Titles and Abstract

Ashay Burungale
Title: On the non-triviality of the p-adic height and p-adic Abel-Jacobi image
Abstract: Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension K. Let p be an odd prime split in K. We describe recent results on the non-triviality of the p-adic height and p-adic Abel-Jacobi image of generalised Heegner cycles over anticyclotomic extensions of K. The later non-triviality is rather general and holds in the non-CM case. The former non-triviality holds in the CM case and is a joint work in progress with D. Disegni.

Ted Chinburg
Title:Second Chern classes in Iwasawa theory
Abstract:Many of the Main Conjectures of Iwasawa theory relate the codimension one behavior of Iwasawa modules to p-adic L-functions. In this talk I will describe work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. Taylor on codimension two behavior when the modules in question are trivial in codimension one. Over imaginary quadratic fields, the second Chern classes of certain Iwasawa modules can be determined using symbols in K_2 groups constructed from Katz p-adic L-functions. By the end of the talk I will discuss a non-commutative version of this result.

Henri Darmon
Title: Stark-Heegner points and generalised Kato classes
Abstract: I will describe a formula relating the two objects in the title, and discuss its applications to the arithmetic of elliptic curves over ring class fields of real quadratic fields. This is joint work with Victor Rotger.

Samit Dasgupta
Title: On p-adic Stark conjectures for real quadratic fields
Abstract: We will recall the statements of various p-adic Stark conjectures for real quadratic fields, and pose a new conjecture for characters of mixed signature.

Mladen Dimitrov
Title: On the exceptional zeros of p-adic L-functions of Hilbert modular forms
Abstract: The use of modular symbols to attach p-adic L-functions to Hecke eigenforms goes back to the work of Manin et al in the 70s. In the 90s, Stevens developed his theory of overconvergent modular symbols, which was successfully used to construct p-adic L-functions on the eigenvariety. In this talk we will present a work in collaboration with Daniel Barrera and Andrei Jorza in which we generalise this approach to the Hilbert modular setting with a view towards applications to the exceptional zero conjecture.

Tim Dokchitser
Title: Growth of Sha in towers for isogenous curves
Abstract: I will discuss the growth of Sha for isogenous abelian varieties in towers of number fields, with particular emphasis on elliptic curves in cyclotomic towers. This is related to the phenomenon of 'positive mu-invariant' that accounts for the exponential growth of the p-part of Sha in the cyclotomic p-tower. This is joint work with Vladimir Dokchitser.

Olivier Fouquet
Title: The Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebra(s).
Abstract: In 1979, Barry Mazur asked whether the special values of the L-functions of two eigencuspforms congruent modulo an ideal m of the Hecke algebra could be computed in terms of the action of the Hecke algebra on the m-torsion of the étale cohomology of the modular curve. The Equivariant Tamagawa Number Conjecture with coefficients in the Hecke algebra provides a subtle and powerful answer to this question for congruent motives occurring in the cohomology of Shimura varieties. I will explain the motivation behind this conjecture, its statement and its proof for GL2/Q under the hypotheses ensuring the existence of a Taylor-Wiles system.

Ming-Lun Hsieh
Title: Anticyclotomic Iwasawa theory for modular forms
Abstract: We will report the recent development of anticyclotomic Iwasawa theory for elliptic modular forms of higher weights. For example, we will discuss an analogue of Mazur's conjecture for modular forms.

Dohyeong Kim
Title: On the transfer congruence between p-adic Hecke L-functions
Abstract: The existence of a non-commutative p-adic L-function predicts a family of congruences between special values of L-functions. We will prove a simplest such congruence of non-abelian nature, namely the transfer congruence, in the case of p-adic Hecke L-functions. We use a mildly improved version of Hsieh's Eisenstein series, and deduce the desired congruence by applying the q-expansion principle.

Guido Kings
Title: Explicit reciprocity laws for Rankin convolutions
Abstract: In our joint work with Loeffler and Zerbes an explicit reciprocity law for Rankin-convolutions of modular forms was proved. The strategy of the proof relies on using non-critical points of the p-adic L-function and a theory of p-adic interpolation of the etale realization of motivic Eisenstein classes. In this talk we explain our approach, discuss other cases where this strategy works and describe a new result, which generalizes the p-adic interpolation of motivic Eisenstein classes.

Preda Mihailescu
Title: On new approaches to classical conjectures in Iwasawa theory

Andreas Nickel
Title: Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture
Abstract: We discuss how the understanding of the structure of Iwasawa algebras can be used to prove the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, we will not assume the vanishing of Iwasawa's $\mu$-invariant. If time permits, we will also discuss applications to the (non-abelian) Brumer-Stark conjecture. This is joint work with Henri Johnston.

Takamichi Sano
Title: Arithmetic properties of zeta elements
Abstract: (Joint work with D. Burns and M. Kurihara) The existence of zeta elements is predicted by the equivariant Tamagawa number conjecture. I will discuss some new arithmetic properties of zeta elements. I will also describe some Iwasawa theoretic aspects of our theory.

Chris Skinner
Title:
Abstract:

Ki-Seng Tan
Title: Iwasawa main conjecture for elliptic curves over global function fields
Abstract: Consider a global function field $K=\mathbb{F}_q(\mathcal{X})$, $\mathcal{X}$ a complete smooth curve over $\mathbb{F}_q$. Let $A$ be an ordinary elliptic curve defined over $K$. A formula of Mazur defines for each $\mathbb{F}_q$-rational effective deivisor $D$ on $\mathcal{X}$, an element $\Theta_D$ in the group ring of the Weil group $W_D, which interpolates special values of $L$-functions associated to $A/K$ and characters of $W_D$.
Let $L/K$ be a $\mathbb{Z}_p^d-extension unramified outside a finite set $S$ consisting of ordinary places of $K$. Denote $\Gamma:=Gal(L/K)$ and $\Lambda:=\mathbb{Z}_p[[\Gamma]]$. Let $X_L$ denote the dual $p$-Selmer group of $A/L$ and let $CH_{\Lambda}(X_L)$ denote the characteristic ideal.
In this talk, we introduce a (modified) $p$-adic $L$-function $\mathcal{L}_{L/K}$ derived from those $\Theta_D$, Supp$D \subset S$, and conjecture that it generates $CH_{\Lambda}(X_L)$. We shall give some evidence of the conjecture.
A pdf can be found here.

Eric Urban
Title:Euler systems, Eisenstein Congruence and Iwasawa Theory.
Abstract:I will discuss a strategy to construct Euler systems using Eisenstein congruences that is potentially applicable to a large number of situations for which no K-theoretic or geometric constructions are available.

Vinayak Vatsal
Title: Families of modular forms of half integral weight.
Abstract: Suppose $F$ and $G$ are holomorphic cuspidal newsforms of even weight and trivial characters of levels M and N respectively, such that $F$ and $G$ are congruent modulo a prime $P$ in the algebraic closure of $\mathbf{Q}$. We can then pose the question of whether or not the modular forms associated to $F$ and $G$ by the Shimura-Waldspurger correspondence are also congruent modulo $P$.One can also ask whether the $P$-adic families associated to $F$ and $G$ survive the Shimura correspondence.
In considering these questions, one quickly realizes that the in the most naive form the answer is negative, but the reason for the failure turns out to be quite subtle. The main point is that the usual Shimura-Waldspurger correspondence does not even yield a canonical bijection on the level of automorphic representations, and existence of half-integral congruences and $P$-adic families depends strongly on the choice of representation in the Waldspurger packet. While the question of p-adic families has been investigated by numerous authors (Hida, Stevens, Ramsey), the results are regrettably incomplete. The question of congruences does not seem to have been studied, beyond some informal speculations (Prasanna).
Our investigation of these questions reveals that the natural normalization for modular forms of half integral weight leads to the appearance of certain Tamagawa factors related to L-invariants of the corresponding integral weight representation. In view of the proof due to Greenberg and Stevens that the L-invariant is the derivative of a p-adic L-function at $s=1$, and the appearance in their proof of a 2-variable p-adic L-function, we are led to examine the properties of p-adic families of modular forms of half integral weight, and to investigate the presence of trivial zeroes and derivatives in these families. We carry out this procedure in the case of families of tame level 1.
The study relies on a detailed analysis of the structure of the Waldspurger packets on the metaplectic group, and on a careful determination of which of the many possible elements of the Waldspurger packet actually show up the families in constructed by Stevens.

Otmar Venjakob
Title: Local Iwasawa cohomology and $(\varphi,\Gamma)$-modules over Lubin-Tate extensions
Abstract: For the $p$-cyclotomic tower of $\mathbb{Q}_p$ one has Fontaine's description of local Iwasawa cohomology in terms of the $\psi$-operator attached to any étale $(\varphi,\Gamma)$-module. In this talk I will report on joint work with Peter Schneider which consists of generalising Fontaine's result to the case of arbitratry Lubin-Tate towers over finite extensions $L$ of $\mathbb{Q}_p$. In particular, we prove a kind of explicit reciprocity law which calculates the related Kummer map using Coleman power series.

Stefano Vigni
Title: Plus/minus Heegner points and Iwasawa theory of supersingular elliptic curves
Abstract: Let E be a rational elliptic curve (without complex multiplication) and let p be a prime of good supersingular reduction for E. Let K be an imaginary quadratic field satisfying a modified "Heegner hypothesis" in which p splits. In this talk I will explain how one can prove that Kobayashi's plus/minus p-primary Selmer groups of E over the anticyclotomic Z_p-extension of K have corank 1 over the corresponding Iwasawa algebra $\Lambda$. Our strategy is based on an extension to the supersingular case of the $\Lambda$-adic Kolyvagin method originally developed by Bertolini in the ordinary setting. Applications to the growth of Selmer groups will be given. This is joint work with Matteo Longo.

A pdf file of titles and abstract for all talks and posters may be downloaded here (though it may not be up to date) and here.