Iwasawa 2019

19 juin 2019, 09:00 → 28 juin 2019, 12:30 Europe/Paris

Grand Amphi de Mathématiques (GAM) (Université de Bordeaux, building A33)

   

In 2019, the international Iwasawa conference takes place in Bordeaux, from June 24 to 28. It will be preceded by four mini-courses (of four hours each) on the topic. 

Please note that registration is free, but required. The deadline is March 31st.
For students applying for financial support, please write us before February 20th.

Mini-courses (June 19 to 22):

1. Venkatesh's conjecture for weight one dihedral modular forms  (Victor Rotger, Universitat Politècnica de Catalunya)
2. p-adic L-functions (Ellen Eischen, University of Oregon)
3. Chern classes and Iwasawa theory (Frauke Bleher, University of Iowa)
4. Overconvergent modular forms and their explicit arithmetic (Jan Vonk, Oxford) 

Conference speakers (June 24 to 28):

Daniel Barrera Salazar (Universidad de Santiago de Chile)
Ted Chinburg (University of Pennsylvania)
Mladen Dimitrov (Université de Lille)
Adrian Iovita (Concordia University and Università degli studi di Padova)
Joaquin Rodrigues Jacinto (Aix-Marseille Université)
Yukako Kezuka (Universität Regensburg)
Guido Kings (Universität Regensburg)
Antonio Lei (Université Laval)
Zheng Liu (McGill University)
David Loeffler(University of Warwick)
Jan Nekovář (Sorbonne Université)
Jishnu Ray (University of British Columbia)
Giovanni Rosso (Concordia University and Cambridge)
Ryotaro Sakamoto (University of Tokyo)
Romyar Sharifi (UCLA)
Florian Sprung (Arizona State University)
Eric Urban (Columbia University and CNRS)
Shunsuke Yamana (Kyoto University). 

Scientific committee: Denis Benois, Henri Darmon, Ming-Lun Hsieh, Masato Kurihara, Otmar Venjakob, Sarah Zerbes.

Organizers: Denis Benois, Pierre Parent. 

mercredi 19 juin

09:00 → 09:30 Registration

09:30 → 10:30 F. BLEHER . Chern classes and Iwasawa theory (1)

10:30 → 11:00 Coffee break

11:00 → 12:00 V. ROTGER. On the arithmetic of elliptic curves via triple products of modular forms (1)

14:30 → 15:30 V. ROTGER. On the arithmetic of elliptic curves via triple products of modular forms(2)

15:30 → 16:00 Coffee break

16:00 → 17:00 E. EISCHEN. p-adic L-functions (1).

jeudi 20 juin

09:00 → 10:00 E. EISCHEN. p-adic L-functions (2).

10:00 → 10:30 Coffee break

10:30 → 11:30 V. ROTGER. On the arithmetic of elliptic curves via triple products of modular forms (3).

14:00 → 15:00 V. ROTGER. On the arithmetic of elliptic curves via triple products of modular forms (4).

15:15 → 16:15 J. VONK. Overconvergent modular forms and their explicit arithmetic (1).

16:30 → 17:30 F. BLEHER Chern classes and Iwasawa theory (2)

vendredi 21 juin

09:00 → 10:00 J. VONK. Overconvergent modular forms and their explicit arithmetic (2).

10:00 → 10:30 Coffee break

10:30 → 11:30 E. EISCHEN. p-adic L-functions (3).

14:00 → 15:00 E. EISCHEN. p-adic L-functions (4).

15:15 → 16:15 F. BLEHER Chern classes and Iwasawa theory (3)

16:30 → 17:30 J. VONK. Overconvergent modular forms and their explicit arithmetic (3).

samedi 22 juin

09:00 → 10:00 F. BLEHER Chern classes and Iwasawa theory (4)

10:00 → 10:30 Coffee break

10:30 → 11:30 J. VONK. Overconvergent modular forms and their explicit arithmetic (4).

lundi 24 juin

09:00 → 09:30 Registration

09:30 → 09:35 Opening

09:35 → 10:35 Ted CHINBURG. Group homology and exterior quotients in Iwasawa theory.

Abstract: Higher codimension Iwasawa theory concerns the support in codimension greater than one of Iwasawa modules. A useful technique when relating this support to p-adic L-functions is to consider the quotient of the top exterior power of an Iwasawa module M of rank r by the sum of the r-th exterior powers of submodules arising from various Panciskin conditions. A natural question is then to give a Galois theoretic interpretation of such exterior quotients.

In this talk I will discuss such an interpretation for r >= 2 involving group homology. The particular homology group involved is H_{r-2}(A,T) when A and T are the first and second graded quotients in the derived series of a pro-p Galois group. One consequence is that the Galois theoretic information provided by second Chern classes in the case of Iwasawa theory over CM fields seems to governed by the first two graded quotients of the derived series, rather than being about higher graded quotients. This is joint work with F. Bleher, R. Greenberg, M. Kakde, R. Sharifi and M. J. Taylor.

10:35 → 11:00 Coffee break

11:00 → 12:00 Sunsuke YAMANA. On central derivatives of (twisted) triple product p-adic L-functions.

Abstract: We will construct twisted triple product p-adic L-functions and discuss its trivial or non-trivial zeros at the center of the functional equation.
In the split and +1 sign case we will determine the trivial zeros of cyclotomic p-adic L-functions associated to three ordinary elliptic curves and identify the double or triple derivatives of the p-adic L-function with the product of the algebraic part of central L-values and suitable L-invariants.
If time permits, we will formulate the p-adic Gross-Zagier formula in the -1 sign case.
This is a joint work with Ming-Lun Hsieh.

14:30 → 15:30 Antonio LEI. Pseudo-null modules and codimension two cycles for supersingular elliptic curves.

Abstract: Let E/ Q be an elliptic curve with supersingular reduction at an odd prime p and a_p(E)=0. Let K be an imaginary quadratic field where p splits and write K_\infty for the compositum of all \mathbb{Z}p-extensions of K. Generalizing Kobayashi's plus and minus Selmer groups over cyclotomic extensions of Q, Kim defined \pm/\pm-Selmer groups for E over K\infty. We present numerical examples where the intersection of a pair of these Selmer groups is pseudo-null. This allows us to give explicit examples which affirm the pseudo-nullity conjecture of Coates and Sujatha. We will also explain how to relate these Selmer groups to Loeffler's 2-variable p-adic L-functions via codimension two cycles. If time permits, we will discuss how our technique can be extended to the setting of tensor products of Hida families. This is joint work with Bharath Palvannan.

15:30 → 16:00 Coffee break

16:00 → 17:00 Florian SPRUNG. Shedding light on Selmer groups for elliptic curves at supersingular primes in Z_p^2-extensions via chromatic Selmer groups.

Abstract: We present some results and techniques concerning Selmer groups in Z_p^2-extensions for elliptic curves at supersingular primes, focusing on the case a_p not equal to 0. In this case, a convenient pair of objects to consider is the 'chromatic Selmer groups' (also called 'signed Selmer groups' when a_p=0).

mardi 25 juin

09:30 → 10:30 Adrian IOVITA. Katz type p-adic L-functions when p is not split in the CM field and applications.

Abstract: With F. Andreatta we constructed p-adic L-functions attached to a triple (F, K, p) where F is a classical, elliptic modular eigenform, K a quadratic imaginary field and p a prime integer, all satisfying certain assumptions of which the most important is that p is not split in K. Such p adic L-functions have been constructed by N. Katz (during the 70') if F is an Eisenstein series and by Bertolini-Darmon-Prasana (2013) when F is a cuspform, when the prime p is split in K. I will also present some arithmetic applications of these constructions.

10:30 → 11:00 Coffee break

11:00 → 12:00 Daniel BARRERA SALAZAR. Triple product p-adic L-functions and Selmer groups over totally real number fields.

Abstract: During the nineties Kato obtained deep results on the Birch and Swinnerton-Dyer conjecture in rank 0 for twists of elliptic curves over Q by Dirichlet characters. More recently, Bertolini-Darmon-Rotger and Darmon-Rotger developed analogous methods to treat twists by certain Artin representations of dimension 2 and 4. The aim of this talk is to explain the main ideas of joint ongoing work with Molina and Rotger which aims to generalize the methods used by Kato, Bertolini-Darmon-Rotger and Darmon-Rotger to totally real number fields, by exploiting the techniques of Andreatta and Iovita.

14:30 → 15:30 Mladen DIMITROV. Geometry of the eigencurve and Iwasawa theory.

15:30 → 16:00 Coffee break

16:00 → 17:00 Zheng LIU. p-adic families of Klingen-Eisenstein series and theta series.

Abstract: p-adic interpolations of Eisenstein series and theta series give explicit examples of p-adic families of automorphic forms. Their congruences with other automorphic forms help show lower bounds of certain Selmer groups. I will first explain the construction of a p-adic Klingen Eisenstein family for symplectic groups, and then discuss its connection with a p-adic family of theta lifts.

17:00 → 17:45 Poster session

mercredi 26 juin

09:00 → 10:00 Jan NEKOVÁŘ. The plectic polylogarithm.

Abstract: We are going to describe the Hodge realisation of the plectic polylogarithm and its relation to special values of L-functions. This is a joint work with A.J. Scholl.

10:15 → 11:15 Guido KINGS. Equivariant motivic Eisenstein classes and a generalization of the Damerell/Shimura/Katz theorem.

(joint with J. Sprang).
Abstract: The equivariant polylogarithm allows to construct in a very general setting cohomology classes of arithmetic groups with values in motivic cohomology. Using the regulator to algebraic de Rham cohomology gives interesting algebraic Eisenstein classes. We use this theory to generalize the results of Damerell, Shimura and Katz on the algebraicity of special values of L-Funktions for Hecke characters for CM fields K to the case of finite extensions L/K over CM fields K.

11:30 → 12:30 Giovanni ROSSO. Families of Drinfeld modular forms.

Abstract: Seminal work of Hida tells us that for eigenforms that are ordinary at p we can always find other eigenforms, of different weights, that are congruent to our given form. Even better, it also says that we can find q-expansions whose coefficients are analytic functions of the weight variable k, that when evaluated at positive integers give the q-expansion of classical ordinary eigenforms.This talk will explain how similar results can be obtained for Drinfeld modular forms. We shall explain how to construct families for Drinfeld modular forms, both ordinary and of positive slope, and how to decide if an overconvergent form of small slope is classical. Joint work with Marc-Hubert Nicole.

jeudi 27 juin

09:30 → 10:30 Eric. URBAN. Towards an Euler system for the standard L-function attached to Siegel modular forms.

10:30 → 11:00 Coffee break

11:00 → 12:00 Joaquin RODRIGUES JACINTO. Norm-compatible cohomology classes in Siegel varieties.

Abstract: We will explain how to construct towers of interesting classes in the cohomology of Siegel sixfolds. We will study their complex regulator and we will give an application to Iwasawa theory. This is joint work with Antonio Cauchi and Francesco Lemma.

14:30 → 15:30 David LOEFFLER. p-adic L-functions and Euler systems for GSp(4).

Abstract: I will explain how the higher Hida theory recently introduced by Pilloni can be used to construct p-adic L-functions interpolating the criticalvalues of the degree 4 (spin) L-functions of automorphic forms on GSp(4), and the degree 8 L-functions of cusp forms on GSp(4) x GL(2).This is joint work with Vincent Pilloni, Chris Skinner and Sarah Zerbes. I will conclude by describing work in progress to relate the GSp(4) p-adic L-function to the images of Euler system classes under the p-adic syntomic regulator map

15:30 → 16:00 Coffee break

16:00 → 17:00 Ryotaro SAKAMOTO. An application of the theory of higher rank Euler, Kolyvagin, and Stark systems.

Abstract: Recently, we established the theory of higher rank Euler, Kolyvagin, and Stark systems when a coefficient ring is Gorenstein. In this talk, I will discuss two applications of this theory.First, I will discuss equivariant BSD conjecture. Second, I will outline the construction of a higher rank Euler system for \mathbb{G}_{m} over a totally real field and explain that all higher Fitting ideals of a certain p-ramified Iwasawa module are described by analytic invariants canonically associated with Stickelberger elements.The first part is joint work with David Burns and Takamichi Sano.

vendredi 28 juin

09:00 → 10:00 Romyar SHARIFI. Eisenstein cocycles in motivic cohomology.

Abstract: I will describe joint work with Akshay Venkatesh on the construction of a 1-cocycle on GL_2(Z) valued in a quotient of a limit of second motivic cohomology groups of open subschemes of the square of G_m over Q. I’ll show how the cohomology class of this cocycle is annihilated by an Eisenstein ideal, and I’ll explain how the cocycle specializes to homomorphisms from first homology groups of modular curves to second K-groups of rings of cyclotomic integers. I also hope to mention a related construction over imaginary quadratic fields.

10:15 → 11:15 Yukako KEZUKA . On the conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication.

Abstract: This talk will describe recent joint work in progress with J. Coates, Y. Li and Y. Tian. Let K be the imaginary quadratic field Q(sqrt{-q}), where q is any prime congruent to 7 modulo 16. Let A be the Gross curve defined over the Hilbert class field H of K, with complex multiplication by the ring of integers of K. In their most recent work, Coates and Li found a large family of quadratic twists E of A whose complex L-series L(E/H,s) does not vanish at s=1. We will discuss the p-part of the Birch and Swinnerton-Dyer conjecture for these curves for every prime p which splits in K (in particular, this includes p=2).

11:30 → 12:30 Jishnu RAY. Selmer groups of elliptic curves and Iwasawa algebras.

Abstract:The Selmer group of an elliptic curve over a number field encodes several arithmetic data of the curve providing a p-adic approach to the Birch and Swinnerton-Dyer, connecting it with the p-adic L-function via the Iwasawa main conjecture. Under suitable extensions of the number field, the dual Selmer becomes a module over the Iwasawa algebra of a certain compact p-adic Lie group over Z_p (the ring of p-adic integers), which is nothing but a completed group algebra. The structure theorem of GL(2) Iwasawa theory by Coates, Schneider and Sujatha (C-S-S) then connects the dual Selmer with the “reflexive ideals” in the Iwasawa algebra. We will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications to the structure theorem of C-S-S. Furthermore, such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. pro-p uniform groups and the pro-p Iwahori of GL(n,Z_p). Alongside Iwasawa theoretic results, we will state results counting the dimension of first cohomology group of the pro-p Iwahori subgroup of any reductive group over Z_p and thus prove the Inverse Galois problem for p-adic Lie extensions. We finally conclude by connecting GL(2) Iwasawa theory of (C-S-S) with PGL(2) Iwasawa theory, thus moving down the Iwasawa theoretic tower, unlike (C-S-S) where their arguments circles on moving up the Iwasawa theoretic tower.