Number Theory Seminars 2008

 Back to Home January 14 Lan, Kai-Wen  (Harvard University) Title: Arithmetic compactifications of PEL-type Shimura varieties I Abstract: In these lectures, we shall explain how to construct the toroidal and minimal compactifications for smooth integral models of PEL-type Shimura varieites (as defined in the works of Kottwitz and many other people), using generalizations of the techniques developed by Mumford and Faltings-Chai. The essentially new ingredient is the emphasis on level structures, realized by a technical calculation of Riemann forms, which enables us to predict the level structures from the degeneration data, and hence enables us to construct the boundary charts over which there does exists a valid level structure. January 17 Choi, Dohoon (Korea Aerospace University) Title: Modularity of traces of singular moduli and its applications Abstract: Zagier proved that the traces of singular values of the j-invariant function are generated by a certain weakly holomorphic modular form of weight 3/2. Recently, Bruinier and Funke extended this result to general weakly holomorphic modular functions. In this talk, we survey their proofs of modularity of traces of singular moduli, which are based on Borcherds lifting and Kudla Milson lifting. We also announce applications to congruences for traces of singular moduli. March 13 Moon, Hyunsuk (Kyungpook National University) Title: Mordell-Weil groups of Jacobians of hyperelliptic curves Abstract: Frey and Jarden have asked whether the Mordell-Weil group of every nonzero abelian variety defined over a number field K has infinite Mordell-Weil rank over the maximal abelian extension of K. They proved that for elliptic curves over Q. Imai and Top generalized independently this result to the Jacobian variety of a hyperelliptic curve defined over Q. In this talk, we give yet another proof of this result and slightly more precise information on the structure of the Mordell-Weil group. March 27 Lee, Yoonjin (Ewha Womans University) Title: Hyperelliptic Function Fields of High Three Rank Abstract: We present several explicit construction methods of hyperelliptic function fields whose Jacobian or ideal class group has large $3$-rank. Our focus is on finding examples for which the genus and the base field are as small as possible. Algorithms, examples, and numerical data will also be shown. April 10 Choi, Dohoon (Korea Aerospace University) Title: Modularity of traces of singular moduli and its applications II Abstract: This is a continuation to the talk given on Jan 17, 2008. April 29 Yoshihiro Ochi (Tokyo Denki University) Title: On the conjectures for fine Selmer groups of elliptic curves Abstract: We discuss the first examples of the conjectures by Coates and Sujatha for fine Selmer groups of elliptic curves over Q. March 27 Najmuddin Fakhruddin (Tata Institute of Fundamental Research) Title: On certain group schemes over discrete valuation rings Abstract: I will describe some classification results for group schemes over discrete valuation rings with smooth generic fibre and non-reduced special fibre. The cases of particular interest are when the reduced fibres are reductive groups (in which case there is a complete classification if the base is strictly henselian) and when the reduced fibres are semi-abelian varieties. June 16 Woo, Jeechul (Harvard University) Title: On elliptic curves of high rank Abstract: The Mordell-Weil rank of elliptic curves over the rational number is conjectured to be unbounded. The conjecture is expected to be true, however it seems we are far from being able to prove it. In the meantime, finding high rank elliptic curves has attracted (in)dependent attention and been (un)successful, with the record of 28. In this talk, I will introduce some techniques involved and current development to this direction June 25 Park, Jeehoon (McGill University) Title: Iwasawa main conjecture for CM elliptic curves over imaginary quadratic fields at supersingular primes Abstract: This talk is based on Joint work with Bei Zhang and Byung Du Kim. We will explain how to construct both the algebraic plus/minus p-adic L-functions and the analytic plus/minus p-adic L-functions of CM elliptic curves over imaginary quadratic fields at supersingular primes and compare them (Iwasawa main conjecture). June 25 Choi, Suh-Hyun (Harvard University) Title: Introduction to the deformation theory of Galois representations Abstract: In 1994, Andrew Wiles showed that R, the universal deformation ring which represents the deformations of an absolutely irreducible mod l Galois representation (of type D) is isomorphic to the Hecke algebra (of type D) T to prove Fermat's last theorem. Since then, showing R=T became one of the basic tools to prove further modularity problems, so it is important to know the structure of universal deformation (lifting) rings. In this talk, I will give an introduction to the Galois deformation theory, and give some examples. June 26 Sander, Zwegers (University College Dublin) Title: Appell-Lerch sums as mock modular forms Abstract: In this talk, we'll discuss various aspects of the so called Appell functions. These functions and their elliptic and modular transformation properties have been studied by Appell (1884), Lerch (1892), Mordell (1920), and others, and more recently by Polishchuk (2001) and Semikhatov-Taormina-Tipunin (2004). Also, they show up in the theory of mock theta functions (Watson, 1936 and Ono-Bringmann, 2006). We'll discuss the elliptic and modular transformation properties of these functions in detail and give some general results. As an example, we get a generalization of a result by Ono-Bringmann, concerning the rank generating function. If time permits we'll also discuss the so called rank-crank PDE, and generalizations thereof. July 10 Jang, Junmyeong (KIAS) Title: Generic p-ranks and the semi-positivity theorem for semi-stable fibrations Abstract: Let \pi : X \to C be a non-isotrivial semi-stable fibration of a proper smooth surface to a proper smooth curve over an algebraically closed field k. If k is a subfield of \C, the field of complex numbers, the semi-positivity theorem states that all the Harder-Narasimhan slopes of the Hodge bundle of \pi are non-negative. But if the characteristic of k is positive, the semi-positivity theorem does not hold in general. In this talk, we will see the semi-positivity theorem holds if the p-rank of the generic fiber is maximal and the semi-positivity theorem fails potentially if the p-rank of the generic fiber is 0. Then we will see some application of these results. July 10 Han, Lin (Inha University) Title: A classification of 2-dimensional l-adic Galois representations Abstract: Continued from the last year talk about l-adic Grothendieck monodromy theorem, in this talk, we briefly review the theorem and will see how the theorem can be used to classify all 2-dim l-adic Galois representations (of local Galois group), equivalently 2-dim Weil-Deligne representations. If time allows, we also relate them with representations of GL_2 (Q_p) as in local Langlands correspondence. We will mainly focus on how we explicitely compute it. August 6 Henry Kim (KIAS/University of Toronto) Title: Analytic properties of $L$-functions 1 Abstract: We give an overview of $L$-functions by several well-known examples such as the Riemann zeta function and $L$-functions attached to elliptic curves. We give applications of $L$-functions to number theoretic problems. August 8 Chang, Seungwhan (KAIST) Title: Filtered modules and Wach modules Abstract: We will talk about filtered phi-modules and Wach modules, linear algebraic data describing crystalline p-adic representations, and about their interconnection via Berger's theory. As an application, we will briefly explain the results of Berger-Li-Zhu and Dousmanis on construction of families of crystalline representations and their mod p reduction. August 20 Henry Kim (KIAS/University of Toronto) Title: Analytic properties of $L$-functions 2 Abstract: We will give basic analytic problems of $L$-functions such as the generalized Riemann hypothesis and Lindelof hypothesis and subconvexity, and moments of $L$-functions. August 28 Andreas Bender (KIAS) Title: On the Goldbach conjecture in the function field case Abstract: I shall present and discuss a recent result of mine on the problem stated in the title. Since the result depends on a former theorem of mine and Olivier Wittenberg, I shall make some remarks on that as well. September 05 Alan Haynes (University of York) Title: An unsolved problem in probabilistic number theory Abstract: In this talk we are going to explain the statement and history of the Duffin-Schaeffer Conjecture. This conjecture is more than 50 years old and has been attempted by several notable mathematicians. It is a problem about how well almost all real numbers can be approximated by rationals, and it combines ideas from probability theory, number theory, and combinatorics. There are also versions of this conjecture which apply to fractals, manifolds, and local fields, and of these all but a set of measure zero are still unsolved. September 25 Han, Lin (Inha University) Title: Semistable abelian varieties and extensions of group schemes Abstract: In this talk, we will see how we can classify strongly ordinary (at some prime) semistable abelian varieties over Q with some additionaly properties. We focus on how we can construct l-adic Tate modules of abelian varieties from possible shapes of finite flat simple group schemes of l-power order which can arise as quotients of semistable abelian varieties. We will start from reminding some basic properties of group schemes of l-power order. October 17 Kang, Soon-Yi (KAIST, ASARC) Title: Galois trace and modular trace of class invariants Abstract: Zagier showed that the traces of singular values of the modular j-invariant are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier and Funke generalized this result by defining modular traces of arbitrary weakly holomorphic modular functions so that they are Fourier coefficients of the holomorphic part of a harmonic weak Maass form. In this talk, we show that the modular traces coincide with Galois traces in the case of Webers' class invariants. As a result, we can show that there is a divisibility property on the Galois trace of the class invariants. This is a joint work with Daeyeol Jeon and Chang Heon Kim. October 31 Lim, Sung-Geun (POSTECH) Title: Generalized Eisenstein series, modular transformation formulas and infinite series identities November 20 Im, Subong (POSTECH) Title: Construction of Jacobi forms associated to indefinite quadratic forms Abstract: Vign{\'e}ras constructs non-holomorphic theta functions according to indefinite quadratic forms with arbitrary signature. I will create examples of non-holomorphic Jacobi forms associated to indefinite theta series by two different methods using Vign{\'e}ras' theta functions. December 11 Im, Bo-Hae (Chung-Ang University) Title: Non-empty projective hypersurfaces over quasi-finite fields Abstract: In this talk, we show that there exists a function $f\colon \bbn\to\mathbb{N}$ such that for every positive integer $d$, every quasi-finite field $K$ and every projective hypersurface $X$ of degree $d$ and dimension $\ge f(d)$, the set $X(K)$ is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois groups. December 18 Keiichi Gunji (POSTECH) Title: On Siegel Eisenstein series of degree 2 for low weights Abstract: In this talk, we consider the Fourier coefficients of Siegel Eisenstein series for low weights, in the case of level $p$. In particular the calculation of the Euler $p$-factor of the Siegel series are given. As an application, we will give the dimension of the space of Siegel Eisenstein series with respect to the principal congruence subgroup of level $p$, in weight 2 case. December 18 Kentaro Ihara (POSTECH) Title: Structure of the algebra of multiple zeta values Abstract: In this seminer, I will talk on a basic problem and properties about the multiple zeta values (MZVs). These values are special values of a multi-variable version of classical Riemann zeta function and they also relate to many mathematical fields, combinatorics, arithmetic geometry and knot theory.... First I will talk about product structure among MZVs, and then discuss the structure of the algebra which generated by MZVs.