9th Kyoto conference on automorphic forms

Program and abstracts

Date: June. 24 – June. 26, 2022
Location: Bld. 3, Room 110
This conference is partially supported by the grant: Grant-in-Aid (C) (20K03512).
June 24, 13:00–14:30 Suzuki Miyu (Kanazawa Univeristy)
Title: Linear periods on representations of GL(n) and its inner forms
Abstract: Let G be an inner form of GL(n) over a local ﬁeld F and H be its symmetricsubgroup which is an inner form of GL(m) over a quadratic extension of F . An Hinvariant linear form on a representation of G is called a linear period. When F is a
non-archimedean local ﬁeld of characteristic 0, Prasad and Takloo-Bighash predictedan ”epsilon dichotomy” for linear periods. That means, they formulated a conjecturalnecessary condition for a representation to have non-zero linear periods in terms oflocal root numbers. In this talk, I will introduce some recent progress made on thisconjecture and related problems. A part of this talk is based on my joint work with
Hang Xue and Hiroyoshi Tamori.
14:45–15:45

Sugiyama Shingo (Nihon University)

Title: The one-level density for Dirichlet L-functions weighted by L-values
Abstract: Katz-Sarnak suggested that low-lying zeros of L-functions in a family are
distributed like eigenvalues of random matrices in a compact matrix group. Inspiredby Kowalski-Saha-Tsimerman, the density of low-lying zeros of L-functions in a family
weighted by central L-values has been studied by Knightly-Reno, the speaker, andFazzari. In this talk, the speaker suggests a conjecture on the density of low-lyingzeros of L-functions weighted by general L-values, and gives a result for a familyof Dirichlet L-functions. This is a joint work with Ade Irma Suriajaya (KyushuUniversity).
16:00–17:00 Katsurada, Hidenori (Hokkaido University)
Title: Estimates for Fourier coeﬃcients of the Duke-Imamoglu-Ikeda lift
Abstract: For a Hecke eigenform h in the Kohnen plus subspace of weight k−n/2+1/2for Γ0 (4), let In (h) be the Duke-Imamoglu-Ikeda lift of h to Sk (Spn (Z)). We givean estimate for the Fourier coeﬃcient cIn (h) (B) of In (h) at a positive deﬁnite halfintegral matrix B, which is better than the usual Hecke’s bound. We also estimate
the cIn (h) (B) in terms of the minors of B. This is a joint work with T. Ikeda.
June 25, 9:30-10:30 Morimoto Kazuki (Kobe University)
Title : On Ichino-Ikeda type formula of Whittaker periods for even unitary groups
Abstract : Lapid and Mao formulated a conjecture on Ichino-Ikeda type formulaof Whittaker periods for quasi-split reductive groups. In the unitary group case,2they showed that their conjecture is reduced to certain local identities. In my previous work, I showed this local identity for even unitary groups at non-split nonarchimedean places. In this talk, I will give a proof of this local identity in the split
case at both non-archimedean and archimedean places. As a consequence, we obtain the formula for irreducible cuspidal generic automorphic representations for evenunitary groups when its local components at real places are discrete series. If timepermits, I will explain an application of this formula to Ichino-Ikeda type formula ofBessel periods for (U (2n), U (1)).
10:45-11:45 Yasuda, Seidai (Hokkaido University)
Title: Local new forms for the general linear groups from topos theoretic viewpoint.
Abstract : In a joint work with Hiraku Atobe and Satoshi Kondo, the speaker constructed a theory of local new forms for irreducible smooth representations of thegeneral linear group over nonarchimedean local ﬁelds of characteristic zero which
are not necessarily generic. In this talk, the speaker will talk about a topos theoretic
background of the theory, explain how the relevant compact open subgroups naturallyarize from the background, and give another proof of a kind of Mackey decomposition
that plays a crucial role in the theory. Moreover, the speaker will explain an explicit
construction, given in terms of the sheaf corresponding to the represenation, of local
newforms for the ladder representations with nontrivial L-factors, and by using this,gives a description of local L and epsilon factors in terns of Hecke operators. Almost
all part of the talk is based on the speaker’s joint work with Satoshi Kondo, and somepart of the talk is based on the speaker’s joint work with Hiraku Atobe and Satoshi
Kondo.
June 26, 9:30–10:30 Ohara Kazuma (University of Tokyo)
Ttitle: On the formal degree conjecture for non-singular supercuspidal representations
Abstract: We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein’s work proving the formal degree conjecture for regularsupercuspidal representations. The main diﬀerence between our work and Schwein’swork is that in non-singular case, the Deligne–Lusztig representations can be reducible, and the S-groups are not necessarily abelian. Therefore, we have to compare
the dimensions of irreducible constituents of the Deligne–Lusztig representations andthe dimensions of irreducible representations of S-groups.
10:45–11:45 Noguchi Hiroshi (Kyoto University)
Title: Genuine characters of local metaplectic groups and Hilbert modular forms ofhalf-integral weight
Abstract: We consider the Hilbert modular analogy of the Dedekind eta function.
Let F be a totally real number ﬁeld and o the ring of integers of F . We study thetafunctions which are Hilbert modular forms of half-integral weight for SL2 (o).
First, we study genuine characters of the metaplectic groups of symplectic groups
G. We have determined somewhat when genuine characters exist. In particular, if

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G = Spn (O), where O is the ring of integers of a 2-adic ﬁeld and n ≥ 2, then genuine
characters do not exist. Second, we deﬁne a multiplier system of half-integral weightand determine when it exists. Then we determine the condition of F that there exists
a theta function which is a Hilbert modular form of half-integral weight for SL2 (o).