Abstracts
Speaker : 박지훈 (Park, Jeehoon)
Title : Modular Symbols with Values in Beilinson-Kato Distributions
Abstract : The Siegel units on the upper half plane studied by K. Kato play a key role in his construction of an Euler system for elliptic curves;
Two key features of the Siegel units are known, namely,
(S-1) the Siegel units satisfy a distribution property;
(S-2) the Siegel units satisfy the Manin relations in the Milnor K_2-group of a modular curve;
This distribution property and the Manin relations suggest the existence of a \GL_2(\Q)-invariant modular symbol with coefficients in $K$-theoretic-valued distributions.
The goal of this talk is to construct such a modular symbol explicitly.
More precisely, for each integer n ≥ 1, we explicitly construct a \GL_n(\Q)-invariant modular symbol \xi_n with coefficients in a space of distributions that takes values in the Milnor K_n-group of the modular function field. Although the various elements that go into this construction appear in various places in the literature, to our knowledge no-one has yet packaged them in this way; we expect that this point of view may be useful for describing new and known Euler systems i.e. “norm-compatible systems of cohomology classes" (especially that of Kato) via the distribution property of our modular symbols, and relating them to computing special values of L-functions and zeta functions.
The talk is based on a joint work with C. Busuioc, O. Patashnick, and G. Stevens.
Speaker : 유명준 (Yu, Myungjun)
Title : The moment method and its application to arithmetic statistics
Abstract : Ellenberg, Venkatesh and Westerland proved the Cohen--Lenstra conjecture for function fields under a mild condition using the moment method. Subsequently, Wood generalized this moment method to study sequences of probability distributions on finite abelian p-groups. In this talk, I will first explain the main ideas behind the proof of the moment theorem. I will then discuss several applications of the moment method.
Speaker : 김민석 (Kim, Minseok)
Title : Increasing the $p$-Selmer rank by twisting
Abstract : We study the $p$-Selmer groups in the family of $p$-twists of an elliptic curve $E$ over a number field $K$.
We prove that if $E/K$ is an elliptic curve over a number field $K$, and if $d$ is congruent to the dimension of the Selmer group of $E/K$ modulo $2$ and is greater than that dimension, then there exist infinitely many characters $\chi \in \Hom(G_K, \mu_p)$ such that $\dim_{\F_p}(\Sel_p(E/K, \chi)) = d$ under certain conditions.
Speaker : 정근영 (Jeong, Keunyoung) (Talk 1)
Title : Distribution of analytic ranks of elliptic curves
Abstract : The minimalist conjecture predicts that among the elliptic curves, 50% have rank 0, 50% have rank 1, and 0% have rank greater than 1. In this talk, we will present results supporting the conjecture for families of elliptic curves with a prescribed torsion structure. The main technique involves computing the 1-level density of low-lying zeros, and counting rational points in the weighted projective space subject to a local condition. This talk is based on joint work with Peter J. Cho and Junyeong Park.
Speaker : 김도형 (Kim, Dohyeong)
Title : Some groups of order eight as Galois groups
Abstract : Wang showed that the rational prime two never remains a prime in a degree eight cyclic extension. Since the field of 2-adic rationals have an unramified cyclic extension of degree eight, it amounts to saying that certain local characters do not extend globally. In this expository talk, we will replace the cyclic group of order eight with a nonabelian group of order eight for which the analogous question has the opposite answer. If time permits, we will speculate on larger groups.
Speaker : 양인규 (Yang, Ingyu)
Title : Splitting of a prime ideal in mod $\ell$ Heisenberg extension
Abstract : For primes $p$ and $\ell$ such that $\ell$ divides $p-1$, Hirano and Morishita constructed a nonabelian Galois extension of the function field $\bff_p(t)$ whose degree is $\ell^3$ and Galois group is of Heisenberg type. Here we analyze how primes of degree one decompose in such extensions. It amounts to investigating the decomposition of the principal ideal $(t-a)$ for $a \in \bff_p-\{0,1\}$ and our main result determines when it decomposes completely in terms of an explicit polynomial in $a$. It is reminiscent of Euler's criterion. The proof relies on both the group structure of the mod-$\ell$ Heisenberg group and the arithmetic of field extensions.
Speaker : 정근영 (Jeong, Keunyoung) (Talk 2)
Title : Distribution of analytic ranks in families of twists
Abstract : Applying the minimalist conjecture to families of quadratic twists of an elliptic curve leads to the Goldfeld conjecture. While the conjecture can be extended to various arithmetic settings, some cases are known to admit counterexamples. In this talk, we present several results that provide evidence, in a weaker sense, for refined versions of the Goldfeld conjecture. The examples discussed include CM elliptic curves defined over their CM field, Jacobians of hyperelliptic curves, and Jacobians of Fermat curves. This talk is based on joint work and an ongoing project with Jigu Kim, Taekyung Kim, Yeong-Wook Kwon, Junyeong Park, and Donggeon Yhee.
Speaker : 윤종흔 (Yoon, Jongheun)
Title : Asymptotics of n-universal lattices over number fields
Abstract : In this talk, we prove an explicit asymptotic formula for the logarithm of the minimal ranks of n-universal lattices over the ring of integers of totally real number fields.
We also show that, for any constant C > 0 and n ≥ 3, there are only finitely many totally real fields with an n-universal lattice of rank at most C, with all such fields being effectively computable. Similarly, for any n ≥ 3, we show that there are only finitely many totally real fields admitting an n-universal criterion set of size at most C, with all such fields likewise being effectively computable.
This is a joint work with Dayoon Park, Robin Visser, and Pavlo Yatsyna.