Number Theory Seminars 2006

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January 12   Oh, Byeong-Kweon (Sejong University)  On quadratic regular forms
         Abstract: In this talk, we present an overview of recent results on positive definite integral quadratic forms which satisfy various regularity properties. In particular, we show some finiteness results on regular forms and almost regular forms, and we also show that the minimal rank of n-regular forms has an exponential lower bound.

January 26   Kim, Chang Heon (Seoul Women's University) Twisted traces of singular values
         Abstract: Let
 be a prime for which the congruence group ( is of genus zero, and  be the corresponding Hauptmodul. We investigate the twisted traces of singular values of  and construct infinite products related to them.

February 9    Cho, Soojin (Ajou University)  Combinatorics of Littlewood-Richardson numbers
         Abstract: There are well known reduction formulas for the universal Schubert coefficients defined on Grassmannians. These coefficients are also known as the Littlewood-Richardson coefficients in the theory of symmetric functions. We restate the reduction formulas combinatorially and provide a combinatorial proof for them.

February 16   Lee, Joongul (Hongik University)  Refinement of a conjecture of Gross

February 16  Chae, Hi-Joon (Hongik University)  L-functions and epsilon factors

March 9    Choi, Dohoon (POSTECH)  The Space of Modular Forms Generated by Eta-quotients
         Abstract: In this talk, we study the spaces of modular forms on
 generated by eta-quotients, where the genus of   is zero or  is a prime. These partly give an answer to the following problem suggested by Ono: Classify the spaces of modular forms which are generated by eta-quotients.

March 23  Kim, Ji Young (SNU) On almost universal forms
         Abstract: We present an overview of results on positive definite integral quadratic forms which satisfy the
-universal property or the almost n-universal property. Furthermore, we classify all almost 2-universal positive definite integral diagonal forms of rank 6 up to isometry. These results imply that there exist infinitely many such forms, up to isometry.

April 13  Choi, Youn-Seo (KIAS)
                      Is it possible to interpret the mock theta function with the theory of partitions?
         Abstract: In this talk, a few elementary theories of partitions are introduced and also the definition of mock theta functions is provided. Then we will discuss that the mock theta functions can be understood by the theory of partitions.

April 27  Choi, So Young (KAIST) The genus of arithmetic curves
         Abstract: For any element
 with  a positive integer, we find the genus of the arithmetic curve  which are independent of , where  is the Fricke involution.
      We obtain that the genus of
 is zero if and only if   or  . As an application, we determine a generator of the function field related to the groups   of genus zero which will be used to generate appropriate ray class fields over imaginary quadratic fields, and show that the fixed point of   in H  is a Weierstrass point of  for all but finitely many .

May 11  Kang, Soon-Yi (KIAS)
                     Generalizations of two fundamental product identities and their applications
         Abstract: Using some standard transformations in the theory of basic hypergeometric series, we first derive a generalization of a reciprocity theorem for a certain q-series found in Ramanujan's lost notebook and utilize it to derive four variable generalizations of Jacobi's triple and quintuple product identities. Then we present some applications of the generalized product identities, including new representations for generating functions for sums of six squares and those for overpartitions.

May 25  Kang, Soon-Yi (KIAS)  The theory of elliptic functions in the signature 3
         Abstract: We first briefly survey the history of elliptic function theory which originated several branches of mathematics such as complex function theory and Galois theory. Then we discuss the main theme of Ramanujan's theory of elliptic functions, including the classical inversion formula which shows the intertwined relations among elliptic functions, theta functions, and hypergeometric functions. Nex, we introduce his discovery of the theory of elliptic functions in alternative bases. In this talk, we focus on the cubic analogue of the theory of elliptic functions and present the recent progress on this topic.

June 8    Takashi Ono (Johns Hopkins University) Arithmetic of Hopf Maps I

June 9    Takashi Ono (Johns Hopkins University) Arithmetic of Hopf Maps II

June 30  Oh, Hee (California Institute of Technology)  Counting rational points of bounded height
         Abstract: A fundamental problem in modern arithmetic geometry is to describe the set of rational points of a projective variety in terms of geometric invariants. One of the main conjectures in this area formulated by Manin in late eighties predicts the asymptotic of the number of rational points of bounded height for Fano varieties. I will discuss a recent proof of Manin's conjecture for the case of the wonderful compactication of semisimple algebraic groups, based on the mixing property of adelic groups (joint work with Gorodnik and Maucourant).

June 30  Andreas Schweizer (University of Exeter)
                     On the m-rank of the class groups of quadratic function fields
         Abstract: Fix a finite field 
of odd characteristic  and an odd integer  that is not divisible by . We are interested in the -rank of the divisor class groups of quadratic extensions  of the rational function field .  Although an analogue of the Cohen-Lenstra heuristics (due to Friedman and Washington) suggests that for each positive integer  a positive proportion of all  should have -rank at least , there are only few results for.
      First we show: If there is one
 with -rank at least , then there are infinitely many  with -rank at least . Then for each  that is congruent to 1 modulo 4 we explicitly construct an  with -rank at least 4. Similar constructions give slightly weaker results if  mod 4 and also for the ideal class group of the integral closure of  in .

July 11   Michael Schlosser (University of Vienna)  Elliptic enumeration of nonintersecting lattice paths
         Abstract: We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's
 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's  and Dougall's  summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the  summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives. (A preprint corresponding to this talk is available at'

July 27   Choi Dohoon (KIAS)
                    Congruence Properties for Fourier Coefficients of Modular Forms of Half Integral Weight 
         Abstract: In this talk, we discuss congruence properties related to Fourier coefficients of modular forms of half integral weight and their applications.

August 10  Andreas Bender (KIAS)  Developments on the Schinzel Hypothesis for Function Fields
         Abstract: This will be a talk about work in progress: I shall review the basic issues, explain what the main theorem says and discuss the picture as it is emerging.

September 14  Lee, Geon-No (Yonsei University)
                     Upper bound of degrees of CM-fields with relative class number one
         Abstract: In 1974, Stark showed that for any fixed integers m and h with m > 2, there exist only finitely many CM-fields of degree 2m with class number h. In 1975, Odlyzko proved that there are only finitely many normal CM-fields of a given class number. In 1979, Hoffstein showed that every normal CM-field of degree greater than 434 has relative class number greater than one. In 2003, Bessassi improved upon the Hoffstein's bound: he proved that every normal CM-field of degree greater than 266 has relative class number greater than one and that if the Generalized Riemann Hypothesis is true then the normal CM-fields with relative class number one are of degrees less than or equal to 164. Now we give the sharper upper bounds for the degrees of CM-fields of relative class number one. We show that the normal CM-fields with relative class number one are of degrees less than or equal to 96 assuming the Generalized Riemann Hypothesis, the CM-fields (not necessarily normal) with relative class number one are of degrees less than or equal to 104 assuming the Generalized Riemann Hypothesis, and the normal CM-fields with relative class number one are of degrees less than or equal to 216 without assuming the Generalized Riemann Hypothesis. By many authors all normal CM-fields of degrees less than or equal to 96 with class number one are known except for the possible fields of degree 64 or 96. Consequently class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.

September 27 Yuichiro Taguchi (Kyushu University)  On extensions of truncated discrete valuation rings
         Abstract: Let 
 be a complete discrete valuation field whose residue field may be imperfect, and let  be an integer . We show that the category of finite separable extensions of  with "ramification bounded by " in the sense of Abbes-Saito depends only on the "length- truncation"  of the integer ring of .

October 12  Kim, Wook (Yonsei University)  On the standard module conjecture
 be a quasisplit algebraic group over a -adic field, and let  be a parabolic subgroup of . For a tempered generic representation  of  and a parameter , an induced module is called standard if  is in the positive Weyl chamber. A standard module has a unique irreducible quotient , the Langlands quotient. The standard module conjecture states that is irreducible if and only if the Langlands quotient is generic. The standard module conjecture together with Shahidi's conjecture on the holomorphy on L-functions implies that normalized intertwining operators associated to a globally generic cuspidal representation are holomorphic. It is called "Assumption A" which has been used to establish functoriality for many cases.
   In this talk we prove the conjecture for general spin groups via the Langlands-Shahidi method and Muic's idea, and put some comments on the general case based on the recent result by Heiermann and Muic.

October 26  Lee, Dong Uk (KIAS)   p-adic monodromy of Picard moduli scheme
         Abstract: Let
 be an imaginary quadratic number field and let  be distinct natural numbers. Suppose a rational prime  splits in .
We show that the naive
-adic monodromy of the ordinary locus of the good reduction over  of Picard moduli scheme of signature  is "as large as possible", i.e. equal to  We discuss an application of this result to the bad reduction of Picard moduli scheme with parahoric level structure. 

October 26   Joseph Hundley (POSTECH)   An Introductory Overview of the Rankin-Selberg Method
        Abstract: Without going into all of the details, we describe the idea of a Langlands
-function and of a Rankin-Selberg integral for one. 
We then discuss why relationships are to be expected between
    * poles of Langlands
    * the conjectural "functorial liftings"
    * nonvanishing of certain "period integrals,"
and how Rankin-Selberg integrals can be used to prove these relationships in some cases. 

November 10   Im, Bo-Hae (Chung-Ang University)  The rank of abelian varieties over cyclic fields
         Abstract: Let
 be a field of characteristic  such that every finite separable extension of  is cyclic.
 be an abelian variety over . We show that if  is infinite but not locally finite, the rank of  over is infinite.

November 23   Seo, Seunghyun  (Seoul National University)
               A generalization of Cayley's formula for trees and parking functions
        Abstract: A leader of a tree
 on  is a vertex which has no smaller descendants in .  In 2005, Gessel and Seo showed

which is a generalization of Cayley formula, where
 is given by  .
In this talk, using a variation of Prufer code which is called a "reverse Prufer code", we give a simple bijective proof of Gessel and Seo's formula. We also discuss some related properties of parking functions and other combinatorial objects.


December 14    Kim, Sangjib  (National University of Singapore)
             Littlewood-Richardson numbers and related combinatorics
          Abstract: Given finite dimensional irreducible representations of
 labeled by their highest weights , , and , Littlewood-Richardson number is the multiplicity of  in the tensor product of  and .
Many combinatorial objects have been introduced to encode these numbers. After reviewing some of them, we will investigate their properties and relations among them.  If time permits, I will also present a new  proof for Littlewood-Richardson rules.


December 28  Yee, Ae Ja  (The Pennsylvania State University)
             Lecture Hall Theorem and a generalization of Euler's theorem
         Abstract: Lecture hall partitions are partitions whose parts satisfy a certain ratio condition. Their enumeration by Bousquet-Melow and Eriksson gives a finite version of a theorem of Euler on strict partitions. In this talk, we will discuss lecture hall theorem and a generalization of Euler's theorem. This is join work with Carla Savage.