Number Theory Seminars 2011

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January 12, 2011 Kim, Dohyeong (POSTECH)

Title: Iwasawa Theory for Elliptic Curves

Abstract: Let $E$ be an elliptic curve with good ordinary reduction at $p$. I will prove an upper bound for the $p$-adic analytic rank of $E$ as $p$ varies while $E$ is fixed. The bound is a constant power of $p$. It will give in turn an upper bound for the $Z_p$-corank of the Tate-Shafarevich group.

January 27, 2011 Lee, Junho (KAIST-ASARC)

Title: Relation between prime producing polynomials and class number 1 problem

Abstract: It has been known for a long time that there exists close connection between prime producing polynomials and class number 1 problem for some number fields. In this talk, we introduce the known results for quadratic fields and the simplest cubic fields, and share a late result related to this problem

February 10, 2011 Yang, Jae-Hyun (Inha University)

Title: Invariant Differential Operators and Siegel-Jacobi Space

Abstract: In this talk I will discuss differential operators on the Siegel-Jacobi space invariant under the natural action of the Jacobi group. and using these invariant differential operators we study Maass-Jacobi forms. Maass-Jacobi forms play an important role in the spectral theory on the Siegel-Jacobi space. I review the works of Hans Maass and Goro Shimura about invariant differential operators on the Siegel space roughly. I will present my results on invariant differential operators on the Siegel-Jacobi space. I also mention the recent results of A. Pitale, K. Bringmann and O. Richter on Maass-Jacobi forms. In the end of my talk, I will propose several important problems that must be investigated in the future.

February 17, 2011 Andreas Schweizer (Academia Sinica, Taipei)

Title: Davenport-Stothers inequalities and elliptic surfaces in positive characteristic

Abstract: If $f$ and $g$ are two polynomials such that $f^3 -g^2$ is not the zero polynomial, can one give a lower bound for the degree of $f^3 -g^2$? In characteristic $0$ the answer was given by Davenport and Stothers. We discuss what happens in positive characteristic. The proof uses elliptic surfaces.

March 10, 2011 Junmyeong Jang (KIAS)

Title: Ordinary reduction problem for an isotrivial elliptic surfaces

Abstract: In this talk, we will see a criterion of an isotrivial elliptic surface and using this, we will show the ordinary reduction theorem for some isotrivial elliptic surfaces defined over a number field.

March 24, 2011 Donggeon Yhee (SNU)

Title: Rank of Quadratic Twists

Abstract: In number theory, an elliptic curves is an interesting object. It has two structures, say algebraic side and analytic side. Over a number field, algebraic side is represented as a finitely generated abelian group and analytic side appears in 'L-function'. Many mathematicians believe that two natures of elliptic curves are related to each other. Goldfeld Conjecture, a conjecture for quadratic twists of ellitic curves, is one of such belief. In this talk, I will introduce the conjecture and partial answers.

April 14, 2011 Daeyeoul Kim (NIMS)

Title: Future Internet and divisor function

Abstract: In this talk, I will state Future internet project at NIMS. And then I will explain properties of the divisor functions with respected to the Weierstrass p-functions.

May 6, 2011 Henry Kim (U.of Toronto)

Title: Application of the strong Artin conjecture to the class number problem

Abstract: We apply the strong Artin conjecture to produce a family of quartic and quintic number fields with largest possible class numbers. This is a joint work with P.J. Cho.

May 11, 2011 (2pm-4pm, #1424) Dalawat (Harish-Chandra Research Institute)

Title: The ramification filtration

Abstract: This will be an introductory talk giving a simple derivation of the ramification filtration on the maximal elementary abelian p-extension of a p-adic field.

May 12, 2011 (4pm-5pm, #1424) Zhang, Yichao (Postech)

Title: Divisor function over quaternion algebras and fourth moments

Abstract: Joint work with Henry H. Kim. In 1988, Duke generalized Potter's second moment result to the case of Dirichlet series of Maass signature. After obtaining an identity for the divisor function over the Hamiltonian quaternion algebra, he was able to give a (sharp) bound for fourth moments of L-functions associated to newforms of level 2 in an average version. In this talk, we shall demonstrate the generalizations of those results, and finally apply them to fourth moments problem in the case of 'square-free' level.

May 12, 2011 (4pm-5pm #1424) Dalawat (Harish-Chandra Research Institute)

Title: Serre's mass formula in prime degree

Abstract: A p-adic field has only finitely many extensions of given degree. Krasner and Serre gave a formula for counting the number of such extensions, each extension being counted with a certain weight depending on its discriminant. We gave a simple proof of this formula in the crucial degree-p case. Our method also works for local fields of characteristic p, even though the number of degree-p extensions is now infinite.

May 13, 2011(5pm-6pm, #1423) Henry Kim (U. of Toronto)

Title: Generalized Kac-Moody algebras and deformation of modular forms

Abstract: We develop an analogue of Gindikin-Karpelevich formula for a family of generalized Kac-Moody algebras, attached to Borcherds-Cartan matrices consisting of only one positive entry in the diagonal. As an application, we obtain a deformation of Fourier coefficients of modular forms such as the modular j-function and Ramanujan tau-function. This is a joint work with K.H. Lee.

May 26, 2011(5pm-6pm, #1424) Byunchul Cha (Muhlenberg College)

Title: Independence of zeta zeros and its applications in analytic number theory

Abstract: It is well known that the zeros of zeta functions of algebraic curves over a finite field $F$, under any complex embedding, have absolute values equal to a square root of the size of $F$. We examine multiplicative relations among these zeta zeros. As an application, we discuss how these relations (or the absence of such relations) can affect asymptotic bounds of partial sums of Moebius functions in a function field setting.

June 7, 2011 Steve Gonek (University of Rochester)

Title: Finite Euler Products and the Riemann Hypothesis

Abstract: We investigate approximations of the Riemann zeta-function by truncations of its Dirichlet series and Euler product, and then construct a parameterized family of non-analytic approximations to the zeta-function. Apart from a few possible exceptions near the real axis, each function in the family satisfies a Riemann Hypothesis. When the parameter is not too large, the functions have roughly the same number of zeros as the zeta-function, their zeros are all simple, and they repel. In fact, if the Riemann Hypothesis is true, the zeros of these functions converge to those of the zeta-function as the parameter increases, and between zeros of the zeta-function the functions in the family tend to twice the zeta function. They may therefore be regarded as models of the Riemann zeta function. The structure of the functions explains the simplicity and repulsion of their zeros when the parameter is small. One might therefore hope to gain insight from them into the mechanism responsible for the corresponding properties of the zeros of the zeta-function.

June 9, 2011 Chonggyu Lee (UI Chicago)

Title: Height functions, Arithmetic dynamics and Commuting maps

Abstract: Fatou and Julia showed that f,g are commuting polynomials if they are common iterates of a polynomial, x^d or Chebychev polynomials. It is quite hard to be commuting maps in general. Using the height function, quite convenient tool in number theory, we can show the similar result in higher dimensions; any endomorphism \phi on projective space over number field has only finitely many maps commuting with \phi.

June 23, 2011 Meesue Yoo (Seoul National University)

Title: Combinatorics of the space of diagonal harmonics and symmetric function operators

Abstract: In this talk, we consider specially defined symmetric functions operators to study the space of diagonal harmonics. They are defined by Haglund, Morse and Zabrocki which play an important role in the study of the combinatorics of the space of diagonal harmonics. We give combinatorial interpretations of those symmetric function operators applied to basis symmetric functions, especially Schur functions. We also relate the operators iteratively applied to 1 and the Hall-Littlewood symmetric functions and derive a new way of calculating the Kostka-Foulkes polynomials.

July 11, 2011 Kai-Man Tsang (The University of Hong Kong)

Title: On a mean value theorem for the Riemann zeta-function

Abstract: The error term E(T) in the mean square formula $\int_0^{\infty} |\zeta(1/2+it)|^2dt=T\log \frac{T}{2 \pi}+(2\ga-1)T+E(T)$ for the Riemann zeta-function on the critical line has been studied extensively alongside other famous error terms in analytic number theory, such as the error term in the Dirichlet divisor problem. The behaviour of E(T) is very interesting and intriguing, and has attracted the attention of many researchers. Asymptotics for the mean square of E(T) has been obtained long time ago, with successive better bounds proved for the error term. In this talk, we shall discuss a new mean value theorem for E(T) and thereby probes into the finer shape of the error term for the mean square of E(T). In particular, we find that the error term in the mean square of E(T) is \\Omega_{-}(x log^2xlog log x)$.

July 11, 2011 Yuk-Kam Lau (The university of Hong Kong)

Title: Average values of divisor sums in arithmetic progressions

Abstract: The divisor function $\tau(n)$ counts the number of positive divisors of an integer n. We are concerned with the sum $S(X,q,b)=\sum_{n \le x, n \cong b \mod q} \tau(n)$. When q=1, Drichlet derived in 1849 a pretty asymptotic formula with elementary methods. For the general case, Selberg and Hooley independently discovered the aymtotic formula $ S(X,q,b)=\frac{1}{\phi(q)}\{XP_q(\log X)+O(X^{1-\delta}\}$ for some $\de >0$ where $\phi(q)$ is the Euler phi function and $P_Q(x)$ is a linear polynomial in x. In this talk, we study the derivation of S(X,q,b) from the main term on average over b. This problem was investigated in a few papers by Banks et al. Blomer, Lu etc. We shall discuss the recent progress, applications and some ideas of proofs.

July 19, 2011 Lapid, Erez (The Hebrew University of Jerusalem)

Title: Multiplicity one and finite multiplicity in representation theory and automorphic form

Abstract: In representation theory a basic problem is to decompose the restriction of an irreducible representation of a group to a subgroup. A basic concept in this context is a 'Gelfand pair'. I will discuss various developments in recent years around this concept, due to many people. My own contribution is joint with Brooke Feigon and Omer Offen.

July 21, 2011 Seokho Jin (POSTECH)

Title: Regularized imaginary quadratic Doi-Naganuma lifting

Abstract: There are several maps from the set of automorphic forms on one group to the set of automorphic forms on another group, often called 'liftings'. We studied, called 'theta liftings' or 'theta correspondences'. Borcherds and Harvey-Moore used this theta correspondence to find a remarkable multiplicative correspondence between classical modular forms with poles at cusps and meromorphic modular forms on complex varieties. The main ingredients of their results were the process of regularization technique. This technique can be used to enlarge the domain of theta liftings. In this talk I'll say about the generalization of the Friedberg's imaginary quadratic Doi-Naganuma lifting to weak Maass forms.

July 28, 2011 Sejeong Bang (Yeungnam University)

Title: The Bannai-Ito Conjecture

Abstract: In their 1984 book gAlgebraic Combinatorics I: Association Schemesh, E. Bannai and T. Ito conjectured that there are only finitely many distance-regular graphs with fixed valency k?3. In the series of papers, they showed that their conjecture holds for k=3, 4, and for the class of bipartite distance-regular graphs. J. H. Koolen and V. Moulton also show that there are only finitely many distance-regular graphs with k=5, 6, or 7, and there are only finitely many triangle-free distance-regular graphs with k=8, 9 or 10. In this talk, we show that the Bannai-Ito conjecture holds for any integer k>2 (i.e., for fixed integer k>2, there are only finitely many distance-regular graphs with valency k). This is a joint work with A. Dubickas, J. H. Koolen and V. Moulton.

August 11, 2011 SugWoo Shin (MIT)

Title: Statistics for families of automorphic forms

Abstract: I will survey on some statistical results due to J.-P. Serre and others on the Hecke eigenvalues for families of classical modular forms (or the Satake parameters for families of automorphic representations of GL(2)). Then I will report on the analogous results for families of automorphic representations for a semisimple group G over Q such that G(R) admits a discrete series. This is a joint work with Nicolas Templier.

September 8, 2011 Toshiro Hiranouchi (Hiroshima University)

Title: A Gillet-Waldhausen type theorem for an exact category with weak equivalences

Abstract: We introduce a notion of quasi-weak equivalences associated with weak-equivalences in an exact category. It gives us a variant of the Gillet-Waldhausen theorem for an exact category with weak equivalences. By-product of the theorem, we obtain a delooping for (idempotent complete and semi-simple) exact categories and a condition that the negative $K$-group of an exact category becomes trivial.

September 29, 2011 Cristian Virdol (Kyushu University)

Title: Special values of L-functions of base change for Hilbert modular forms

Abstract: I will generalize some results, obtained by Shimura and Yoshida on the critical values of L-functions of l-adic representations attached to Hilbert modular forms, to the critical values of L-functions of arbitrary base change to totally real number fields of l-adic representations attached to Hilbert modular forms.

October 27, 2011 Kyu-Hwan Lee (University of Connecticut)

Title: Root multiplicities of hyperbolic Kac-Moody algebras and Fourier coefficients of modular forms

Abstract: In this talk, we will consider the hyperbolic Kac-Moody algebra associated to a certain rank 3 Cartan matrix. The denominator function of the hyperbolic Kac-Moody algebra is not an automorphic form. However, Gritsenko and Nikulin extended it to a generalized Kac-Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as a product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac-Moody algebra which contains the hyperbolic Kac-Moody algebra and whose denominator function is related to a product of Dedekind eta-functions. In particular, root multiplicities of the generalized Kac-Moody algebra are determined by Fourier coefficients of a modular form. We will compute asymptotic formulas for these Fourier coefficients using the method of Hardy-Ramanujan-Rademacher, and obtain an asymptotic bound for root multiplicities of the hyperbolic Kac-Moody algebra. Our method can be applied to other hyperbolic Kac-Moody algebras and to other modular forms. This is a joint work with Henry Kim.

November 24, 2011 Bo-Hae Im (Chung-Ang University)

Title: Rank of elliptic curves over some special types of infinite algebraic extensions

Abstract: We prove that the rank of elliptic curves over some special types of infinite algebraic extensions is infinite and we will talk about the historical progress of Larsen's conjecture. A part of the results is a joint work with Prof. Larsen.

November 30, 2011 Heesung Shin (Inha University)

Title: On Entringer families

Abstract: Andr\'e proved that the number of down-up permutations on $\{1, 2, ..., n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting down-up permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. Sequentially, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in down-up permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of $E_{n,k}$ in the model of trees. The aim of this talk is to provide bijections between the different models for $E_{n,k}$ as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's down-up permutation model and Poupard's increasing tree model.