Number Theory Seminars 2012

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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 Prufer domain or a GCD-domain. As a corollary, we study a Bezout 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).

June 28, 2012 Lee, Sangjune (Emory University)

Title: The number of Sidon sets in $[n]$ and the maximum size of Sidon subsets contained in a random subset of $[n]$

Abstract: A set $A$ of positive integers is called a Sidon set if all the sums $a_1+a_2$, with $a_1\leq a_2$ and $a_1 ,a_2 \in A$, are distinct. In this talk we deal with results on the number of Sidon sets in $[n]$ and the maximum size of Sidon sets in sparse random subsets of $[n]$.

The first question in this talk was suggested by Cameron--Erd$\ddot{o}$s in 1990. They proposed the problem of estimating the number of Sidon sets contained in $[n]$. Results of Chowla, Erd$\ddot{o}$s, Singer, and Tur$\acute{a}$n from the 1940s imply that the maximum size of Sidon sets in $[n]$ is $\sqrt{n}(1+o(1))$. From this result, one can trivially obtain that the number of Sidon sets contained in $[n]$ is between $2^{(1+o(1))\sqrt{n}}$ and $n^{c\sqrt{n}}$ for some absolute constant $c$. We obtain an upper bound $2^{c\sqrt{n}}$ on the number of Sidon sets which is sharp up to a constant factor in the exponent when compared to the previous lower bound $2^{(1+o(1))\sqrt{n}}$.

Next, we investigate the maximum size of Sidon sets contained in sparse random sets $R\subset [n]$. Let $R=[n]_m$ be a uniformly chosen, random $m$-element subset of $[n]$. Let $F([n]_m)=\max\{|S|\colon S\subset[n]_m\hbox{ is Sidon}\}$. Fix a constant $0\leq a\leq1$ and suppose $m=(1+o(1))n^a$. We show that there is a constant $b=b(a)$ for which \begin{equation}\label{eq:b(101)} F([n]_m)=n^{b+o(1)} \end{equation} almost surely and we obtain what $b=b(a)$ is. Surprisingly, between two points $a=1/3$ and $a=2/3$, the function $b=b(a)$ is constant.

This is joint work with Kohayakawa, R$\ddot{o}$dl, and Samotij.

July 12, 2012 Cho, Jaehyun (Fields Institute)

Title: Simple zeros of automorphic $L$-functions on $\Gamma_0(4)$

Abstract: Let $f$ be a Hecke modular form or Maass form on $\Gamma_0(4)$ and $L(f,s)$ is the attached $L$-function. We show that existence of one simple of $L(f,s)$ implies an infinitude of simple zeros.

July 26, 2012 Yhee, Donggeon (Seoul National University)

Title: From torsion to Gross and Zagier

Abstract: Torsion structure of elliptic curves plays an important role in some problems on arithmetic of elliptic curves. In this talk, I will describe the role for a conjecture of Gross and Zagier. B. Gross and D. Zagier suggest a conjecture in their paper in 1986 ; when Heegner point has infinite order, the index $[E(K):\mathbb Z P_K]$ can be written in a product of Manin constant, Tamagawa numbers, and square root of Shafarevich-Tate group. As a weak result, the product would be divisible by the number of torsion points in $E(K)$. We will study how the number of torsion points divides each terms in the product.

July 26, 2012 Jung, Junehyuk (Princeton University)

Title: On the sparsity of positive-definite automorphic forms within a family

Abstract: In 1990, Baker and Montgomery prove that almost all Fekete polynomials with respect to a certain ordering have at least one zero on the interval $(0,1)$. Fekete polynomial has no zero on the interval $(0,1)$ if and only if the corresponding automorphic form is positive-definite. Generalizing their result, I'm going to prove for various families of automorphic forms that almost every automorphic forms within them are not positive-definite. These families include the family of holomorphic cusp forms, the family of the Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves.

August 9, 2012 Pinter, Akos (University of Debrecen)

Title: Modular forms and diophantine equations

Abstract: The application of modular forms for certain diophantine equations provides several attractive results. In this talk the speaker shows some recent finiteness theorems related to the power values of power sums, power values in the terms of an arithmetical progression, Fermat-like ternary equations and binomial Thue equations.

August 9, 2012 Pinter, Akos (University of Debrecen)

Title: Counting polynomials and diophantine equations

Abstract: We report on some recent diophantine results concerning classical counting polynomials including binomial coefficients and power sums.

September 12 (Wed.), 2012 Go Yamashita (The thinktank at TOYOTA Central R& D)

Title: Reductions of two dimensional crystalline representations and Hypergeometric polynomials

Abstract: The reductions of crystalline representations are not understood so much even in the two dimensional case. The case of large $v_p(a_p)$ can be attacked by the integral $p$-adic Hodge theory (e.g., Berger-Li-Zhu, Berger-Breuil). The case of small $v_p(a_p)$ can be attacked by the $p$-adic local Langlands (e.g., Berger-Breuil, Buzzard-Gee). However, the intermediate case was unattackable so far. In this talk, we calculate the reduction the case with small and intermediate $v_p(a_p)$ if the weight is less than or equal to $(p^2 +1)/2$ by the method of the integral $p$-adic Hodge theory. The hypergeometric polynomials mysteriously appeared in the calculation, and such phenomena had not been observed so far. This is a joint work with S. Yasuda(Osaka).

September 26 (Wed.), 2012 Kwon, Jae Hoon (sungkyunkwan University)

Title: Littlewood identity and crystal base

Abstract: We give a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type $B$ and $C$ in terms of classical Young tableaux. We then obtain a new description of its Littlewood-Richardson rule and a maximal Levi branching rule in terms of classical Littlewood-Richardson tableaux, which extends in a bijective way the well-known stable formulas at large ranks.

October 11 (Thur.), 2012 Lee, Jung-Jo (Yonsei University)

Title: Dirichlet series and Hyperelliptic Curves

Abstract: For a fixed hyperelliptic curve $C$, we study the variation of rational points on the quadratic twists of $C$. We will associate a certain Dirichlet series to the family of quadratic twists of $C$, and study this series to obtain information about the curves. We will also consider related problems with information obtained in this way. This is a joint work with Ram Murty.

November, 1 (Thur.), 2012 Kim, Jon-Lark (Seogang University)

Title: Codes, Rings and Lattices

Abstract: Codes over rings have been used in the construction of interesting Eucildean or Hermitian lattices.We begin with basic definitions of codes and lattices. It is known that there are exactly three commutative rings with 1 of order $p^2$ and characteristic $p$, where $p$ is prime. We describe that how these rings can be related to certain quotient rings of the ring of algebraic integers of an imaginary quadratic number field. Then we construct Hermitian lattices from codes over these rings. We only assume graduate level abstract algebra course. Some part is based on the joint work with Y. Lee and S. Dougherty.

November, 15 (Thur.), 2012 Kim, Na Young (Seoul National University)

Title: Elliptic curves with constant root number for all cubic twists

Abstract: We consider elliptic curves $E$ over a number field $K$ with the property that the root number $w(E_d/K)$ is constant for all twists $E_d$ of $E$. In quadratic twist case, T. Dokchitser showed that it is equivalent to the analogous property over local fields. We prove that in cubic twist case, it is completely determined by whether K contains $\sqrt{-3}$.

November, 29 (Thur.), 2012 Kim, Ji Young (Seoul National University)

Title: Strictly regular positive definite quaternary forms

Abstract: A positive denite quadratic $\mathbb{Z}$-lattice is said to be strictly regular if it primitively represents all positive integers that are primitively represented by its genus. It will be shown that there exist only finitely many isometry classes of primitive integral positive definite quaternary quadratic $\mathbb{Z}$-lattices that are strictly regular. The complete enumeration of the diagonalizable lattices having this property will be described. As a consequence, all one- class genera of diagonal quaternary quadratic forms are determined.

December, 13 (Thur.), 2012 Zhi-Hong Sun (Huaiyin Normal University)

Title: Legendre polynomials and supercongruences

Abstract: Let $p$ be an odd prime. Recently Zhi-Wei Sun made lots of conjectures on $$\sum_{k=0}^{p-1} \frac{{ 2k \choose k}^2}{m^k}, \sum_{k=0}^{p-1} \frac{{ 2k \choose k}^3}{m^k}, \sum_{k=0}^{p-1} \frac{{ 2k \choose k}{ 3k \choose k}}{m^k}, \sum_{k=0}^{p-1} \frac{{ 2k \choose k}{ 4k \choose 2k}}{m^k},$$ $$\sum_{k=0}^{p-1} \frac{{ 2k \choose k}^2 { 3k \choose k }}{m^k}, \sum_{k=0}^{p-1} \frac{{ 2k \choose k}^2{ 4k \choose 2k}}{m^k}, \sum_{k=0}^{p-1} \frac{{ 3k \choose k} { 6k \choose 3k}}{m^k}, \sum_{k=0}^{p-1} \frac{{ 2k \choose k} { 3k \choose k} { 6k \choose 3k}}{m^k},$$ modulo $p^2$, where $m$ is an integer not divisible by $p$. For instance, he conjectured $\sum_{k=0}^{p-1}{ 2k \choose k}^3 \equiv 0 $($\mod p^2$) for $p \equiv 3, 5, 6$ ($\mod 7$). Let $\left\{ P_n(x)\right\}$ be the Legendre polynomials. In this talk we reveal connections between Legendre polynomials and cubic Jacobsthal sums. For example, we have $P_{[\frac{p}{6}]} (t)\equiv - ( \frac{3}{p}) \sum_{x=0}^{p-1} (\frac{x^3 - 3x + 2t}{p}) $ ($\mod p$ ), where $[\cdot ]$ is the greatest integer function, $(\frac{a}{p})$ is the Legendre symbol and $t$ is a rational $p$-adic integer. Using Legendre polynomials, ellipitic curves with complex multiplication and combinatorial identities proved by WZ method we solve many of Zhi-Wei Sun's conjectures on supercongruences involving the above sums. We also present some new conjectures on supercongruences.

December, 27 (Thur.), 2012 Lee, Dong-Uk (KAIST)

Title: Non-emptiness of Newton strata of Shimura varieties

Abstract: The Newton stratification on a special fiber of a Shimura variety (of abelian type) is a stratification defined in terms of F-isocrystals with G-structure. There is a conjectural group-theoretic description of the F-isocrystals that are expected to show up in a given Shimura variety. We discuss two results confirming this conjecture, first one on the mu-ordinary locus for simply-connected abelian type, second one on certain strata (including mu-ordinary and basic ones) for simply-connected PEL-type. We also explain how the Langlands-Rapoport conjecture, which is established in the PEL-type cases, leads to such non-emptiness results.