### Schedules

운영위원
2/20(Mon) 2/21(Tue) 2/22(Wed)
9:30 - 10:10   김병두 (V.U.W.) 이현석 (연세대학교)
10:30 - 11:10   김찬호 (KIAS) 임수봉 (성균관대학교)
11:30 - 12:10   최도훈 (한국항공대) Free Discussion
12:30 - 14:00   Lunch
14:00 - 14:40 최준화 (POSTECH) 박철 (KIAS)
15:00 - 15:40 김광섭 (KIAS) 양재현 (인하대학교)
16:00 - 16:40 박지훈 (POSTECH) 권용재 (인하대학교)
17:00 - 17:40   주장원 (서울대학교)
18:00 -    Banquet

# 2/20 (Mon)

## Title : Quadratic twists of Gross curves

Abstract :  Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime $q\equiv 7\pmod{8}$, and write $\mathcal{O}$ for the ring of integers of $K$. Let $H=K(j(\mathcal{O}))$ denote the Hilbert class field of $K$, where $j$ is the classical $j$-function. There is a unique elliptic curve defined over $\mathbb{Q}(j(\mathcal{O}))$, which is called the Gross curve, whose $j$-invariant is equal to $j(\mathcal{O})$, whose ring of $H$-endomorphisms is equal to $\mathcal{O}$, whose minimal discriminant ideal in $H$ is equal to $(q^{3})$, and which is isogenous to all of its conjugates under the action of the Galois group of $H$ over $K$. In this talk, we will discuss the Iwasawa theory of quadratic twists of these Gross curves, especially the non-vanishing theorem on Iwasawa $\mu$-invariants.

## Title : The minimal number of Frobenius elements of $G_{K,S}$ whose conjugacy classes generate the whole group $G_{K,S}$

Abstract : Assume that $K$ is a number field and $S$ is a finite set of primes of $K$. Let $G_S(K)$ be the Galois group $\mbox{Gal}(K_{S}/K)$, where $K_S$ is the maximal extension of $K$, which is unramified outside the primes in $S$. Suppose that the rank of $G^{ab_{K,S}}$ is $r$. In this talk, we use topological methods to show that $G_{K,S}$ can be generated by $r$ (or 1 if $r = 0$) Frobenius classes and $r$ is minimal.

## Title : Inverse values of the modular $j$-function and homotopy Lie theory

Abstract : The goal of this talk is to give a simple arithmetic application of the enhanced homotopy (Lie) theory for algebra varieties developed by myself and my self and my collaborators (Dokyung Kim, Yesule Kim, and Jae-Suk Park). Namely, we compute the inverse values of the modular $j$-function by using deformation theory for period matrices of elliptic curves based on homotopy Lie theory. The key other ingredient in our approach is J. Carlson and P. Griffiths' explicit computation regarding infinitesimal variations of Hodge structures. This is a joint work with Kwang Hyun Kim and Yesule Kim.

# 2/21 (Tue)

## Title : The Mordell-Weil ranks over cyclotomic extensions of abelian varieties defined over number fields whose primes above $p$ are totally ramified

Abstract : Continuing Kim's earlier work ("Ranks of the Rational Points of Abelian Varieties over Ramified Fields, and Iwasawa Theory for Primes with Non-Ordinary Reduction," preprint) on the Mordell-Weil ranks over cyclotomic extensions of elliptic curves over number fields whose primes above $p$ are totally ramified, we study the Mordell-Weil ranks of abelian varieties under similar conditions.

Our model is Barry Mazur's Rational Points of Abelian Varieties with Values in Towers of Number Fields'' (Inventiones math. 18, 183--266 (1972)), and Bernadette Perrin-Riou's Theorie d'Iwasawa p-adique locale et globale'' (Invent. math. 99, 247-292 (1990)). There are two main difficulties: good non-ordinary reduction, which is generally harder than good ordinary reduction, and the ramification of the field over which the abelian varieties are defined.

We follow Kim's earlier work, which in turn adopted and further developed Perrin-Riou's ideas. However, what we have to overcome is not trivial at all. In particular, whereas elliptic curves have only one kind of good non-ordinary reduction--supersingular reduction, abelian varieties can have a mix of ordinary and non-ordinary reduction types, and since abelian varieties generally have higher dimensions, they have multiple logarithm generators, and therefore, the local points we generate (mimicking universal norms) can have different norm relations. Finally, our technique involves constructing a power series which functions as a logarithm much as Shinichi Kobayashi (Iwasawa theory for elliptic curves at supersingular primes. Inventiones Mathematicae 152 (2003), no.1, 1-36) does, but finding the right constant becomes a non-trivial matter if we work at the level of generality in our work.

Overcoming all these issues, we generalize Kim's work (which means we also generalize Mazur's work, and Perrin-Riou's work in our context), and in particular, obtain bounds for the Mordell-Weil ranks of abelian varieties defined over the above-mentioned number fields. In particular, we obtain the Mordell-Weil ranks over cyclotomic extensions of the Jacobian varieties of hyperelliptic curves of large $p$-power exponents.

## Title : An anticyclotomic Mazur-Tate conjecture for modular forms

Abstract :  We discuss what the Mazur-Tate conjecture and its anticyclotomic variant are and how to approach to the the anticyclotomic one under certain assumptions.

## Title : 자기 동형 표현의 유리성 체에 관한 산술

Abstract : 주어진 자기 동형 표현에 관해 동형 사상을 주는 갈루아 표현들의 고정되는 복소수 상에 가장 작은 체를 자기 동형 표현의 유리성 체라 한다자기 동형 표현과 혹은 특수한 자기 동형 표현의 모임들에 대한 유리성 체에 관한 산술에 대해 발표한다

## Title : A first step towards mod $p$ Langlands program for $\mbox{GL}_{n}(\mathbb{Q}_{q})$

Abstract : Let $F/\mathbb{Q}$ be a CM field in which $p$ splits completely and $\overline{r}:\mbox{Gal}(\overline{\mathbb{Q}}/F)\rightarrow\mbox{GL}_{n}(\overline{\mathbb{F}}_{p})$ a continuous automorphic Galois representation. We assume that $\overline{r}|_{\mbox{Gal}(\overline{\mathbb{Q}}_{p}/F_{w})}$ is an ordinary representation at a place $w$ above $p$. In this talk, we discuss a problem about local-global compatibility in the mod $p$ Langlands program for $\mbox{GL}_{n}(\mathbb{Q}_{p})$. It is expected that if $\overline{r}|_{\mbox{Gal}(\overline{\mathbb{Q}}_{p}/F_{w})}$ is tamely ramified, then it is determined by the set of modular Serre weights and the Hecke action on its constituents. However, this is not true if $\overline{r}|_{\mbox{Gal}(\overline{\mathbb{Q}}_{p}/F_{w})}$ is wildly ramified, and the question of determining $\overline{r}|_{\mbox{Gal}(\overline{\mathbb{Q}}_{p}/F_{w})}$ from a space of mod $p$ automorphic forms lies deeper than the weight part of Serre's conjecture. We define a local invariant associated to $\overline{r}|_{\mbox{Gal}(\overline{\mathbb{Q}}_{p}/F_{w})}$ in terms of Fontaine-Laffaille theory, and discuss a way to prove that the local invariant associated to $\overline{r}|_{\mbox{Gal}(\overline{\mathbb{Q}}_{p}/F_{w})}$ can be obtained in terms of a refined Hecke action on a space of mod $p$ algebraic automorphic forms on a compact unitary group.

## Title : Stable Jacobi forms and the universal Jacobian locus

Abstract : In this talk, I will explain the notion of the stability of Jacobi forms, present some properties of stable Jacobi forms and discuss the relations between stable Jacobi forms and the universal Jacobian locus. I also give a brief review on the work on stable modular forms that had been done by E. Freitag forty years ago, and the recent work of G. Codogni and N. I. Shepherd-Barron. Finally I present some interesting open problems.

## Title : The rank of an elliptic curve

Abstract : In this talk, we survey the recent works about the rank of an elliptic curve and the Selmer groups of an elliptic curve that were done by Manjul Bhargava and his colleagues. In the end of this talk, we make a comment on the new directions to the study of the rank of an elliptic curve.

## Title : A generalization of Gauss' triangular theorem

Abstract : A quadratic polynomial $\Phi_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $\Phi_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any nonnegative integer $n$. In this talk, we show that if $(a,b,c)=(2,2,6), (2,3,5)$ or $(2,3,7)$, then $\Phi_{a,b,c}( x,y,z)$ is universal. These were conjectured  by Zhi-Wei Sun. This is a joint work with Byeong-Kweon Oh.}

# 2/22 (Wed)

## Title : MZV (Multiple Zeta Values)

Abstract : L. Euler (1707-1783) investigated the values of the numbers

$$\zeta(s)=\sum_{n\geq 1}\frac{1}{n^{s}}$$

for $s$ a rational integer, and B. Riemann (1826-1866) extended this function to complex values of $s$. For $s$ a positive even integer, $\zeta(s)/\pi^{s}$ is a rational number. Our knowledge on the values of $\zeta(s)$ for $s$ a positive odd integer is extremely limited. Recent progress involves the wider set of numbers

$$\zeta(s_{1},\ldots,s_{k})=\sum_{n_{1}>\cdots>n_{k}\geq 1}\frac{1}{n_{1}^{s_{1}}\cdots n_{k}^{s_{k}}}$$

for positive integers with $s_{1},\ldots,s_{k}$ positive integers with $s_{1}\geq 2$.

## Title : Limits of traces of singular moduli

Abstract :  Zagier proved that traces $\mbox{Tr}_{d}(j_{1})$ of singular moduli for $d<0$ are coefficients of a weakly holomorphic modular form $g_{1}$ of weight $3/2$. Duke, Imamoglu, and Toth defined a modular trace $\mbox{Tr}_{d}(j_{1})$ for $d>0$ by using the cycle integral of $j_{1}$, and showed that its generating function is a mock modular form whose shadow is $g_{1}$. In this talk, we introduce connections between $\mbox{Tr}_{d}(j_{1})$ for $d>0$ and $\mbox{Tr}_{d}(j_{1})$ for $d<0$ by considering a certain asymptotic behavior of twisted sums of $\mbox{Tr}_{d}(j_{1})$ over $d>0$ and $d<0$, respectively.