Abstract
Jan. 23 (Mon.)
13:20 – 14:20 Takao Komatsu (Zhejiang Sci-Tech University)
Title: Arithmetic approach to p-numerical semigroups
Abstract: The p-numerical semigroup is a generalization of the algebraically deﬁned numerical semigroup by focusing on the number of solutions of the corresponding linear diophantine equation. Numerical semigroups have been studied from various angles, such as algebraic curves and commutative
algebra, but in this talk I will give their properties from the number-theoretic side.
14:40 – 15:40 Koji Tasaka (Aichi Prefectural University)
Title: Spherical designs and modular forms of the D4 root lattice
Abstract: We will talk about a spherical design, one of important combinatorial structures, of theshells of the D4 lattice and its application to the theory of modular forms and Hecke operators on
the space of harmonic polynomials. This is a joint work with Masatake Hirao and Hiroshi Nozaki.
16:00 – 17:00 Kohta Gejima (Osaka Metropolitan University)
Title: Determining cusp forms by critical values of Rankin-Selberg L-functions
Abstract: In this talk I will describe that a holomorphic cusp form g is uniquely determined bycertain critical values of the family of Rankin–Selberg L-functions L(f ⊗ g, s), where f runs over aﬁxed orthogonal basis of the space of cusp forms of weight k. The ideas employed are simpler thanthose used in the previous results for the central value.
Jan. 24 (Tue.)
9:20 – 10:20 (EST 19:20 – 20:20, Jan. 23 (Mon.)) Peter Humphries (University of Virginia)
Title: Newform Theory
Abstract: When do there exist automorphic forms that are invariant under distinguished congruencesubgroups? When is a period integral of automorphic forms exactly equal to an L-function? How can
one quantify the complexity of an automorphic representation? We shall discuss three interrelated
notions in the theory of automorphic forms and automorphic representations: newforms, L-functions,and conductors. In particular, we cover how to deﬁne the newform associated to an automorphic
representation of GLn , how to realise certain L-functions as period integrals involving newforms,
and how to quantify the ramiﬁcation of an automorphic representation in terms of properties of thenewform. A key emphasis is the union of approaches to deﬁning newforms in both nonarchimedean
and archimedean settings. Finally, we will brieﬂy discuss notions of newforms for groups other thanGLn .
10:40 – 11:40 Eric Stade (University of Colorado Boulder)
Title: An asymptotic orthogonality relation for GL(n, R) and recurrence relations for Mellin transformsof GL(n, R) Whittaker functions
Abstract: We discuss joint work with Dorian Goldfeld and Michael Woodbury on an asymptotic
orthogonality relation, with a power-savings error term, for Fourier coeﬃcients of GL(n, R) Maasscusp forms.
Our proof of this orthogonality relation relies on two conjectures, one concerning lower bounds
for certain Rankin-Selberg L-functions, and the other concerning recurrence relations for Mellintransforms of GL(n, R) Whittaker functions. We further discuss joint work with Taku Ishii towards
the second conjecture.
1

13:20 – 14:20 Kenichi Namikawa (Tokyo Denki University)
Title: An integrality of critical values of Rankin-Selberg L-functions for GL(n) × GL(n − 1)
Abstract: By using the generalized modular symbol method due to Kazhdan-Mazur-Schmidt, Raghuram proved the algebraicity of ratios of period integrals of Rankin-Selberg L-functions for GL(n) ×GL(n − 1) to Whittaker periods. In this talk, we consider a reﬁnement of the generalized modular
symbol method and we prove the algebraicity of critical values of Rankin-Selberg L-functions for
GL(n) × GL(n − 1) if the base ﬁelds are totally imaginary. Key ingredients of the proof are a description of local systems on GL(n) by Gel’fand-Tsetlin basis and an explicit formula for the archimedean
local zeta integrals. Furthermore, we also propose a formulation for p-integral properties of these
critical values. This is a joint work with Takashi Hara (Tsuda university) and Tadashi Miyazaki(Kitasato university).
14:40 – 15:40 Kazuki Morimoto (Kobe University)
Title: On Ichino-Ikeda type formula of Bessel periods for (U(2n), U(1)) and (GL(2n), GL(1))
Abstract: Under certain assumptions at archimedean places, we prove Ichino-Ikeda type formulaof Bessel periods for (U(2n), U(1)) conjectured by Y. Liu. We prove it by combining theta lifts
from U(2n) to U(2n) and Ichino-Ikeda type formula of Whittaker periods for U(2n). In a similarargument, we also show Ichino-Ikeda type formula of Bessel periods for (GL(2n), GL(1)) for any
irreducible cuspidal tempered automorphic representations. This talk is based on a joint work with
Masaaki Furusawa.
Jan. 25 (Wed.)
9:20 – 10:20 Yugo Takanashi (University of Tokyo)
Title: Parity of conjugate self-dual representations of inner forms of GLn over p-adic ﬁelds
Abstract: There are two parametrizations of discrete series representations of GLn over p-adic ﬁelds.
One is the local Langlands correspondence, and the other is the local Jacquet-Langlands correspondence. The composite of these two maps the discrete series representations of an inner form of GL n toGalois representations called discrete L-parameters. On the other hand, we can deﬁne the parity foreach self-dual representation depending on whether the representation is orthogonal or symplectic.
The composite preserves the notion of self-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed the transformation law, and Mieda proved its conjugate
self-dual analog under some conditions on groups and representations. We will talk about the proofof the general case of this analog. We use the globalization method, as in the proof of Prasad andRamakrishnan.
10:40 – 11:40 Shuji Horinaga (Nippon telegraph and telephone corporation)
Title: Cuspidal components of Siegel modular forms for large discrete series representations of Sp 4 (R)Abstract: As far as I know, there are no known results of an explicit description of cuspidal components
for non-holomorphic automorphic forms except for nearly holomorphic modular forms. In this talk, weinvestigate the cuspidal components and the structures for automorphic forms on Sp 4 which generatelarge discrete series representations by the explicit formulas of degenerate Whittaker functions oflarge discrete series representations. This talk is based on the joint work with Hiro-aki Narita.
13:20 – 14:20 Takuya Yamauchi (Tohoku University)
Title: Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect
Abstract: In this talk, I will explain my recent work with Kim and Wakatsuki on equidistribution
theorems for holomorphic Siegel cusp forms of general degree in the level aspect.
14:40 – 15:40 Kazunari Sugiyama (Chiba Institute of Technology)
Title: The modularity of Siegel’s zeta functions
2

Abstract: Siegel deﬁned zeta functions associated with indeﬁnite quadratic forms, and proved theiranalytic properties such as analytic continuations and functional equations. Coeﬃcients of these zetafunctions are called measures of representations, and play an important role in the arithmetic theoryof quadratic forms. In a 1938 paper, Siegel made a comment to the eﬀect that the modularity of hiszeta functions would be proved by a suitable converse theorem. Later in a 1951 paper, Siegel provedthat the measures of representations appear as Fourier coeﬃcients of some real analytic automorphic
forms. Further, an explicit formula for Siegel’s zeta functions is proved by Ibukiyama. On the other
hand, it is only recently that papers on Weil-type converse theorems for Maass forms have appeared.
The purpose of the present talk is to accomplish Siegel’s original plan by using a new conversetheorem. It is also shown that ”half” of Siegel’s zeta functions correspond to holomorphic modularforms.
16:00 – 17:00 Masao Tsuzuki (Sophia University)
Title: A relative trace formula on GL(n) and its application
Abstract: The Whittaker functional and the Rankin-Selberg integral are two of the most importantand basic period integrals for automorphic forms on general linear groups. Taking the rational
number ﬁeld for the base ﬁeld, we develop a version of relative trace formulas on GL(n), whose
spectral side involves two of these period integrals with the minimal Eisenstein series being placed
as the GL(n − 1) integrand of the Rankin-Selberg period. On the geometric side, we observe thata simplication is brought by choosing test functions with suﬃciently large mirabolic levels. If time
permits, as an application, we shall report a new non-vanishing result for central standard L-values
for a family of GL(n) Hecke Maass cusp forms with growing prime mirabolic levels and nontrivial
central characters.
Jan. 26 (Thu.)
9:20 – 10:20 Shingo Sugiyama (Nihon University)
Title: Resolvent trace formulas of Hecke operators and optimal estimates of the Hurwitz class numbers
Abstract: The Eichler-Selberg trace formula for a Hecke operator has many applications to dimension
formulas and equidistributions of Hecke eigenvalues. In this talk, we give a resolvent trace formulaof a Hecke operator. We also give an application of the resolvent trace formula to optimal estimatesof an average of the Kronecker-Hurwitz class numbers. This is a joint work with Masao Tsuzuki(Sophia University).
10:40 – 11:40 Yoshinori Mizuno (Tokushima University)
Title: On Hutchinson’s conjecture
Abstract: Let j(τ ) be the elliptic modular function. Following Gross and Zagier, we deﬁneJ(d1 , d2 ) := ∏∏(j(τ1 ) − j(τ2 ))

4
w1 w2

,

[τ1 ],disc(τ1 )=d1 [τ2 ],disc(τ2 )=d2

where di is a negative discriminant, wi is the number of units in the quadratic order Odi and theproduct is taken over all quadratic irrationals τi of discriminant di in the upper-half plane modulo
the action of SL2 (Z). In 1985, Gross and Zagier established a closed formula of J(d1 , d2 )2 when d1 ,d2 are relatively prime negative fundamental discriminants. In 1998, Tim Hutchinson presented aconjectual extension of the closed formula to the case when d1 and d2 are not necessarily fundamental,and gcd(d1 , d2 ) is a power of a prime not dividing the product of the conductors of d1 and d2 . In thistalk, we give a proof of this conjecture when d1 , d2 are fundamental and gcd(d1 , d2 ) is a power of aprime. The proof proceeds along the lines of the second proof given in Gross-Zagier’s paper. Butto do so, we need to study the class number h(d1 , d2 , δ) of pairs of positive-deﬁnite binary quadratic3forms of discriminant d1 , d2 with codiscriminant δ, when gcd(d1 , d2 ) is a power of a prime and δ isarbitrary. We give a closed formula of the number h(d1 , d2 , δ) and relate it to the Fourier coeﬃcientsof Hilbert-Eisenstein series of weight 1 associated to the quadratic order Od1 d2 . Ideal theory of Od1 d2(d d )and the genus character χd11,d22 (of non-fundamental discriminant d1 d2 ) together with an explicit(d d )

formula of the L-function of χd11,d22 are indispensable.
13:20 – 14:20 Shotaro Kimura (Waseda University)
Title: On constructions of modular diﬀerential equations
Abstract: A modular diﬀerential equation is a diﬀerential equation whose solution space satisﬁes
modular invariance. The studies of modular diﬀerential equations appear in a variety of ﬁelds beyond number theory. For example, it has applications to vertex operator algebras, conformal ﬁeld
theory, elliptic genera, and so on. Thus it is an interesting problem to generalize modular diﬀerential equations to various modular forms. In this talk, we introduce the construction and propertiesof the modular diﬀerential equation for skew-holomorphic Jacobi forms which are not holomorphic.
Moreover, we show a uniﬁed construction method of higher-order modular diﬀerential equations forseveral modular forms by Rankin-Cohen brackets.
14:40 – 15:40 Siegfried Böcherer (University of Mannheim)
Title: On equivariant holomorphic diﬀerential operators starting from vector-valued cases
Abstract: The theory of Rankin-Cohen bilinear holomorphic diﬀerential operators is well explored for
scalar-valued cases, mainly by the work of Ibukiyama. Not so much is known when we start from
vector-valued automorphy factors. We will describe some constructions starting from nonholomorphic operators of Maaß-Shimura type. We focus on operators of order one, but by some compatibility
with tensor products we can cover more general situations. For the case of symmetric tensor representations we can however give quite complete results by a direct approach. Some parts of the talkare based on the Mannheim PhD-thesis 2021 by Julia Meister.
Jan. 27 (Fri.)
9:20 – 10:20 Keiichi Gunji (Chiba Institute of Technology)
Title: Simple functional equations satisﬁed by Siegel Eisenstein series of level p
Abstract: Let p be an odd prime number. We consider the functional equations of the Siege Eisensteinseries of level p with trivial or quadratic character. As is well-known, for degree n case, the dimensionof the space of Siege Eisenstein series is (n + 1)-dimensional, thus the functions equations are written
by matrix form. This matrix seems to become complicated, however we can choose simple equations
by considering the U (p)-eigen functions. In this talk we ﬁrst study the U (p)-action on the spaceof Siegel Eisenstein series to get the U (p)-eigen functions. Next we shall give the explicit form of
the functional equations for eigen functions, as a consequence we explain that one can write therepresentative matrix of the functional equations by the products of rather simple matrices.
10:40 – 11:40 Tomoyoshi Ibukiyama (Osaka University)
Title: Diﬀerential operators on Siegel modular forms and Laplace transform
Abstract: The talk has two aims. One is to give a new special basis of the polynomial ring C[T ] incomponents of n × n symmetric matrix T . The basis( Pν (T) ) is characterized by the action of Pν (∂Z )∂son automorphy factors det(CZ + D) , where ∂Z = ∂zij for the vaiable Z of the Siegel upper half
space Hn . (Our theory is quite diﬀerent from the well-known theory by Shimura.) The basis is given
as coeﬃcients of certain explicitly described generating series. The second aim is to apply this to theso-called pullback formula of the Siegel Eisenstein series of degree 2n and determine all necessary
constants appearing there.

4