Number Theory Seminars 2010

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January 7, 2010 Lim, Subong (POSTECH)

Title: Heat operators and Mock Jacobi forms

Abstract: We study a sufficient and necessary condition for Jacobi integrals to have a dual Jacobi form. Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form. We discuss Lerch sums as a typical example of mock Jacobi form.

January 19, 2010 Kim, Byoung Du (Victoria University of Wellington)

Title: Iwasawa theory for modular forms at supersingular primes

Abstract: We will briefly overview past and current trend of Iwasawa theory for modular forms. Our focus will be on supersingular primes.

January 25, 2010 Jeng-Daw Yu (National Taiwan University)

Title: Ordinary crystals with lograithmic poles

Abstract: We study the abstract formalism of ordinary crystals with logarithmic poles over a smooth affine base and give some of their properties. In particular, the canonical coordinates of an ordinary crystal are obtained. The canonical group structure on the moduli will also be addressed if time permits.

February 11, 2010 Kim, Ho-il (Kyungbook National University)

Title: Automorphic forms, generalized Kac-Moody algebras, and BPS states.

Abstract: There have been many works on the automorphic forms, generalized Kac Moody algebras and BPS states (physically stable states?) by many mathematicians and physicists for more than a decade. Moore, Harvey, Borcherds, Nikulin, Kontsevich, Verlinde are some of them. They showed deep relation among number theory(modular forms), Lie algebra, geometry(K3, Calabi Yau varieties) and physics (string theory). In this lecture, we introduce what have been discussed so far and what remain to be settled. This is a self contained talk, focusing on mathematical results even though we mention from what kind of physics they evolved. In the first half of the talk, we give some introduction on generalized Kac-Moody super algebras with examples involving K3 surfaces and in the second half of the talk we apply these tools to understand BPS states in N=4 case. If time permits we explain more on lifting methods and BPS states in N=2 case.

February 25, 2010 Shin-ya Koyama (Toyo University)

Title: Absolute zeta functions

Abstract: For proving the Riemann type hypothesis, it is considered that additive structure for zeros of zeta functions are crucial. In 1995, Manin pointed out that we may use the Kurokawa tensor product for zeta functions, which was the tensor product over ``the field with one element." In this talk, we first give a survey on a recent progress on the Kurokawa tensor product. In the later half in this talk we introduce the absolute zeta functions, which possess additive structure of zeros.

March 11, 2010 Satoshi Kondo ( IPMU (Institute for the Physics and Mathematics of the Universe))

Title: On the rational K-group of an elliptic surface over a finite field

Abstract: This is a joint work with Seidai Yasuda (RIMS). We compute the second rational K-group of an elliptic surface over a finite field. When the generic fiber of the elliptic surface is an elliptic curve, we show that the dimension is greater than or equal to the number of places of split multiplicative reduction.

March 25, 2010 Yoonbok Lee (POSTECH)

Title: Extended Selberg Class

Abstract: The class of functions having Dirichlet series, analytic continuation and functional equation (roughly speaking) is called the extended Selberg class. We study its structure for small degrees and examine zeros of its functions on the rectangular region $\sigma_1 < \Re s < \sigma_2 $ , $0< \Im s < T$ for $ \frac{1}{2}< \sigma_1 <\sigma_2 $. My recent work with Haseo Ki in 2010 is about the zeros of degree one $L$-functions in the extended Selberg class.

April 8, 2010 Hae-Sang Sun (KIAS)

Title: Borel's conjecture and the transcendence of the Iwasawa power series

Abstract: Emil Borel conjectured that every irrational algebraic (p-adic) numbers have base-g-expansion for each positive integer g, each of which digit appears in the expansion with equal frequency. For p-adic version, one considers the p-adic expansion. In the talk, we will present that Borel's conjecture implies the transcendence of the Iwasawa power series, which is essentially the Kubota-Leopoldt p-adic L-function.

April 22, 2010 Lin Han (Inha University)

Title: Non-vanishing for central values of modular L-functions modulo $\ell$ and extension groups

Abstract: Let $\ell\geq 5$ be a prime and $E$ an elliptic curve over $\mathbb{Q}$ of odd square free conductor. Denote by $E_D$ the $D$-quadratic twist of $E$ for a fundamental discriminant $D$ and by $\Sha(E_D)$ the Tate-Shafarevich group of $E_D$. This presentation shows an instance of application of Galois representation to the study of algebraic parts of central critical values of twisted L-series modulo primes $\ell$. This also gives us information on $\Sha(E_D)$. Under the assumption of BSD conjecture, if the order of $\Sha(E_D)$ is not divisible by a prime $\ell\geq 11$ for some negative fundamental discriminant $D$, then there are infinitely many negative fundamental discriminant $D$ such that $\ell \nmid |\Sha(E_D)|$.

May 13, 2010 Bumkyu Cho (POSTECH)

Title: Zagier duality between modular forms of integral weight

Abstract: We show the existence of ``Zagier duality'' between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades. Concerning the isomorphism between the spaces of scalar and vector valued harmonic weak Maass forms of integral weight, Zagier duality between scalar valued ones is derived.

May 27, 2010 Dohoon Choi (Korea Aerospace University)

Title: Exponents of modular forms and harmonic weak Maass forms of weight 0

Abstract: In this talk, we study connections between exponents of modular forms and harmonic weak Maass forms of weight 0 and their applications.

June 9,10, 2010 Kenichi Bannai (Keio University)

Title: $p$-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields

Abstract: This is a joint work in progress with G. Kings. The $p$-adic Beilinson conjecture as formulated by Perrin-Riou gives the precise conjectural interpolation property of $p$-adic $L$-functions, even at points which are non-critical. In these series of talks, assuming certain conditions (namely that the imaginary quadratic field $K$ is of class number one and that $p$ is ordinary), we will show that the $p$-adic $L$-function for Hecke characters associated to imaginary quadratic fields constructed by Vishik-Manin and Katz satisfies at non-critical values the interpolation property predicted by the $p$-adic Beilinson conjecture. Our main tool is the explicit calculation of the $p$-adic realization of the Eisenstein class.

June 24, 2010 Kentaro Ihara (POSTECH)

Title: Period relations on multiple Hecke L-values

Abstract: Around 2005, Manin introduced an iterated integral version of period integral of modular forms. In the talk we discuss the Mellin transformation of the integral then relates the multiple Hecke L-function. Also the linear relations among the special values of the L-function, which generalize the classical period relation, are described explicitly.

July 8, 2010 Chul-hee Lee (UC Berkeley)

Title: Nahm's conjecture

Abstract: The Rogers-Ramanujan identities provide examples of q-hypergeometric series with modularity. Finding the overlap between the classes of q-hypergeometric series and modular functions in general is not a well understood problem. Nahm's conjecture relates this problem to the Bloch group in algebraic K-theory. I will give a brief introduction to this conjecture.

July 15, 2010 Masanori Morishita (Kyushu University)

Title: The universal deformation and the associated homological invariants for hyperbolic knots

Abstract: I will talk about the deformation theory for group representations with its application to knot theory. In particular, I will produce the universal deformation for holonomy representations and the associated homological invariants for hyperbolic knots, which are regarded as analogues of Hida's universal deformation for ordinary modular Galois representations and the associated homological invariants.

August 12, 2010 Seungwhan Chang (Ewha Womans University)

Title: Extensions of rank one (phi, Gamma)-modules and crystalline representations

Abstract: I will talk about extensions of rank one (phi, Gamma)-modules (corresponding to 2-dimensional reducible mod p Galois representations of an absolutely unramified p-adic field) with certain boundedness property and explain how they are related in a certain precise sense to reductions of crystalline extensions of crystalline characters.

August 26, 2010 Wook Kim (KIAS)

Title: Generic unitary representations of general spinor groups

Abstract: First, we review the general aspect of admissible representations and then explain the relations between admissible (local) and automorphic (global) representations. The main part is to consider the problem of the classification of the generic unitary representations of general spinor groups.

September 9, 2010 Lee, Jung-Jo (Yonsei University)

Title: Kummer theory of local fields

Abstract: Let $K|\bQ_p(\zeta)$ be a finite extension where $p$ is a fixed prime number and $\zeta$ is a primitive $p$-th root of 1. The filtration $(U_n)_{n>0}$ on $K^{\times}$ by units of various levels induces a filtration on the $\mathbb{F}_p$-space $\overline{K^{\times}}=K^{\times}/K^{\times p}$ denoted by $(\overline{U}_n)_{n>0}$. Let $M=K(\sqrt[p]{K^\times})$ be the maximal elementary abelian $p$-extension of $K$, and let $G =\Gal(M|K)$, endowed with the ramification filtration $(G^u)_{u \in [-1,+\infty[ }$ in the upper numbering. I will explain the relationship between $(G^u)_u$ and $(\overline{U}_n)_{n>0}$. This relationship allows us to compute the discriminant of any elementary abelian $p$-extension of local fields, without invoking class field theory, etc. This talk will be about general background materials rather than new results.

October 7, 2010 Lim, Subong (POSTECH)

Title: Theta lifting on weak Maas forms

Abstract: We construct theta liftings from half-integral weight weak Maass forms to even integral weight weak Maass forms. Moreover it gives an extension of Niwa's theta liftings on harmonic weak Maass forms. And we obtain similar result with Niwa.

October 28, 2010 Lee, Yoonbok (KIAS)

Title: Zeta Functions off the Critical Line

Abstract: The prime number theorem was first proved in 1896 using the analytic properties of the Riemann zeta function. There are various zeta functions containing arithmetic informations which have similar properties as the Riemann zeta function and have an important role in (analytic) number theory. Their behavior inside the critical strip is mysterious and the truth of many conjectures are still far beyond. In this talk, we introduce various classical zeta functions and some interesting results concerning their behavior off the critical line.

November 25, 2010 Kim, Byungchan (SNUST)

Title: Overpartition function and its analogues modulo powers of 2

Abstract: An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. Overpartitions have been used to interpret or to prove identities arising from basic hypergeometric series. In this talk, we will investigate behavior of overpartition function and its analogues modulo powers of 2. For this, we will use various tools including arithmetic of quadratic forms, combinatorics and modular forms.

December 23, 2010 Hu, Su (KAIST)

Title: On $p$-adic Hurwitz-type Euler zeta functions

Abstract: Henri Cohen and Eduardo Friedman constructed the $p$-adic analogue for Hurwitz zeta functions, and Raabe-type formulas for the $p$-adic gamma and zeta functions from Volkenborn integrals satisfying the modified difference equation. In this paper, we define the $p$-adic Hurwitz-type Euler zeta functions. Our main tool is the fermionic $p$-adic integral on $\mathbb Z_p$. We find that many interesting properties for the $p$-adic Hurwitz zeta functions are also hold for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. This is a joint work with Dr. Min-Soo Kim.