we construct an explicit isomorphism of these two progenerators. We also explain that the induced isomorphism between the endomorphism algebras is compatible with their descriptions in terms of affine Hecke algebras.

• Steven Spallone: Residues of Intertwining Operators for Classical Groups

Abstract: A deeper understanding of parabolically induced representations comes from their interrelationships through meromorphically defined intertwining operators.  A certain residue of these operators is essential to understand. It was studied via harmonic analysis by several authors under the direction of Shahidi. We survey the results of this program.

• Steven Spallone: Stiefel-Whitney Classes of Representations 

Abstract: An orthogonal representation of a finite group has invariants living in the (mod 2) group cohomology, called Stiefel–Whitney Classes (SWCs). The first two SWCs vanish iff the representation lifts to the spin group. If a vector bundle arises from balancing a G-bundle with a representation of G, then its SWCs can be deduced from the SWC of the representation. This talk surveys my research group's work in computing SWCs for familiar groups, as well as the spinoriality problem.

• Shuichiro Takeda: On dual groups of symmetric varieties and distinguished representations of p-adic groups.

Abstract: Let X=H\G be a symmetric variety over a p-adic field, where we assume G is split. In this talk, I will introduce a complex group G_X^\vee, which naturally gives rise to a map from G_X^\vee\time SL_2 to the Langlands dual group G^\vee. Further, I will talk about how this map is related to H-distinguished representations of G through numerous examples, especially in relation to the theory of Kato-Takano on H-relative matrix coefficients.