Program
초록은 발표 순서대로 쓰여져 있습니다.
Title : Anti-Mordell-Weil fields and the infinite rank of abelian varieties over fields with finitely generated Galois groups
Speaker : Bo-Hae Im (KAIST)
Abstract : We give the definition of an anti-Mordell-Weil field and introduce some related conjectures on the fields with finitely generated Galois groups. Various methods have been applied to prove these conjectures and they are introduced. At last we prove that the trivial representation of certain free groups appearing on a representation of the absolute Galois group on an abelian variety A has infinite multiplicity if and only if A over the fixed field under a given automorphism in the absolute Galois group, which recovers the proof of Frey and Jarden's result.
Title : 2-Selmer ranks of elliptic curves in the family of quadratic twists
Speaker : Myungjun Yu (KIAS, CMC)
Abstract : Let E be an elliptic curve over a number field K. The Mordell-Weil rank is defined by the number of (linearly independent) Z-copies in the group E(K). We often study Selmer groups to compute the Mordell-Weil rank. A quadratic twist of E is an elliptic curve, which is geometrically the same with E, but can be arithmetically very different from E. In this talk, we explain how 2-Selmer ranks vary in the family of quadratic twists. We will also discuss more generally the Jacobian of hyperelliptic curve case.
Title : Distribution of Bianchi modular symbols
Speaker : Dohyeong Kim (SNU)
Abstract : Over the rationals, arithmetic properties of elliptic curves admit concrete analytic descriptions in terms of modular symbols. Over imaginary quadratic fields, Bianchi modular symbols are expected to play similar roles. We use tools from dynamical analysis to investigate distributions of Bianchi modular symbols, focusing on the case of Gaussian integers. It boils down to a distributional analysis of the complexity of the Euclidean algorithm for Gaussian integers, which might be of independent interest. It is a report on the work in progress with Jungwon Lee and Seonhee Lim.
Title : Modular symbols and Selmer groups
Speaker : Chan-Ho Kim (KIAS, CMC)
Abstract : We discuss a more refined relationship between modular symbols and Selmer groups of elliptic curves from the viewpoint of refined Iwasawa theory.
Title : Rational cuspidal subgroup of J_0(N)
Speaker : Hwajong Yoo (SNU)
Abstract : We construct modular functions on J_0(N) using variants of Siegel units. Then we prove that the rational cuspidal subgroup of J_0(N) is equal to the rational cuspidal divisor class group of X_0(N) when N=p^2M for any squarefree integer M.
Title : On the 2-adic valuations of L-values of elliptic curves
Speaker : Junhwa Choi (KIAS)
Abstract : In this talk, I will discuss some analytic results about the 2-part of the Birch and Swinnerton-Dyer conjecture for an infinite family of special elliptic curves defined over the Hilbert class field of an imaginary quadratic field.
Title : Average analytic rank of elliptic curves with prescribed torsion
Speaker : Peter Jaehyun Cho (UNIST)
Abstract : We show that the average analytic rank of elliptic curves with prescribed torsion $G$ is bounded for every torsion group $G$ under GRH for elliptic curve $L$-functions.
Title : The algebraic parts of the central values of quadratics twists of modular L-functions modulo l
Speaker : Youngmin Lee (KAIST)
Abstract : In this talk, we find the set of fundamental discriminants D such that l-adic valuation of algebraic part of central value of D-quadratic twists L-function is zero. These apply to indivisibility of the order of the Shafarevich-Tate group of an elliptic curve over $\mathbb{Q}$. To do this, we based on refinement of Waldspurger's formula and properties of mod l modular forms of half-integral weight whose the Fourier coefficients are supported on finitely many square classes. This is a joint work with Dohoon Choi.