Number Theory Seminars 2005

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March 10   Kwon, Soun-Hi (Korea University) Class numbers of number fields
         Abstract: We will discuss the class number one problem for normal totally imaginary number fields.


March 15   Andreas Bender (Kyoto University)  The Schinzel hypothesis in the function field case
         Abstract: The Schinzel hypothesis is a farreaching generalization of the twin prime conjecture, with applications in diophantine geometry. We shall look at a very weak form of this conjecture which can be proved if the ground ring is
 rather than .


March 24   Pierre Matsumi (KIAS)
                  Introduction to a cohomological hasse principle for arithmetic varieties
         Abstract: In 20th century, Hasse, Noether... proved the famous local-global short exact sequence for Brauer groups over number fields. This was, from the cohomological viewpoint, reinterpreted by Kato, who generalized the above short exact sequence to a beautiful complex for arithmetic varieties called Kato-complex. In this talk, I will explain the brief history on it until now and talk on my progress during my KIAS fellowship.


April 14   Seo, Soogil (KIAS)  Cohomology groups of the class groups  
         Abstract: We compute cohomologies of the class groups in terms of special units. The motivation starts from comparison between the group structures of the class groups and special units of Euler systems. The method to compute cohomology of these for p-parts goes back to Iwasawa, Gold, Kim and etc. More recently Schoof made an interesting way to compute it for non-p parts over
. We will introduce his method and extend it to  extension using spectral sequence of Hochchild-Serre and results of Iwasawa, Gold, Kim.


April 19   David Burns (King's College London)
Congruences between derivatives of abelian L-functions


April 21   David Burns (King's College London)
Congruences between derivatives of abelian L-functions II


April 28   Ki, Haseo (Yonsei Univ.)  Zeros of approximations of Zeta functions  


May 12   Lee, Joung-Yeon (Seoul National University)
Class number one problem of Richaud-Degert Type
         Abstract: A certain type of real quadratic fields
 is Richaud-Degert (,  is not 5, positive square free and such that  divides  and ). When  or , this is called narrow-Richaud-Degert. Originally, Chowla conjectured that  for . And Yokoi conjectured that  if . Both conjectures were proved by Biro. The rest part was conjectured  for , by Mollin. We prove this conjecture by calculating the special values of some Zeta functions. This completes the class number one problem for real quadratic fields of narrow-Richaud-Degert type.


May 17   Solomon Friedberg (Boston College)  Sums of L-functions and applications to number theory


May 17   Solomon Friedberg (Boston College)  Theta functions on Odd Orthogonal Groups  
         Abstract: Given an
-homomorphism , the Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of  to those of . In the case that the adelic points of the algebraic groups ,  are replaced by their covers, one may specify an analogue of the -group (depending on the cover), and then one may still expect the existence of a correspondence. In this talk we describe joint work with Profs. D. Bump (Stanford) and D. Ginzburg (Tel-Aviv University) in which we construct such a map for the double cover of the split special orthogonal groups, raising the genuine automorphic representations  to those of . To do so we use as integral kernel a of rather exotic theta representation, exotic since, in contrast to the classical theta correspondence, this representation is not minimal in the sense of corresponding to a minimal coadjoint orbit.


May 26   Kim, Jon-Lark (University of Nebraska at Lincoln)
Capacity-Approaching Low-Density Parity-Check Codes
         Abstract: Suppose we want to send a message to a receiver over a noisy channel, for example, a pair of wires, a band of radio frequencies, a beam of light, and generally, magnetic storage devices, compact discs, etc. The fundamental problem in communication is to determine what message was sent on the basis of the received message. In 1948 Claude Shannon introduced a number called the capacity of the channel and showed in a nonconstructive way, that arbitrary reliable communication is possible at any rate below the channel capacity. The design of codes with efficient encoding and decoding algorithm which approach the capacity of the channel is one of main areas in coding theory. Coding theorists have been led to search for codes that have considerable algebraic or combinatorial structures.
          Low-density parity-check (LDPC) codes, originally introduced by Gallager in 1962, have been one of the hottest topics in coding theory since MacKay and Neal in 1996 demonstrated that some long LDPC codes approach near Shannon limit under the sum-product algorithm. In this talk, we overview LDPC codes, Tanner graphs, and discuss our results on the two constructions of LDPC codes. One is based on finite groups, generalizing the idea of Lubotzky, et al and independently Margulis' construction of Ramanujan graphs. The other is based on algebraically defined q-regular bipartite graphs suggested by Lazebnik and Ustimenko.


June 9    Lee, Hyang-Sook (Ewha Womans University)
Pairing based Cryptosystem and Vector Decomposition Problem
         Abstract: Pairings were first used in cryptography as a cryptanalytic tool for reducing the discrete log problem on some elliptic curves to the discrete log problem in a finite field. Positive cryptographic applications based on pairings arose from the work of Joux, who gave a simple one round tripartite Diffie-Hellman protocol on supersingular curves. Many cryptographic schemes based on the pairings have been developed recently, such as identity based encryption, identity based signature schemes and identity based authenticated key agreements and identity based signcryption. We discuss some pairing-based cryptographic schemes in this talk. More recently the cryptography based on the vector decomposition problem(VDP) was suggested and it was proved that the VDP in a two dimensional vector space is at least hard as the computational Diffie-Hellman problem in a one dimensional subspace under some condition. However we present the VDP can be solved for a certain basis although the conditions are satisfied.


June 23  Shim, Kyungah  (Ewha Womans University)
A New Class of Problems for Cryptographic Schemes
         Abstract: We introduce a new class of problems which can be considered as a  composition of a computation problem and a decision problem. We provide a new hard problem which belongs to the new class. As the gap problems which deal with the gap of difficulty between computation problems and decision problems, we define the gap problems based on the gap of difficulty between composition problems and computation problems. Finally, we discuss their applications to cryptography.


July 14   Im, Bo-Hae (University of Utah)
The rank of elliptic curves and infinite multiplicity of roots of unity on elliptic curves
         Abstract: Let
 be a number field,  an algebraic closure of  and  an elliptic curve defined over Let  be the absolute Galois group of  over . We prove that there is a subset  of Haar measure 1 such that for every , the spectrum of  in the natural representation  of  consists of all roots of unity, each of infinite multiplicity. In particular, we discuss the infinite multiplicity of the eigenvalue 1 case which shows the infinite rank of a given elliptic curve and the openness condition of such a subset for each root of unity.


July 28   Yang, Jae-Hyun (Inha University)  Harmonic Analysis on Homogeneous Spaces I
         Abstract: I give a survey talk on recent progress in harmonic analysis on semisimple or reductive symmetric spaces reviewing classical harmonic analysis.


July 28   Andreas Bender (KIAS)  The Schinzel hypothesis in the function field case
         Abstract: The Schinzel hypothesis is a farreaching generalization of the twin prime conjecture, with applications in diophantine geometry. We shall look at a very weak form of this conjecture which can be proved if the ground ring is 
 rather than


August 4    Yang, Jae-Hyun (Inha University)  Harmonic Analysis on Homogeneous Spaces II
         Abstract: I give a lecture of harmonic analysis on homogeneous spaces of non-reductive type that are important arithmetically and geometrically. I present some new results and compare these results with those obtained in the case of reductive symmetric spaces.


August 4   Lee, Kyu-Hwan (University of Toronto)
Iwahori-Hecke algebras of SL2 over 2-dimensional local fields


August 11   Shin, Sug Woo (Harvard University)  Shimura varieties and Langlands Correspondence
         Abstract: Shimura varieties often arise as moduli spaces of abelian varieties with additional structures. Some examples are elliptic modular curves and Hilbert modular varieties. On the other hand, mathematicians believe that there is Global Langlands Correspondence, namely a correspondence between Galois representations and automorphic representations. Local Langlands Correspondence can also be formulated and are believed to be compatible with global correspondence. Shimura varieties are especially interesting because Langlands Correspondence is presumably realized in their cohomology spaces. This idea is encoded in the recent proof of Local Langlands Conjecture by Harris and Taylor.
      With these generalities in mind, I will focus on easy and primitive examples of Shimura varieties as cyclotomic fields, imaginary quadratic fields and elliptic modular curves. We will see how Langlands Correspondence can be seen in their cohomologies. Adelic point of view of these objects will be emphasized on the way.


August 11   Lee, Kyu-Hwan (University of Toronto)
Spherical Hecke algebras of GLn over 2-dimensional local fields
         Abstract: After considering construction of an invariant measure on GLn over 2-dimensional local fields, we will define spherical Hecke algebras of these groups. Then we will discuss Satake isomorphism.


August 17   Henry Kim  (University of Toronto)  Langlands functoriality conjecture


August 18   Yang, Jae-Hyun (Inha University)  Siegel modular forms and Jacobi forms
         Abstract: I give a talk about the theory of Siegel modular forms and Jacobi forms in the aspects of arithmetic, geometry and representation theory.


August 25   Kim, Tae-Kyun (Kongju University)  On Iwasawa p-adic L-functions


September 8    Kim, Donggyun (Korea University)
Elementary introduction to local Langlands correspondence for GL(2)


September 22   Park, Poo-Sung (Seoul National Univ.)  2-Universal Hermitian Forms
         Abstract: A positive definite hermitian lattice is said to be 2-universal if it represents all positive definite binary hermitian lattices. We find all ternary and quaternary 2-universal hermitian lattices over imaginary quadratic fields and provide the 15-theorem type of criteria for 2-universality of hermitian lattices. We also investigate asymptotic behavior of minimal ranks of 2-universal hermitian lattices over imaginary quadratic fields. As an application we discuss the solvability of certain types of Diophantine equations.


October 13  Kim, Ho-il (Kyungbook National University)  Quantum Theta Functions
         Abstract: We want to describe the quantum abelian varieties by using two lattice structures representing complex structures and noncommutative structures. Then we analyze the symmetry on them, leading to the orbifolds of them.


October 27  Park, Jinsung (KIAS)  On the zeta regularized determinant and the Selberg zeta function
         Abstract: In this talk, I will explain the relation of the zeta regularized determinant of the Laplacian and the Selberg zeta function for the congruence subgroup of


November 10  Jeong, Kyeonghoon (Seoul National Univ.)
Some Lie algebra applications to number theory


November 17  Sohn, Jaebum (Yonsei University)
Equivalent continued fractions and acceleration of its convergence
         Abstract: The Bauer-Muir transformation is useful to prove equivalence of continued fractions. With appropriate choice of modifying factors
, it gives equivalent continued fraction up to modify sense. In this talk, we first examine equivalent continued fractions up to modified convergence of several Ramanujan type continued fractions that include the Rogers-Ramanujan continued fraction and the Ramanujan's cubic continued fraction by using this transformation. And then we comment about the acceleration of convergence of these continued fractions.


November 17  Kim, Seon-Hong (Chosun University)
Densely algebraic bounds for the exponential function
         Abstract: An upper bound for
 that implies the inequality between the arithmetic and geometric means is generalized with the introduction of a new parameter . The new upper bound is smoothly and densely algebraic in , and valid for  for arbitrarily large positive  provided that  is sufficiently close to . The range of its validity for negative  is investigated through the study of the zero distribution of a certain family of quadrinomials.


December 8  Kim, Daeyeoul (Sogang University)  Algebraic integrers for basic Appell series
         Abstract: In this talk, we state properties for basic Appell series. And then we prove algebraic properties for them.


December 22  Lee, Yoonjin (Simon Fraser University)
The structure of the class groups of global function fields
         Abstract: The problem of determining the structure of the class group dates back to Gauss. In this talk we discuss the structure of the class groups of global function fields. Let
 be a finite field and  a  transcendental element over .
      We show an explicit method of constructing, for positive integers
 and with , infinitely many global function fields K of degree m over  such that  has a given unit rank and the ideal class group of  contains a subgroup isomorphic to .