김완수(KAIST)

Title: Unit-root L-functions and p-adic differential equations

Abstract: We start with reviewing Dwork’s seminal work on a certain p-adic hypergeometric function, which has an application to the unit-root L-function of the Legendre family of elliptic curves in characteristic p>2. Then I would like to overview what can be said about unit-root L-function of the family of abelian varieties over a curve, and discuss its potential applications.

박철(UNIST)

Title: Fontaine--Laffaille modules and their mod-$p$ local-global compatibility

Abstract: Let $K$ be a finite extension of $\mathbb{Q}_p$. It is believed that one can attach a smooth $\overline{\mathbb{F}}_p$-representation of $\mathrm{GL}_n(K)$ (or a packet of such representations) to a continuous Galois representation $\mathrm{Gal}(\overline{\mathbb{Q}}_p/K)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ in a natural way, that is called mod $p$ Langlands program for $\mathrm{GL}_n(K)$. This is known only for $\mathrm{GL}_2(\mathbb{Q}_p)$: one of the main difficulties is that there is no classification of such smooth representations of $\mathrm{GL}_n(K)$ unless $K=\mathbb{Q}_p$ and $n=2$. However, for a given continuous Galois representation $\overline{\rho}_0:\mathrm{Gal}(\overline{\mathbb{Q}}_p/K) \rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$, one can define a smooth $\overline{\mathbb{F}}_p$-representation $\overline{\Pi}_0$ of $\mathrm{GL}_n(K)$ by a space of mod $p$ automorphic forms on a compact unitary group, which is believed to be a candidate on the automorphic side corresponding to $\overline{\rho}_0$ for mod $p$ Langlands correspondence in the spirit of Emerton. The structure of $\overline{\Pi}_0$ is very mysterious as a representation of $\mathrm{GL}_n(K)$, but it is conjectured that $\overline{\Pi}_0$ determines $\overline{\rho}_0$, which is called mod-$p$ local-global compatibility. In this talk, we discuss a way to prove this conjecture in the case that $\overline{\rho}_0$ is Fontaine--Laffaille. More precisely, we prove that the tamely ramified part of $\overline{\rho}_0$ is determined by the Serre weights attached to $\overline{\rho}_0$, and the wildly ramified part of $\overline{\rho}_0$ is obtained in terms of refined Hecke actions on $\overline{\Pi}_0$. This is based on a joint work with Daniel Le, Bao Le Hung, Stefano Morra, and Zicheng Qian.

유명준(KIAS)

Title: Selmer near-companion curves for cyclic extensions

Abstract: Let $E$ be an elliptic curve over a number field $K$ and let $p$ be a prime. To each cyclic extension $L/K$ of degree $p$, we attach the $L/K$-twist of $E$, which is an abelian variety of dimension $p-1$. Then one can study how Selmer ranks vary in the family of such twists. In this talk, I will introduce a certain condition on the Selmer ranks of these twists that determines the $p$-torsion Galois module of $E/K$.

유화종(SNU)

Title : The rational torsion subgroup of J_0(N)

Abstract : In this talk, we try to compute the rational torsion subgroup of the modular Jacobian J_0(N).

Let N be a positive integer. For a prime p whose square does not divide 12N, we determine the structure of

the p-primary part of the rational torsion subgroup of J_0(N) by proving a generalized version of the conjecture of Ogg.

이재훈(KAIST)

Title: Torsion subgroups of elliptic curves over $\mathbb{Z}_p$-extensions of a local field

Abstract: Let L be a finite extension of $\mathbb{Q}_l$ for some odd prime l and let $L_{\infty}$ be a $\mathbb{Z}_p$-extension of L for some odd prime p. For an elliptic curve E over L, we will determine the cases that the torsion subgroup of E($L_{\infty}$) is finite .

최도훈(Korea Univ.)

Title: 라마누잔 추측에 관한 소개

Abstrat: 라마누잔 추측을 설명하고 그 추측에 관한 최근의 진전을 소개하고자 한다.

Clifford Blakestad(Postech)

Title: On the relationship between division polynomials and p-adic sigma functions on abelian varieties

Abstract: In 1991 Mazur and Tate published a paper on p-adic sigma functions attached to elliptic curves with ordinary reduction, producing a p-adic analog of the Weierstrass sigma function attached to a complex elliptic curve. For the construction of this sigma function, they generalized the classical theory of division polynomials defined in connection to the m-torsion of an elliptic curve E[m], to instead be defined for an arbitrary isogeny of elliptic curves E->E'. In this talk, we will review these constructions and discuss the correct way to generalize them to principally polarized abelian varieties of arbitrary dimension, in turn allowing us to define a higher dimensional notion of p-adic sigma functions.