Mini-workshop for young number theorists




November 10 - 11                 KIAS 7323, 8101

Title/Abstract Home > Title/Abstract

Ryota Shii (Kyushu University)

Title: On non-trivial $Lambda$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $mathbb{Z}_{p}$-extension at inert primes 

Abstract:  In Iwasawa theory, it is a fundamental problem that a Selmer group of elliptic curves has no non-trivial $Lambda$-submodules with finite index. Regarding this problem, there is some research by R. Greenberg, B. D. Kim, and T. Kitajima--R. Otsuki. We solved this problem under some assumptions for the anticyclotomic $mathbb{Z}_{p}$-extension of an imaginary quadratic field in which $p$ is inert. In this talk, we present the background of this research and explain the sketch of the proof.


Taiga Adachi (Kyushu University)

Title: The 2-adic valuations of the algebraic central L-values of quadratic twists of weight 2 modular forms (Joint work with Keiichiro Nomoto (Koden Electronics Co., Ltd.) and Ryota Shii (Kyushu university))

Abstract : Recently, Shuai Zhai, Chao Li and Li Cai calculated the 2-adic valuations of the algebraic central L-values for quadratic Dirichlet characters and weight 2 modular forms by using modular symbols. They imposed some conditions on the conductors of characters and Fourier coefficients of modular forms. We succeeded in obtaining the 2-adic valuations for a new class of quadratic twists. This talk is based on a joint work with Keiichiro Nomoto (Koden Electronics Co., Ltd.) and Ryota Shii (Kyushu university).


Ryota Tajima (Kyushu University)

Title: The $p$-adic constant for mock modular forms associated to CM forms

Abstract: Let $g in S_{k}(Gamma_{0}(N))$ be a normalized newform and $f$ be a harmonic Maass form that is good for $g$. The holomorphic part of $f$ is called a mock modular form and denoted by $f^{+}$. For odd prime $p$, K. Bringmann, P. Guerzhoy, and B. Kane obtained a $p$-adic modular form of level $pN$ from $f^{+}$ and a certain $p$-adic constant $alpha_{g}(f)$. When $g$ has complex multiplication by an imaginary quadratic field $K$ and $p$ is split in $mathcal{O}_{K}$, it is known that $alpha_{g}(f)$ is zero. On the other hand, we do not know much about $alpha_{g}(f)$ for an inert prime $p$. In this talk, we prove that $alpha_{g}(f)$ is a $p$-adic unit when $p$ is inert in $mathcal{O}_{K}$ and $dim_{mathbb{C}}S_{k}(Gamma_{0}(N))=1$.


Eunju Shin (Seoul National University)

Title: Decidability for the equivalence of integral solutions of character varieties on a four-punctured sphere

Abstract: In this talk, we will consider integral solutions of character varieties on a four-punctured sphere. It is well known that the mapping class group acts on these solutions. We will then introduce graphs associated with a height function in a one-to-one correspondence with mapping class group orbits and finally prove that it is decidable whether or not any two integral solutions are in the same mapping class group orbit.


Yong-Gyu Choi (KAIST)

Title: On degenerations of D-shtukas over ramified legs

Abstract: Moduli spaces of shtukas on smooth projective curves over finite fields are analogues of Shimura varieties in positive characteristic. We provide a necessary and sufficient condition for moduli spaces of shtukas for central division algebras to be proper over the whole curve. This enhances the previous work of Lau which supplied a properness criterion over the unramified locus of the curve. Somewhat surprisingly, the properness criterion over the whole curve and the one over the unramified locus are essentially different. This is a joint work with Wansu Kim and Junyeong Park.



GyeongWon Oh (UNIST)

Title: Number of cyclic extensions with local conditions and its applications.

Abstract: Many mathematicians have studied counting number fields of a fixed degree with respect to discriminants or conductors. Let l be an odd prime number. In this talk, I introduce how to count cyclic extensions of degree l over Q satisfying a finite number of local conditions with a power- saving error term in the conductor aspect. It generalizes some of Maki’s work about the number of abelian extensions in the conductor aspect without any local conditions. After that, I explain its application in number theory, such as the average of residue and distribution of L-values in the critical strip. This talk is based on a joint work with my advisor P.J.Cho.