January 10, 2012 Lee, Min (Brown University)

Title: Effective count for Apollonian circle packings

Abstract: We obtain an effective count for circles in Apollonian packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume whose fundamental group has critical exponent bigger than 1.

January 30, 2012 Bartel, Alex (POSTECH)

Title: Integral Galois modules I

Abstract: Galois groups of number fields naturally act on many interesting number theoretic objects, e.g. on integral units in number fields, or on Mordell-Weil groups of elliptic curves. In these talks, we will investigate the structure of these objects as Galois modules. The first talk will introduce the main questions, give concrete examples and state the main results. After that, we will present the techniques for obtaining those results, which are mostly representation theoretic, and will offer some open problems. Part of the work to be presented is joint with Bart de Smit.

February 1, 2012 Bartel, Alex (POSTECH)

Title: Integral Galois modules II

Abstract: Galois groups of number fields naturally act on many interesting number theoretic objects, e.g. on integral units in number fields, or on Mordell-Weil groups of elliptic curves. In these talks, we will investigate the structure of these objects as Galois modules. The first talk will introduce the main questions, give concrete examples and state the main results. After that, we will present the techniques for obtaining those results, which are mostly representation theoretic, and will offer some open problems. Part of the work to be presented is joint with Bart de Smit.

February 23, 2012 Ahlgren, Scott (University of Illinois at U-C)

Title: Mock modular grids and Hecke relations for mock modular forms

Abstract: We study infinite grids of mock modular forms which contain a generating function of number-theoretic or combinatorial interest as the first entry (for example, the 'smallest parts' partition function of Andrews, or a mock theta function of Ramanujan). By studying the action of the Hecke operators on these grids, we obtain a number of interesting corollaries for these functions. (Joint with Byungchan Kim).

March 8, 2012 Lim, Jung Wook (Sogang University)

Title: Characterizations of some integral domains in special pullbacks

Abstract: Let $D \subsetneq E$ be an extension of integral domains, $K$ be the quotient field of $D$, $\Gamma$ be a nonzero torsion-free grading monoid with $\Gamma \cap -\Gamma=\{0\}$, $\Gamma^*=\Gamma \setminus \{0\}$, $E[\Gamma]$ be the semigroup ring of $\Gamma$ over $E$, and $D+E[\Gamma^*]=\{f \in E[\Gamma] \mid f(0) \in D\}$ be the composite semigroup ring. Pullback constructions have for many years been an important tool in the arsenal of commutative algebraists because of their use in producing many examples. In this talk, we investigate the $D+E[\Gamma^*]$ construction which is a nice example of them. In fact, we show when the composite semigroup domain $D+E[\Gamma^*]$ is a Pr\"ufer domain or a GCD-domain. As a corollary, we study a B\'ezout domains of the form $D+E[\Gamma^*]$. We also completely characterize when the domain $D+E[\Gamma^*]$ is a generalized Krull domain or a generalized unique factorization domain.

March 22, 2012 Eum, Ick Sun (KAIST)

Title: Representations by $x_1^2+2x_2^2+x_3^2+x_4^2+x_5^2+x_1x_2+2x_2x_4+x_3x_4-x_3x_5-x_4x_5$

Abstract: Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by the quintic quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_5^2+x_1x_2+2x_2x_4+x_3x_4-x_3x_5-x_4x_5$. Then the associated theta function is a modular form of weight $5/2$ for $\Gamma_0(8)$ associated with the character $\left(\frac{8}{\cdot}\right)$. We express this theta function as a linear combination of Hecke eigenforms and find the general formula of the representation number $r_Q(n)$.

April 12, 2012 Kim, Dohyeong (POSTECH)

Title: Non-commutative $p$-adic $L$-functions and algebraic K-theory

Abstract: The main conjectures of non-commutative Iwasawa theory predicts that the critical values of Artin twists of modular $L$-functions satisfy certain congruences. In this talk, I will explain how one can prove such congruences for simplest modular forms and Artin representations. Namely, the modular form corresponds to a power of Hecke character of an imaginary quadratic field obtained from Tate module of a CM elliptc curve, and the Artin representation factors through so called false Tate curve extension. I will also explain how computations in K theory predicts this type of congruence.

April 26, 2012 Kim, Daeyeoul (NIMS)

Title: Curious convolution formula for Divisor functions

Abstract: In this talk we will deal with divisor functions arising from $q$-series and theta functions. Let $\sigma_{s}(N)$ denote the sum of the $s$-th power of the positive divisors of $N$ and $ \sigma_{s,r} (N; m) =\sum_{d|N , d\equiv r \bmod m}d^s$ with $N,m,r,s,d\in \mathbb{Z}$, $d,s>0$ and $r\geq 0$. In a celebrated paper [Ram], Ramanujan proved $ \sum_{k=1}^{N-1}\sigma_{1}(k)\sigma_{1}(N-k) = \frac{5}{12}\sigma_{3}(N)+ \frac{1}{12}\sigma_{1}(N) -\frac{6}{12}N\sigma_{1}(N)$ using elementary arguments. The coefficients' relation in this identity ($\frac{5}{12} +\frac{1}{12}-\frac{6}{12}=0$) motivated us to write this article. In this talk, we presented the convolution sums $\sum_{k< N / m} \sigma_{1,i}(dk;2)\sigma_{1,j}(N-mk;2)$ for odd and even divisor functions with $i,j=0,1$, $m=1,2,4$, and $d|m$. If $N$ is an odd positive integer, $i,j=0,1$, $m=1,2,4$, $s=0,1,2$, and $d|m|2^s$, then there exist $u, a,b,c\in\Bbb Z$ satisfying $\sum_{k<2^s N/m} \sigma_{1,i}(dk;2)\sigma_{1,j}(2^s N-mk;2)=\frac{1}{u}[a\sigma_3(N) +bN\sigma_1(N) +c\sigma_1(N)]$ with $a+b+c=0$ and $(u,a,b,c)=1$. We also give an elementary problem (O) and solve special cases of them in (O) . Finally, we proposed a curious conjecture drawing inspiration from Ramamnujan as follows:

Conjecture: If $N$ is an odd positive integer, $d, m,n\in\Bbb Z^+$ and $d|m|n$, then there exist $u, a,b,c\in\Bbb Z$ satisfying $\sum_{k< nN /m } \sigma_{1,i}(dk;2)\sigma_{1,j}(n N-m k;2)=\frac{1}{u}[a\sigma_3(N) +bN\sigma_1(N) +c\sigma_1(N)]$ with $a+b+c=0$ and $(u,a,b,c)=1$.

May 10, 2012 Park, Jeehoon (POSTECH)

Title: $p$-adic local Langlands for dihedral Galois representations

Abstract: The goal of this talk is to construct the $p$-adic Local Langlands correspondence for $GL_2$ for the 2-dimensional irreducible dihedral $p$-adic Galois representations. P. Colmez constructed such a correspondence for any $p$-adic Galois representation via the theory of $(\phi,\Gamma)$-modules. Our method is completely different from his and we use more representation theoretic technic not using any $(\phi,\Gamma)$-modules. We do this by constructing a $p$-adic theta correspondence between $p$-adic characters of a quadratic extension field $E$ of $Q_p$ and admissible $p$-adic Banach representations of $GL_2(Q_p)$.

May 25, 2012 Park, Jeongho (POSTECH)

Title: From class numbers to elliptic curves

Abstract: The distribution of class numbers of real quadratic fields will be discussed in an analtic point of view. Some of convincing conjectures on it and the behavior of fundamental units will lead us to stride over continued fractions and modular arithmetic, where a family of elliptic curves appears.

There is no seminar on 14th June for 2012 KIAS-POSTECH Workshop (L-series).