Frank Thorne (Univ. South Carolina)
|Newforms for ramified U(2,1)|
|Exponential sums associated to prehomogeneous vector spaces over finite fields|
|Symmetric determinantal representation of plane curves over global fields|
|A refinement of Zagier’s conjecture in terms of partial derivatives of Shintani L-functions|
|Dirichlet series of 3 variables and Koecher-Maass series of non-holomorphic Siegel-Eisenstein series|
Newforms for ramified U(2,1)
Let G be a unitary group in three variables defined over a p-adic field. If G is associated to a quadratic unramified extension E over F, a theory of newforms for G and its application to Rankin-Selberg integral are known. In particular, zeta integrals of newforms attain L-factors. In this talk, we consider the case when E/F is ramified.
Exponential sums associated to prehomogeneous vector spaces over finite fields
Let (G, V) be a prehomogeneous vector space. When (G, V) is defined over a finite field, it is natural to associate a multidimensional Gauss sum -- i.e., a Fourier transform associated to the characteristic function of any of the G-orbits.
I will spend the first part of the talk explaining how they arise and how previous work on these sums has led to arithmetic applications. I will then discuss ongoing joint work with Takashi Taniguchi; we have developed a simple method to evaluate these sums and have succeeded in applying it in a variety of cases. I will then speculate about potential future applications.
Symmetric determinantal representation of plane curves over global fields
Symmetric determinantal representations (SDPs) of plane curves are one of the classical subjects in algebraic geometry. Over the algebraically closed field of characteristic zero, it is known that SDPs of smooth plane curves corresponds to the special kind of line bundles, non-effective theta characteristics on the curve. In this talk, we give a general treatment of SDPs regardless of the base fields, and show applications on the local- global principles of SDPs.
A refinement of Zagier’s conjecture in terms of partial derivatives of Shintani L-functions
Zagier’s conjecture on the special values of the partial zeta functions of an abelian extension of number fields is an analog of the celebrated Stark’s conjecture. While Stark’s conjecture deals with the special value at s=1, Zagier’s conjecture deals with the special values at s≧2, where the Bloch groups and the polylogarithm functions play similar roles for the multiplicative group and the logarithm function. Moreover, Zagier constructed a complex-valued ideal class invariants whose real part equal to the partial zeta values, and gave a refined version of his conjecture for imaginary quadratic number fields. I will give an explanation of Zagier’s new class invariants in terms of the Shintani L-functions, and generalize the refined version of his conjecture to number fields with a single complex place.
Dirichlet series of 3 variables and Koecher-Maass series of non-holomorphic Siegel-Eisenstein series
As we can see from the Shintani zeta functions of symmetric matrices, some exceptional treatments are required for the study of the Dirichlet series associated with 2 by 2 indefinite symmetric matrices. Analogous problem arises when we try to define a Dirichlet series associated with non-holomorphic Siegel modular forms. The main goal of this talk is to define a Koecher-Maass series of indefinite Fourier coefficients of the degree 2 non-holomorphic Siegel-Eisenstein series reasonably in the sense that the associated Koecher-Maass series has a meromorphic continuation and a functional equation.
本研究集会は科学研究費補助金（課題番号 25707002, 24654005） から補助を受けています．