The 26th Autumn Workshop on Number Theory 

• Youngmin Lee: Exceptional and Equidistributed Eigenvalues of Automorphic Representations of GL(2)

Abstract: The Ramanujan-Petersson conjecture for GL(2) predicts that cuspidal automorphic representations of GL(2) are tempered at all places.  This conjecture remains open for automorphic representations of GL(2) whose archimedean components are principal series.  

In this talk, I will present two results related to the Ramanujan-Petersson conjecture for GL(2) over arbitrary number fields whose local components are principal series at every archimedean place.  

The first part concerns the archimedean places, where I will discuss an upper bound on the number of automorphic representations that do not satisfy the Ramanujan-Petersson conjecture at all archimedean places.  

The second part focuses on the non-archimedean places, where I will introduce the vertical Sato–Tate law describing the equidistribution of Hecke eigenvalues at a fixed prime ideal, and discuss its application to the Ramanujan–Petersson conjecture at non-archimedean places.  

These results are obtained using the Arthur-Selberg trace formula.  This is joint work with Dohoon Choi, Min Lee, and Subong Lim.

• Yuchan Lee: Diophantine analysis and Arthur's trace formula

Abstract: A Diophantine problem investigating the asymptotic behavior of integral points of a variety $X$ has been thoroughly studied over $\mathbb{Q}$. When $X$ is a homogeneous space of a simply connected and semisimple group, the line of study has been further extended to general number fields. On the other hand, Arthur's trace formula - one of the central tools in the Langlands program - provides a powerful means of accessing information about automorphic representations.

In this talk, I will explain my recent result that connects these two areas; Diophantine analysis and the geometric side of Arthur's trace formula. As an application, I will present an asymptotic formula for the number of integral matrices with a fixed characteristic polynomial.

• Qirui Li: Hecke Operators in Fundamental Lemmas for Gross–Zagier Type Formulas on GL(2n)

Abstract: Fundamental lemmas play a central role in the Langlands program and in applications to the study of L-functions via the relative trace formula. Guo proposed a fundamental lemma generalizing Waldspurger’s formula to general linear groups GL(2n), and proved it for the unit element by establishing an equivalence with the quadratic base-change fundamental lemma. For decades, this result was thought to hold only in the unit case. We show that the method extends to a slightly broader class of Hecke functions, though these functions reveal the natural limits of the approach. Our construction also provides fundamental tools for studying general Hecke functions in the arithmetic setting, which are essential for proving Gross–Zagier–type formulas for GL(2n).

• Weixiao Lu: The global GGP conjecture for Fourier-Jacobi period on unitary group

Abstract: The Gan-Gross-Prasad(GGP) conjectures predict a relation between certain periods on the classical group and the central value of L-function. In this talk, we will explain the proof of the GGP conjecture for the Fourier-Jacobi period on the unitary group, based on the comparison of relative trace formulas. We will explain the key global and local harmonic analysis input to achieve this comparison.

• Jiajun Ma: Theta Correspondence and Endo-parameters

Abstract: The theory of semisimple characters and endo-parameters provides a powerful framework for studying smooth representations of p-adic classical groups over fields of characteristic different from p and 2 (where p ≠ 2). This theory was initiated by Bushnell, Kutzko, and Henniart for general linear groups and extended to classical groups through the work of Kurinczuk, Skodlerack, and Stevens. Endo-parameters can be viewed as restrictions of Langlands parameters to the wild inertial group, and they provide a decomposition of the category of smooth representations into subcategories that are coarser than the classical Bernstein decomposition.  

In this talk, I will establish a theta correspondence for semisimple characters and demonstrate that the theory of endo-parameters is compatible with theta correspondence. I will then present two applications of this result, including an explicit description of the correspondence between supercuspidal representations and a reduction-to-depth-zero theorem for theta correspondence.  

The talk is based on ongoing work joint with Loke, Stevens, and Trias.

• Yoichi Mieda: On supercuspidal representations in the l-adic cohomology of some local Shimura varieties

Abstract: A local Shimura variety is a p-adic analogue of a Shimura variety. Its l-adic cohomology is expected to be related to the local Langlands correspondence. In the Lubin-Tate case, a supercuspidal representation appears only in the middle degree. In the GSp(4) case, a supercuspidal representation appears only in the degrees 3,4, and it appears in the degree 4 if and only if its L-parameter has non-trivial SL(2)-part. In this talk, I will explain my recent results in this direction for more general cases.

• Yuki Nakata: Local Langlands correspondence for covering groups of tori

Abstract: We determine the group S indexing the representations in a packet in the local Langlands correspondence for a certain covering group of an algebraic torus. This correspondence is proposed by Weissman for the functorial covers of a torus defined by Brylinski and Deligne. Inspired by Weissman's project to pursue the Langlands program for covering groups, we extend his result on the local correspondence so that we can admit some ramification. To achieve this, we apply an orthogonal relation in the local Tate duality.

• Jhan-Cyuan Syu: Categorical Local Langlands Program and Perverse Sheaves

Abstract: Based on Fargues-Scholze geometrization program over non-archimedean local fields, the conjectural set-theoretical local Langlands correspondence can be extended to a conjectural equivalence of two categories of sheaves on some moduli stacks. I will first introduce the main conjecture in the categorical local Langlands program and then discuss perverse sheaves both on the automorphic side and the spectral side in this framework.

• Yugo Takanashi: On Sauvageot’s Density Theorem

Abstract: We prove a new result on the trace Paley–Wiener theorem for p-adic reductive groups, extending a work of Muić. Our statement gives finer control on the image of the Hecke algebra under the Fourier transform. As an application, we obtain a complete proof of Sauvageot’s density result for Fourier transforms of Hecke–algebra functions in an ambient function space,  closing the gaps highlighted by Nelson–Venkatesh and others. 

• Geo Kam Fai Tam: Endoscopic liftings of supercuspidal representations for classical groups

Abstract: Let $G$ be a classical group (orthogonal, symplectic, or unitary) over a $p$-adic field $F$, and $\pi$ be a supercuspidal representation of $G(F)$. We propose a general strategy to express the Langlands parameter of $\pi$, only requiring the residual characteristic $p$ to be odd, with proven examples by the speaker (partly joint with Corinne Blondel) and others. To briefly explain the strategy, we apply M{\oe}glin’s theory of Jordan blocks and Bushnell-Kutzko’s theory of covering types to determine the endoscopic lift of $\pi$ into the general linear group whose dual expresses the dual group of $G$ as a complex matrix group. The cuspidal support of this lifted representation is expressed in terms of the inducing type of $\pi$ defined by Stevens’ construction of supercuspidal representations using skew semi-simple strata. This strategy is completely on the representation side of the LLC.