Number Theory Seminars 2009
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September 10, 2009 MingLun Hsieh (National Taiwan University) Title: Eisenstein congruence and Iwasawa main conjecture for CM fields Abstract: We prove Iwasawa main conjecture for CM fields in certain cases by the study of Eisenstein congruence on the quasisplit unitary group of degree three.I will talk about some ingredients in the proof from results on the nonvanishing of Hecke Lvaleus modulo p proved by Hida September 10, 2009 Lee, Yoonbok (Yonsei University) Title: On the zeros of Epstein zeta functions Abstract: We investigate the zeros of Epstein zeta functions associated with positive definite quadratics form with rational coefficients whose class number is bigger than 1. In this talk, we discuss previous results of Davenport and Heilbronn, Voronin, Bombieri and Mueller, and improve their results. This talk will be the same as in the conference "Zeta function days in seoul" in Yonsei University. September 23, 2009 Andreas Bender (KIAS) Title: Quantitative aspects of the Goldbach and Twin Prime conjectures in F_q[t] Abstract: Heuristic considerations lead to a quantitative version of the Goldbach conjecture, predicting in how many ways an even number larger than 2 can be written as the sum of two primes. We shall see that with the integers replaced by F_q[t], such a conjecture can be proved. An analogous result about the twin prime conjecture will also be discussed. September 24, 2009 Yoo, SangBum (Seoul National University) Title: Stringy Efunction and intersection cohomology of the moduli space of Higgs bundles over an algebraic curve Abstract: Let M be the moduli space of semistable rank 2 Higgs bundles with trivial determinant over a smooth projective curve X of genus g greater than or equal to 2. We provide explicit formulae for the stringy Efunction and the intersection cohomology of M. A formula for the intersection cohomology of M can be obtained under some assumption about the equivariant cohomological formula for the blowup of a singular variety. In this talk, we will introduce the moduli space of semistable Higgs bundles, the stringy Efunction and the intersection cohomology. And then we will explain the main idea of computations for these invariants.
October 9, 2009 Lee, Jung Jo (NIMS) Title: A work of Kolyvagin on Birch and SwinnertonDyer conjecutre Abstract: A basic Galois cohomology will be explained in the first part. Then the work of Kolyvagin on Birch and SwinnertonDyer conjecutre will be explained in the second part. The idea of Euler system coming from Heegner points will be introduced.
October 22, 2009 Lee, Jungyun (ASARC, KAIST) Title: Behavior of Hecke Lfunction of real quadratic fields at s=0 and related problems Abstract: (A part of the abstract. Here is the complete one in pdf format.) In this talk, I am going to explain how this property is applied in solving class number problems of real quadratic fields, caliber number problems and in determination of all Rabinowitch polynomials. Furthermore, we give a criterion for (K_n, \chi_n) to yield the linearity property of the special value of Hecke Lfunction using ZagierYamanoto version of Kronecker limit formula.
November 19, 2009 Kang, SoonYi (ASARC, KAIST) Title: Mock modular forms with small halfintegral weights Abstract: A mock modular form is the holomprphic part of a harmonic weak Maass form. In particular, Ramanujan's mock theta function and the generating series of the traces of singular moduli are mock modular forms of weights 1/2 and 3/2, respectively. In this talk, we will survey some recent progress in these mock modular forms.
November 26, 2009 Jun, Byungheup (Konkuk University) Title: Beside Minkowski theory December 3, 2009 Park, Seungkook (KIAS) Title: Coset bounds for algebraic geometric codes Abstract: We use the coset bounds to find the minimum distance of the twopoint Hermitian code.
December 10, 2009 Lee, Jun Ho (ASARC, KAIST) Title: Evaluation of Dedekind zeta functions at s = 1 of the simplest quartic fields Abstract: The simplest quartic field was introduced by M.Gras and studied by A. J. Lazarus. In this talk, we will evaluate the values of the Dedekind zeta functions at s = 1 of the simplest quartic fields. We first introduce Siegel's formula for the values of the Dedekind zeta function of totally real number field at negative odd integers, and will apply Siegel's formula to the simplest quartic fields. As a result, we compute the values of the Dedekind zeta function at s = 1 of the simplest quartic fields. Finally,we talk about the divisibility of this zeta values.
