Number Theory Seminars 2014

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January 9, 2014 Chia-Fu Yu (Academia Sinica)

Title: Density of the ordinary locus in the Hilbert-Siegel moduli spaces

Abstract: In this talk we shall address some background about the problem of the density of the ordinary locus. The main part of this talk will explain the ideas and proof of the density of the Hilbert-Siegel moduli spaces.

January 9, 2014 Kim, Chang Heon (Hanyang University)

Title: Introduction to Zagier lifts

Abstract: In this talk I will discuss the Zagier lifts of weakly holomorphic modular forms and harmonic weak Maass forms, and their applications.

January 23, 2014 Choi, Soohak (Sogang University)

Title: Local duality theorem for $q$-ary $1$-perfect codes

Abstract: In this talk, we derive the relationship between local weight enumerator of $q$-ary $1$-perfect code in a face and that in the orthogonal face. As an application of our result, we compute the local weight enumerators of a shortened, doubly-shortened, and triply-shortened $q$-ary $1$-perfect code.

February 13, 2014 Kim, Kwang-Seob (POSTECH)

Title: A construction of nonabelian simple étale fundamental groups

Abstract: Under the assumption of the GRH(Generalized Riemann Hypothesis), we show that there is a real quadratic field K such that the étale fundamental group of the spectrum of the ring of integers of K is isomorphic to $A_5$. To the best of the author's knowledge, this is the first example of a nonabelian simple étale fundamental group in the literature under the assumption of the GRH.

February 27, 2014 Choi, Dohoon (Korea Aerospace University)

Title: Mock modular forms and Eichler integrals

Abstract: In this talk, I will talk about connection between mock modular forms and Eichler integral. This talk is focused on application of this connection.

February 27, 2014 Kwon, DoYong (Chonnam National University)

Title: Power series whose coefficients are Sturmian sequences

Abstract: Sturmian sequences are the simplest sequences among aperiodic ones. We consider power series whose coefficients are Sturmian sequences, and investigate their analysis and number theory. Some of their examples in dynamical viewpoint are also presented.

March 13, 2014 Nam, Ha Yan (Yonsei University)

Title: A Generalized Pentagonal Number Theorem and Colored Partitions

Abstract: We introduce the fixed-order preserving map to prove some identities involving the pentagonal number theorem. We extend this idea to get some new identities and a generalized version of pentagonal number theorem. We also give combinatorial interpretations in terms of colored partitions.

March 27, 2014 Eum, Ick Sun (KAIST)

Title: Some applications of eta-quotients

Abstract: We show that every modular form on $\Gamma_0(2^n)$ ($n\geq2$) can be expressed as a sum of eta-quotients. Furthermore, we construct a primitive generator of the ring class field of the order of conductor $4N$ ($N\geq1$) in an imaginary quadratic field in view of the special value of certain eta-quotient.

Apr 24, 2014 Yoo, Hwajong (University of Luxembourg)(This seminar starts at 11:00am)

Title: Rational torsion points of $J_0(pq)$

Abstract: In this talk, we discuss the cuspidal group of $J_0(pq)$ and the rational torsion points of $J_0(pq)$. We prove the following statement. If a prime $\ell$ does not divide 6pq*gcd(p-1, q-1)*gcd(p-1,q+1)*(q-1,p+1), then the $\ell$-primary part of the rational torsion subgroup of $J_0(pq)$ is isomorphic to the $\ell$-primary subgroup of the cuspidal group.

May 8, 2014 Henry H. Kim (University of Toronto)

Title: Artin representations for $GSp_4$ attached to real analytic Siegel cusp forms of weight $(2,1)$

Abstract: Let F be a vector-valued real analytic Siegel cusp eigenform of weight $(2,1)$ with the eigenvalues $-5/12$ and $0$ for the two generators of the center of the algebra consisting of all $Sp_4(R)$-invariant differential operators on the Siegel upper half plane of degree $2$. Under natural assumptions in analogy of holomorphic Siegel cusp forms, we construct a unique symplectically odd Artin representation $\rho_F: G_Q \rightarrow GSp_4(C)$ associated to $F$. We explain that holomorphic Siegel cusp forms never give rise to Artin representations. This is a joint work with T. Yamauchi.

May 8, 2014 Haan, Jaeho (Seoul National University)

Title: The Bessel Period of U(3) and U(2) involving a non-tempered representation

Abstract: In 2012, Neal Harris has given a refined Gross-Prasad conjecture for unitary group as an analogue of Ichino and Ikeda's paper concerning special orthogonal groups. In his paper, he stated a conjecture under the assumption that the pair of given representations should be tempered. In this paper, we consider a specific pair involving a non-tempered one. In this case, an analogous formula still exists but the central critical L-value is slightly different with the one in the conjecture. This verifies that the tempered condition is indispensable in formulating his conjecture.

May 15, 2014 Henry H. Kim (University of Toronto)(This seminar starts at 15:30)

Title: Central limit theorem for Artin $L$-functions

Abstract: We show that the trace of Frobenius elements of Artin $L$-functions formed from a family of cubic fields obeys the Gaussian distribution. The essential ingredient is the result of Taniguchi and Thorne on counting cubic fields with given splitting types at finitely many primes. This is a joint work with P.J. Cho.

May 15, 2014 Henry H. Kim (University of Toronto)(This seminar starts at 16:45)

Title: Ikeda lift for exceptional group of type $E_7$

Abstract: Ikeda constructed a cusp form on $Sp_{2n}$ (rank $2n$) from Hecke eigenform on the upper half plane which has been conjectured by Duke and Imamoglu and Ibukiyama. We give a similar construction on the $27$-dimensional exceptional tube domain where the exceptional group of type $E_7$ acts. This is a joint work with T. Yamauchi.

June 26, 2014 Lee, Yoonbok (University of Rochester)

Title: SIMPLE ZEROS OF PRIMITIVE DIRICHLET L-FUNCTIONS AND THE ASYMPTOTIC LARGE SIEVE

Abstract: (Joint work with Chandee, Liu and Radziwill) Assuming the generalized Riemann hypothesis, we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet L-functions are simple. This improves on earlier work of ?zl?k which gives a proportion of at most 86%. We further compute the q-analogue of the Pair Correlation Function F() averaged over all primitive Dirichlet L-functions in the range || < 2. Previously such a result was available only when the average included all the characters . As a corollary of our results, we obtain an asymptotic formula for a sum over characters similar to the one encountered in the Barban?Davenport?Halberstam Theorem.

July 10, 2014 Park, Chol (University of Toronto)

Title: Reduction modulo $p$ of certain semi-stable representations

Abstract: In this talk, we will fi nd Galois stable lattices in the 3-dimensional semi-stable representations of $G_{\mathbb{Q}_{p}}$ with Hodge--Tate weights $(0,1,2)$ by constructing their strongly divisible modules. We will also compute the Breuil modules corresponding to the mod $p$ reductions of the strongly divisible modules, and determine which of the representations has an absolutely irreducible mod $p$ reduction.

July 24, 2014 Cha, Byungchul (Muhlenberg College)

Title: Mobius Disjointness in function fields

Abstract: Sarnak's Mobius Disjointness Conjecture expresses the randomness of Mobius function in the language of dynamical systems. In particular, the conjecture states that the Mobius flow is disjoint from any deterministic flow. In this talk, we present a function field analog of Sarnak's conjecture and study the case of Kronecker flow. This is joint work with Dong Han Kim.

September 25, 2014 Choi, Dohoon (Korea Aerospace University)

Title: Quantum modular forms and Mock modular forms

Abstract: In his last letter to Hardy, Ramanujan described 'mock theta functions'.In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic part of a harmonic Maass form. On the other hand, motivated by a number of examples from quantum invariants of 3-manifolds, Vassiliev invariants of knots, and period functions of Maass wave forms, Zagier defined a weight quantum modular form as a complex-valued function on the set of rational numbers.In this talk, I will talk about connections between these two modular objects. More precisely, there is a linear injective map from the space of mock modular forms to quantum modular forms and that this linear map gives expressions for "Ramanujan's radial limits" as L-values.

September 25, 2014 Tasaka, Koji (PMI,POSTECH)

Title: Multiple zeta values related to modular forms

Abstract: The multiple zeta values (abbreviated MZVs) are multivariate generalisations of the values of the Riemann zeta function at positive integers. These real number are known to be related with number theory, knot theory, quantum field theory, arithmetic geometry and so on. Our interest in the study of MZVs is a connection with the theory of elliptic modular forms (or their period polynomials), which was first discovered by Don Zagier and then investigated in depth by Gangl, Kaneko and Zagier in the case of depth 2. In my talk, we will provide this connection for arbitrary depths through the study of linear relations among MZVs at the sequences indexed by odd integers greater than 1, modulo lower depth and $\zeta(2)$. This work is motivated by a certain dimension conjecture proposed by Francis Brown. We finally present an affirmative answer to his dimension conjecture in the case of depth 4.

October 23, 2014 Kim, Dohyeong (IBS-CGP)

Title: Integer values of binary cubic forms: a modular approach

Abstract: Representing an integer by a binary forms has a long history. We begin with a binary cubic form $h(x,y)$ and a finite set $S$ of prime numbers. We will be concerned with the set of pairs $(a,b)$ of relatively prime integers for which $h(a,b)$ is an $S$-integer. Using a modular approach, namely a variant of the Frey curve trick, we show the finiteness of such pairs. Furthermore, it provides an algorithm to search for such pairs, which we will illustrate by examples. By specializing cubic form $h$, we recover Siegel's finiteness result on the unit equation, and Ramanujan-Nagell equation, among others.

November 27, 2014 Ha, Junsoo (KIAS)

Title: Smooth Polynomial solutions to $X+Y=Z$ over finite fields.

Abstract: We say an integer is $y$-smooth if all of its prime factors are less than or equal to $y$. We consider the Diophantine equation $a+b=c$ where all variables are $y$-smooth and $(a,b,c)=1$. A recent work of Lagarias and Soundararajan showed that this equation has at least exp$(y^{1/\kappa})$ solutions for $\kappa>8$ when $y$ is large. In this talk, I will describe some recent progress in this problem and an analogous theorem for the polynomial rings over finite fields.

November 27, 2014 Yhee, Donggeon (PMI,POSTECH)

Title: Koszul-Tate resolution for Batalin-Vikovisky algebra.

Abstract: J. Tate made systematic use of commutative differential graded algebras over a commutative noetherian ring $R$ and showed that there always exists a free resolution $X$ of the residue class ring $R=M$ which is a commutative differential graded algebra over $R$. I want to introduce a generalization of the result in a suitable way to a case when $M$ is not necessarily an ideal of $R$. It turns out that homotopy Lie theory, so called $L_{\infty}$-homotopy theory, plays an essential role.

December 23, 2014 Lee, Chul-hee (The University of Queensland)

Title: Rogers-Ramanujan identities in affine Kac-Moody algebras.

Abstract: The character of an irreducible representation with dominant integral highest weight of an affine Lie algebra can be written as a finite sum of theta functions with coefficients called string functions. There are still many aspects of string functions that are not well understood. In this talk I will review the basic properties of them and introduce the conjecture of Kuniba-Nakanishi-Suzuki on generalized Rogers-Ramanujan identities related to them.

December 23, 2014 Kim, Chan-Ho (University of California, Irvine)

Title: Variation of anticyclotomic Iwasawa invariants in Hida families.

Abstract: This is ongoing joint work with Francesc Castella and Matteo Longo. We investigate how congruences of modular forms affect Iwasawa invariants of their anticyclotomic $p$-adic $L$-functions. This work can be regarded as an application of the idea of Greenberg-Vatsal and Emerton-Pollack-Weston to the anticyclotomic setting. In this talk, we will mainly focus on the weight 2 case to illustrate the application to Iwasawa theory and also give an explicit example at the end.

December 24, 2014 Kim, Seungki (Stanford University)

Title: The shape of a random lattice.

Abstract: This is a non-technical introduction to a study of statistics on random lattices. In particular, we will focus on the distribution of lengths of short vectors of a random lattice. The first theorem in this direction is a well-known theorem by Siegel on the Haar measure of $SL_n(R)$. The development of the theory culminates in the 1950's with the works of Rogers and Schmidt, but the field comes to a mysterious demise since 1960. However, a few years ago, Sodergren proved a theorem that, for a fixed $k$, the lengths of the first $k$ shortest vectors of a random lattice converges in distribution to the first $k$ points of a Poisson process as dimension $n$ goes to infinity. In fact, now we know that this $k$ can grow like $O(\sqrt{n})$, and expect that it can grow like $O(n)$; but there are still many mysteries left. The connections and applications to analytic number theory, automorphic forms, and dynamics will also be discussed.