第25回整数論オータムワークショップ

• Noriyuki Abe: Irreducibility of p-adic Banach principal series

Abstract: In this talk, I give a criterion for the irreducibility of p-adic Banach principal series. As an application I give an exact condition for the irreducibility of the representation induced by a finite-dimensional unitary representation. The talk is based on a joint work with Florian Herzig.

• Yongqi Feng: On formal degrees of unipotent representations

Abstract: Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation V can be expressed in terms of the adjoint gamma factor of the enhanced Langlands parameter of V. In joint-work with Eric Opdam and Maarten Solleveld, we have proved this conjecture for unipotent G-representations. In this talk, I will explain the strategy we used in the proof with explicit examples.  

• Jungtaek Hong: Orbital integrals and ideal class monoids for a Bass order

Abstract: A Bass order is an order of a number field whose fractional ideals are generated by two elements. Majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is 4th-power-free in $\mathbb{Z}$, is a Bass order.

In this talk, I will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of $R$. I will also explain explicit enumeration of all orders containing $R$. Our method is based on local-global argument and exhaustion argument, by using orbital integrals for $\mathfrak{gl}_n$ as a mass formula. This is joint work with Sungmun Cho and Yuchan Lee.

• Naoki Imai: Local Langlands correspondence for p-adic covering groups

Abstract: Recently, Fargues-Scholze constructed the local Langlands correspondence for p-adic reductive groups and formulated the categorical conjecture. In this talk, we discuss its generalization to covering groups of p-adic reductive groups.

• Hiroshi Ishimoto: Local newforms for the rank one metaplectic group

Abstract: The theory of local newforms is a local analogue of that of newforms for modular forms. In 2011, Roberts and Schmidt obtained an explicit formula for the number of local newforms for the rank one metaplectic group over a nonarchimedean local field of characteristic zero, without considering the conductors.

In this talk, I will introduce an explicit formula for the conductor for the metaplectic group under the assumption that the residual characteristic is odd. Moreover, I will talk about the nonvanishing of Whittaker functional on the space of local newforms, to give a uniqueness property of local newforms.

• Edmund Karasiewicz: The stable wave front set of theta representations

Abstract: The Fourier coefficients of theta functions have featured prominently in numerous number theory applications and constructions in the Langlands program. For example, they play an important role in the recent work of Friedberg-Ginzburg generalizing the theta correspondence to higher covering groups. For their construction one wants to know the wave front set of the theta representations, i.e. the largest nilpotent orbit with nonvanishing Fourier coefficient.

To investigate these Fourier coefficients it can be valuable to study the analogous local question. In this talk we consider local theta representations and describe how to compute their stable wave front set. This is joint work with Emile Okada and Runze Wang.

• Yukako Kezuka: Non-commutative Iwasawa theory of abelian varieties

Abstract: Non-commutative Iwasawa theory has emerged as a powerful framework for understanding deep arithmetic properties over number fields contained in a p-adic Lie extension and their precise relationship to special values of complex L-functions. This talk aims to explore non-commutative Iwasawa theory over global function fields. We consider an abelian variety A defined over various base fields F, and discuss its arithmetic over the cyclotomic Z_p-extension and more general p-adic Lie extensions. After reviewing some known results over number fields, we shift our focus to the case of global function fields. In this context, we investigate the arithmetic of A over different p-adic Lie extensions without assuming the finiteness of the Selmer group of A over the base field F, as well as its relation to the order of vanishing of the L-function of A/F at s=1.

• Seiji Kuga: An asymptotic formula of special average of central L-values on GSp_4 for square free level

Abstract: Andrianov suggested the Dirichlet series constructed from the Fourier coefficients of Siegel cusp forms for $Sp_4(\mathbb{Z})$ and proved its functional equation by computing the Rankin-Selberg integral. In this talk, we consider the Rankin-Selberg integral for Siegel cusp forms for square-free levels of degree 2 and give its explicit formula. As an application, we will give an asymptotic formula of central values of the spinor L-function for $GSp_4$ by means of a kind of relative trace formula. This is a joint work with Masao Tsuzuki (Sophia University).

• Erez Lapid: Recent progress on zeta functions of prehomogeneous vector spaces

Abstract: The study of prehomogeneous vector spaces and their zeta functions was initiated by Sato and Shintani over 50 years ago. Among other things, they play a crucial role in the contribution of the unipotent conjugacy classes to Arthur's trace formula. I will report on a joint work with Tobias Finis which aims to provide some conceptual understanding of these zeta functions.

• Hengfei Lu: The Prasad-Takloo-Bighash conjecture

Abstract: Let E be a quadratic field extension over F. Let A be a central simple algebra over F containing E. Let B be the centralizer of E inside A. Then (GL(1,A),GL(1,B)) forms a symmetric pair, called the Prasad-Takloo-Bighash pair. Given a character \chi of E*, it will induce a character on GL(1,B), still denoted by \chi. In this talk, we will show that Hom(\pi,\chi) is at most 1-dimensional for any irreducible representation of GL(1,A). Furthermore, we will show that \pi is \chi-self-dual. When E=F+F, we will show that \dim Hom(\pi,\chi) is at most 1 for any irreducible representation of GL(2n,F) and any character \chi of GL(n,F) x GL(n,F). This is a joint work with Nadir Matringe.

• Nadir Matringe: On local converse theorems 

Abstract: We propose a local converse theorem statement for Langlands parameters of quasi-split groups over non Archimedean local fields with acceptable Langlands dual group.  We will prove it when the group is split. We will also discuss the local converse theorems for classical groups from the point of view of Langlands parameters.

• Beth Romano: A Fourier transform for unipotent representations of p-adic groups

Abstract: In the representation theory of finite reductive groups, an essential role is played by Lusztig's nonabelian Fourier transform, an involution on the space of unipotent characters of the group. For reductive p-adic groups, the unipotent local Langlands correspondence gives a natural parametrization of irreducible smooth representations with unipotent cuspidal support. However, many questions about the characters of these representations are still open. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we propose a potential lift of Lusztig's Fourier transform to the setting of split p-adic groups and their pure inner twists. Our work generalizes a construction of Moeglin--Waldspurger for orthogonal groups. In my talk, I will introduce some of these ideas via examples.

• Yugo Takanashi: On the formal degree conjecture for G_2

Abstract: In 2008, Hiraga, Ichino and Ikeda proposed a conjecture on the explicit description of the Plancherel measures of reductive groups over local fields assuming the existence of conjectural local Langlands correspondences. This conjecture is called the formal degree conjecture.

In 2021, Beuzart-Plessis announced the proof of this conjecture for classical groups, using the Arthur's endoscopic character relations for classical groups. In this talk, we will explain how the endoscopic character relations for triality PGSO_8 and G_2 imply the formal degree conjecture for G_2.

• Yiyang Wang: Formal degrees and parabolic induction: the maximal generic case

Abstract: The formal degree conjecture of Hiraga-Ichino-Ikeda expresses the formal degree of a discrete series  of a reductive p-adic group in terms of arithmetic invariants of its Langlands parameter. A natural question is the compatibility of this conjecture with the parabolic induction process. In this talk I will explain how to use local harmonic analysis to deal with the simplest nontrivial case, together with future problems in more general cases.

• Jiandi Zou: Simple type theory for metaplectic covers of GL(r) and applications

Abstract: Let $F$ be a non-archimedean local field whose residue field is of cardinality $q$, and $G=\mathrm{GL}_r(F)$ and $\widetilde{G}$ an $n$-fold metaplectic cover of $G$ with $n$ dividing $q-1$. Following the type theoretical method of Bushnell--Kutzko, we construct simple types $(\widetilde{J},\widetilde{\lambda})$ of $\widetilde{G}$ and show that they are types related to certain Bernstein blocks of $\mathrm{Rep}(\widetilde{G})$ (in the $n=1$ case, they relate to the Bernstein blocks containing discrete series representations). In particular, we show that every cuspidal representation of $\widetilde{G}$ (or more generally, of a Levi subgroup $\widetilde{M}$ of $\widetilde{G}$) contains a maximal simple type and is constructed via the compact induction of a related extended maximal simple type. We also describe the Hecke algebra $\mathcal{H}(\widetilde{G},\widetilde{\lambda})$ of $(\widetilde{J},\widetilde{\lambda})$ and show that it is indeed an affine Hecke algebra of type A if $\widetilde{G}$ is either a Kazhdan--Patterson cover or the Savin's cover. Finally, we give certain applications of the above theory, for instance, we calculate the Whittaker dimension of an irreducible representation of $\widetilde{G}$ for the above two special covers.