Karl Rubin

Title: Higher rank Kolyvagin systems

Abstract:  Both a rank-one Euler system and a rank-one Kolyvagin system consist of families of cohomology classes with appropriate properties and interrelationships. Given such an Euler system, Kolyvagin's derivative construction produces a rank-one Kolyvagin system, and a rank-one Kolyvagin system gives a bound on the size of a Selmer group. Ideas and conjectures of Perrin-Riou show that in some situations (for example, starting with an abelian variety of dimension r, or the global units in a totally real field of degree r) an Euler system is more naturally a collection of elements in the r-th exterior powers of cohomology groups. In this situation, Barry Mazur and I define a Kolyvagin system of rank r also to be a suitable collection of elements in r-th exterior powers, and we show how a Kolyvagin system of rank r bounds the size of the corresponding Selmer group.

Chang Heon Kim

Title: Families of elliptic curves over cubic number fields with

prescribed torsion subgroups (joint work with Daeyeol Jeon and Yoonjin Lee)

Abstract: Eight years ago Jeon, Kim and Schweizer determined those finite groups which

appear infinitely often as torsion groups of elliptic curves over cubic number fields.

In this talk we will construct such elliptic curves and cubic number fields for each possible torsion group.

Hershy Kivilevsky

Title: Mordell-Weil groups of elliptic curves under field extension.

Abstract: Let E be an elliptic curve defined over the rational field \Q with L-function

L(E,s).

We are interested in studying E(K) as K varies over finite extensions of \Q.

Analytically this questions translates (under the Birch and Swinnerton-Dyer

conjecture) into when L(E,1, \chi)=0 for Artin characters \chi.

For [K:\Q]=2, there is an extensive literature on this question. We present

our results and conjectures when [K:\Q]>2.

Daniel Delbourgo

Title: Congruences between Hasse-Weil L-functions

Keisuke Arai

Title: Algebraic points on Shimura curves of $\Gamma_0(p)$-type

 Abstract: We classify the characters associated to algebraic points on
Shimura curves of $\Gamma_0(p)$-type, and over number fields
(not only quadratic fields but also fields of higher degree)
we show that there are few points on such a Shimura curve
for every sufficiently large prime number $p$.

We also obtain an effective bound of such $p$.
This is an analogue of the study of rational points or points
over quadratic fields on the modular curve $X_0(p)$ by Mazur
and Momose.

Massimo Bertolini

Title: p-adic Rankin L-series and algebraic cycles

Abstract: I will report on recent work, in collaboration with Darmon and Prasanna, relating values of Rankin p-adic L-series to the p-adic Abel-Jacobi image of certain algebraic cycles. Arithmetic applications will be discussed.

Mihran Papikian

Title: Optimal quotients of Mumford curves and component groups

Abstract: Let $X$ be a Mumford curve. We say that an elliptic curve

is an optimal quotient of $X$ is there is a finite morphism

$X\to E$ such that the homomorphism $\pi: Jac(X)\to E$

induced by the Albanese functoriality has connected and

reduced kernel. We consider the functorially induced map

$\pi_\ast: \Phi_X\to \Phi_E$ on component groups of the

Neron models of $Jac(X)$ and $E$. We show

that in general this map need not be surjective, which answers negatively

a question of Ribet and Takahashi. Using rigid-analytic

techniques, we give some conditions under which $\pi_\ast$

is surjective, and discuss arithmetic applications to

modular curves. This is a joint work with Joe Rabinoff.

Shun Ohkubo

Title: On differential modules associated to de Rham representations in 
the imperfect residue field case



Abstract: Let K be a CDVF of mixed characteristic (0,p), whose residue 
field admits a finite p-basis. Denote the absolute Galois group of K by 
G. For a given de Rham representation V of G, we will construct a 
differential module N_dR(V), which is a generalization of Laurent Berger
's N_dR(V) in the perfect residue field case. We also explain some 
properties of this differential module.

Dae Yeol Jeon

Title: Families of elliptic curves over quartic number fields with prescribed torsion subgroups (joint work with Chang Heon Kim and Yoonjin Lee)

Abstract: Jeon, Kim and Schweizer determined those finite groups which appear infinitely often as torsion groups of elliptic curves over quartic number fields. In this talk we will construct infinite families of elliptic curves over quartic number fields for each possible torsion group.

Byoung Du Kim

Title: Iwasawa theory for elliptic curves for non-ordinary/supersingular primes over imaginary quadratic fields.

Abstract: Often in Iwasawa theory, (for a fixed prime $p$) we want to relate some kind of $p$-Selmer groups (which can be roughly considered as the set of rational points plus the Shafarevich-Tate group) on the algebraic side and some kind of $p$-adic $L$-function (which can be thought of as a $p$-adic power series that incorporates the special values of an $L$-function) on the analytic side. It is well-known that the Iwasawa theoretic properties of the conventional Selmer groups and p-adic L-functions break down when the prime $p$ is non-ordinary/supersingular. Kobayashi and Pollack proposed the plus/minus Selmer groups and plus/minus $p$-adic $L$-functions respectively as alternatives, and it is known that they work well over the cyclotomic extensions of the rational number field $\mathbb Q$. By building upon their ideas, the works of Katz and Hida, and our previous works, we construct two-variable p-adic L-functions, and $\pm/\pm$-Selmer groups over the $\mathbb Z_p^2$-extension of imaginary quadratic fields where $p$ is non-ordinary/supersingular, and splits completely over the imaginary quadratic field. We will argue that these are good objects to study by illustrating their properties. We also present a conjecture in the spirit of the main conjecture of Iwasawa theory, which hypothetically connects the algebraic and analytic properties of elliptic curves by way of relating the two objects we construct.

Cristian Popescu

TBA