Automorphic Forms and Related Zeta Functions

Transcription

1 Automorphic Forms and Related Zeta Functions Jan. 20 (Mon) 13:30-14:30 Hiro-aki Narita (Kumamoto University) Title: Lifting from Maass cusp forms for Γ 0 (2) to cusp forms on GL(2) over a division quaternion algebra Abstract: For a division quaternion algebra D with the discriminant two, we construct Maass cusp forms on GL 2 (D) explicitly by some lifting from Maass cusp forms for Γ 0 (2). This is an analogue of the Saito-Kurokawa lifting. We expect that this lifting leads to a construction of a CAP representation of GL 2 (D), which is an inner form of GL(4). The aim of the talk is to report recent progress of our study on this. This is a joint work with Masanori Muto and Ameya Pitale. 14:45-15:45 Ameya Pitale (University of Oklahoma) Title: Test vectors and central L-values for GL(2) Abstract: In this talk, I will present some recent joint work with Daniel File and Kimball Martin on test vectors for the Waldspurger functional for local representations of GL(2). The study of these test vectors was first done by Gross and Prasad and their results apply to the case where there is no joint ramification. We have obtained these test vectors in the case of joint ramification under an assumption. These test vectors form the input of a relative trace formula identity due to Jacquet and Chen, which helps obtain explicit central L-value formulas for L-functions of GL(2) twisted by characters of a quadratic extension. This leads to some non-vanishing and non-vanishing mod p results. 16:00-17:00 Minoru Hirose (Kyoto University) Title: On the normalized Shintani L-function and Hecke L-function of totally real fields Abstract: The normalized Shintani L-function is a several variable generalization of a classical Hurwitz-Lerch zeta function. It is a modification of the Shintani zeta function. The normalized Shintani L-function is holomorphic and satisfies a good functional equation. We proved that any Hecke L-function of a totally real field can be expressed as a diagonal part of some normalized Shintani L-function. This gives a good several variable generalization of a Hecke L-function of a totally real field which has desirable trivial zeros. This also gives a new proof of the functional equation of the Hecke L-functions of totally real fields. Our result is in the paper On the theory of normalized Shintani L-function and its application to L-function available at Jan. 21 (Tue) 9:45-10:45 Neven Grbac (University of Rijeka) Title: Analytic properties of automorphic L-functions and Arthur classification Abstract: Langlands spectral theory describes the residual spectrum of a reductive group in terms of intertwining operators and analytic properties of automorphic L-functions.

2 On the other hand, according to Arthur s endoscopic classification of automorphic representations of classical groups (due to Mok for quasi-split unitary groups), an Arthur parameter is attached to every residual representation. Comparing the two approaches yields information on the analytic properties of automorphic L-functions. In the talk we explain how this general idea is applied to prove the holomorphy in the critical strip of the complete symmetric and exterior square, and Asai L-functions. 11:00-12:00 Yoshi-Hiro Ishikawa (Okayama University) Title: Towards rationality of critical value of the standard L-function for U(2, 1) Abstract: We report a rationality result on special value of the standard L-function. The method is to study a zeta integral for generic cusp forms on U(2, 1). We do not put assumption on non-vanishing of Archimedean component. 13:30-14:30 Keiju Sono (Tokyo University of Agriculture and Technology) Title: Pair correlation of low-lying zeros of quadratic L-functions Abstract: We consider the pair correlation of the zeros of quadratic L-functions near the real axis. Assuming the Generalized Riemann Hypothesis (GRH), we give an asymptotic formula involving non-trivial zeros of quadratic L-functions. We also introduce several results on the multiplicity of zeros of quadratic L-functions at s = 1/2 and the average distance of low-lying zeros. 14:45-15:45 Shingo Sugiyama (Osaka University) Title: Relative trace formulas and subconvexity estimates of L-functions for Hilbert modular forms Abstract: Ramakrishnan and Rogawski gave an asymptotic formula for a mean of central L-values attached to the imaginary quadratic base change of elliptic holomorphic cusp forms with prime level as the level tends to infinity. Feigon and Whitehouse extended their result to the case of Hilbert cusp forms with square free level when the quadratic extention involved is totally imaginary. In this talk, we report our recent generalization of their asymptotic formula for Hilbert cusp forms to arbitrary levels, dropping the totally imaginary condition of the quadratic extension at the same time. As an application of this, we obtain a subconvexity bound of quadratic base change L-functions for Hilbert cusp forms in the weight aspect. This is a joint work with Masao Tsuzuki (Sophia University). 16:00-17:00 Masao Tsuzuki (Sophia University) Title: Equidistribution of Hecke operators on special cycles of compact unitary Shimura varieties Abstract: In a hermitian space over a totally real number fied, codimension one subspaces perpendicular to anisotropic vectors yield cycles on the Shimura variety of unitary group in different levels, which, as a whole, span a subspace of the derham cohomology group stable under the Hecke operators. When the Shimura variety is compact, we obtain an asymptotic formula of matrix coefficients of such Hecke operators on the special cycles as the level grows along a tower of pricipal congruence subgroups. We also report an explicit computation of relative trace formula which is used to prove the asymptotic formula.

3 Jan. 22 (Wed) 9:45-10:45 Kazunari Sugiyama (Chiba Institute of Technology) Title: Automorphic pairs of distributions and its application to explicit constructions of Maass forms Abstract: Firstly, after T. Suzuki (1979), we define automorphic pairs of distributions on (GL(1),V(1)), and associated Dirichlet series. This Dirichlet series satisfy the same functional equation as that satisfied by L-functions associated with Maass wave forms. Secondly, we explain that the automorphic pairs correspond to (the dual of) the principal series representations of SL(2, R), and the principal series representations correspond to the space of the eigenfunctions of an invariant differential operator via the Poisson transform. These correspondences give rise to a converse theorem for Maass forms. Finally, by using the zeta functions associated with a certain prehomogeneous vector space related with quadratic forms, we construct Dirichlet series whichi satisfy the assumptions of our converse theorem. This talk is based on a joint work with Fumihiro Sato, Keita Tamura (Rikkyo University), Tadashi Miyazaki (Kitasato University), and Takahiko Ueno (St. Marianna University School of Medicine). 11:00-12:00 Tadashi Miyazaki (Kitasato University) Title: Automorphic pairs of distributions on (GL(1) SO(q),V(q)) Abstract: Fumihiro Sato and Keita Tamura defined automorphic pairs of distributions on a prehomogeneous vector space of commutative parabolic type, and proved that their Dirichlet series have meromorphic continuations and satisfy functional equations. In this talk, we will discuss automorphic pairs of distributions on (GL(1) SO(q),V(q)). We will introduce the converse theorem for them, and the relation with automorphic forms on SO(1,q+ 1). This is a joint work with Fumihiro Sato. 13:20-14:20 Yumiko Hironaka (Waseda University), Yasushi Komori (Rikkyo University) Title: On spherical functions on the space of p-adic unitary hermitian matrices in the case of odd size Abstract: We consider the space of unitary hermitian matrices of odd size with respect to an unramified quadratic extension of p-adic field of odd residual characteristic. We determine the Cartan decpomposition, all the spherical functions, and the Plancherel formula. We use our previous general expression formula to obtain the explicit formula of spherical functions, whose main terms can be expressed by a kind of Macdonald polynomials. We compare the results to those for our previous even size case. 14:35-15:35 Seyfi Turkelli (West Illinois University) Title: Lefschetz Numbers and the Cohomology of Bianchi Groups Abstract: Bianchi groups are the congruence subgroups of SL(2,O) where O is the ring of integers of an imaginary quadratic field. In other words, they are imaginary quadratic field analogs of the classical modular groups, i.e. congruence subgroups of SL(2, Z). As in the classical case, the cohomology of Bianchi groups with certain coefficient modules are in

4 fact the space of certain automorphic forms of cohomological type. Many problems about the cohomology of these arithmetic groups that are answered in their classical setting are still open. One of the fundamental open problem is to give a closed formula for the dimension of these cohomology spaces for a given Bianchi group (i.e. level) and coefficient system (i.e. weight). We will address this problem in the talk. After introducing Bianchi groups briefly, I will talk about a recent result on the dimension of the cohomology of Bianchi groups. More precisely, I will talk about a method, originally due to Harder, and how it is used to give a lower bound for the dimension of these cohomology spaces. This is a joint work with M. H. Sengun. 15:50-16:50 Tomoyoshi Ibukiyama (Osaka University) Title: Higher Spherical Polynomials (joint work with Don Zagier) Abstract: Since Don Zagier and I finished recently the first part of the paper on works lasting from 1990, I would like to report the outline of that work. Related with differential operators on Siegel modular forms of general degree n which behaves well under restriction to the diagonals, we can define a new special polynomials defined by a system of differential operators, which is a generalization of Legendre or Gegenbauer polynomials which appears for n = 2. We developed a theory on these polynomials independently from the relation with modular forms. This includes, for example, a general theory on dimensions, scalar product and orthogonality, related Lie algebras and two canonical basis, and explicit theory on polynomials (particularly for n = 3), and construction of universal generating functions for general n, their algebraicity and non-algebraicity for small n. 16:50 - On the next RIMS conference 18:00 - Banquet Jan. 23 (Thu) 9:45-10:45 Tomoya Machide (National Institute of Informatics) Title: Results of double zeta values related to modular forms on full modular group and Ramanujan s formula for Bernoulli numbers Abstract: It is known that the upper bound of the dimension of the vector space spanned by double zeta values of a given weight can be written in terms of the dimension of the space of modular forms of the weight. In this talk, we give specific sets of generators whose number equals the upper bound. We also introduce identities of double zeta values which are related to identities of Bernoulli numbers proved by Ramanujan. 11:00-12:00 Takashi Nakamura (Tokyo University of Science) Title: Zeros of the derivatives of zeta functions Abstract: We show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. By using this result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many

5 complex zeros off the critical line. Zeros of the derivatives of these zeta functions are studied as well. 13:30-14:30 Ren He Su (Kyoto University) Title: Fourier coefficients for Eisenstein series in the Kohnen plus space Abstract: The Hilbert modular forms in the Kohnen plus space have half-integral weights and are characterized by the behavior of its Fourier coefficients. In the talk, we construct an Eisenstein series in the space and give an explicit formula of its Fourier coefficients. This gives a generalization of the modular forms H r introduced by Cohen in :45-15:45 Sho Takemori (Kyoto University) Title: A congruence property of Siegel modular forms of degree 2 and weight 47, 71, 89 Abstract: For k =35, 47, 71, 89, we put p =(2k 1)/3. Let X k be a suitable normalization of a Hecke eigenform of degree two, level 1 and weight k. For a half integral semipositive definite matrix T, a(t ; X k ) denotes the T th Fourier coefficient of X k. Kikuta, Kodama and Nagaoka proved that (det T )a(t ; X 35 ) 0 mod 23 for all T. In this talk, I introduce that there exist higher weight examples. More precisely, I introduce that (det T )a(t ; X k ) 0modP for k =47, 71, 89, where P is a prime above p. I also introduce computational examples that suggest relation between Hecke polynomials of X k and an elliptic cusp form of weight (k + 1)/3 modulo p. Jan. 24 (Fri) 9:45-10:45 Hiraku Atobe (Kyoto University) Title: Pullbacks of hermitian Maass lifts Abstract: Ichino gave an explicit formula for pullbacks of Saito-Kurokawa lifts in terms of central critical values of L-functions for SL 2 GL 2. Ichino and Ikeda gave an explicit formula for the restriction of hermitian Maass lifts of degree 2 to the Siegel upper half space of degree 2 in terms of central critical values of triple product L-functions. These pullbacks have been used to study the algebraicity of critical values of certain automorphic L-functions. In this talk, we consider pullbacks of hermitian Maass lifts to the space of diagonal matrices. By using these pullbacks, we give an explicit formula for the central values of L-functions for GL 2 GL 2. 11:00-12:00 Masaaki Furusawa (Osaka City University) Title: On special values of certain L-functions Abstract: (joint work with Kazuki Morimoro) In this talk, we would like to discuss an algebraicity result concerning special values at critical points, in the sense of Deligne, of tensor product L-functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite, and, cuspidal representations of GL(2) corresponding to primitive cusp forms, over totally real number fields. Our result conforms with the conjecture of Deligne on special values of motivic L-functions, according to the computation of the Deligne periods in this case by Kazuki Morimoto.