Supported by the Royal Society, the London Mathematical Society, Glasgow Mathematical Journal, DPMMS and Emmanuel College, Cambridge
Organisers: Tim Dokchitser, Vladimir Dokchitser, David Loeffler, Sarah Zerbes

Invited participants

Massimo Bertolini (Duisburg-Essen)      Kazuya Kato (Chicago)      Romyar Sharifi (Arizona)
Thanasis Bouganis (Durham)      Minhyong Kim (Oxford)      Ye Tian (Beijing)
John Coates (Cambridge)      Guido Kings (Regensburg)      Jacques Tilouine (Paris)
John Cremona (Warwick)      Masato Kurihara (Keio)      Otmar Venjakob (Heidelberg)
Takako Fukaya (Chicago)      Barry Mazur (Harvard)      Andrew Wiles (Oxford)
Haruzo Hida (UCLA)      Karl Rubin (Irvine)      Christian Wuthrich (Nottingham)
Mahesh Kakde (King's College)      Peter Schneider (Münster)

The conference marks the 70th birthday of John Coates and aims both to explain the state of our knowledge and to reflect on the modern approaches to the arithmetic of elliptic curves, modular forms and Iwasawa theory. It follows a Royal Society Kavli Centre workshop on March 23-24 on the same topic, but while the workshop is limited to 20 participants due to the Royal Society restrictions and is fully booked, the conference is open for everyone to attend. We particularly welcome graduate students and young researchers to participate, and we offer financial support to UK graduate students.


The talks will take place in Mill Lane Lecture Room 9.
10:30 Coffee and Registration (Mill Lane Lecture Room 7)
11:40 Welcome
11:45-12:45 Andrew Wiles - Class numbers of imaginary quadratic fields

Abstract. I will discuss the existence of imaginary quadratic fields with class numbers prime to l and given splitting conditions at a finite set of primes.

Lunch break
14:15-15:15 John Cremona - Some density results in number theory

Abstract. I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families has a solution either locally (over the reals or the p-adics), everywhere locally, or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

15:45-16:45 Barry Mazur - Diophantine Stability

Abstract. An irreducible algebraic variety V over a field K will be said to be diophantine-stable for a field extension L/K if V has no new L-rational points; that is, if V(L)=V(K). If V contains a curve birationally equivalent to the projective line over K, then V fails to be diophantine-stable for any nontrivial extension L/K. What happens if V doesn't contain such a curve? I will talk about results related to diophantine-stability obtained jointly with Karl Rubin. One application of our work is that for K a number field, the only smooth irreducible projective curve over K that is not diophantine-stable for any nontrivial extension L/K is the projective line over K.

17:00-18:00 Minhyong Kim - Diophantine Geometry, fundamental groups, and non-abelian reciprocity laws

Abstract. We state a new collection of reciprocity laws with coefficients in a hyperbolic curve and give examples of associated explicit formulas.

9:30-10:30 Peter Schneider - The Auslander-Gorenstein formalism for Hecke algebras

Abstract. I will start by explaining the relevance of Hecke algebras for the representation theory of p-adic groups. After briefly describing the homological Auslander-Gorenstein condition for noncommutative rings I will state a theorem that Iwahori-Hecke algebras are Auslander-Gorenstein. The main part of the talk will be devoted to the discussion of consequences of this property for the module theory of Hecke algebras. This is joint work with R. Ollivier.

11:00-12:00 Thanasis Bouganis - p-adic measures for Hermitian modular forms and the Rankin-Selberg method

Abstract. After giving a short introduction to the theory of hermitian modular forms (modular forms associated to unitary groups), we will discuss algebraic and p-adic properties of special values of L-functions attached to a Hecke eigenform. For the study of these special values there exist two general methods, the "Doubling-Method" and the "Rankin-Selberg method". In this talk we will focus on the construction of p-adic measures using the second method.

Lunch break
13:30-14:30 Ye Tian - Congruent numbers and Heegner points

Abstract. Heegner proved that any prime congruent to 5, 7 modulo 8 or twice prime congruent to 6 modulo 8 is a congruent number. With general Gross-Zagier formula and Waldspuger formula, we are able to extend Heegner's result to the case with many prime factors by an induction method.

15:00-16:00 Haruzo Hida - Geometric control of modular Jacobians

Abstract. Analyzing known elementary relations between U(p) operators and Picard/Albanese functoriality of the Jacobians of each tower of modular curves of p-power level, we get fairly exact control of the ordinary part of the limit Barsotti-Tate groups and the (p-adically completed) limit Mordell-Weil groups with respect to the weight Iwasawa algebra. Computing Galois cohomology of these controlled Galois modules, we hope to get good control of the (ordinary part of) limit Selmer groups and limit Tate-Shafarevich groups. We will describe the ideas and hopefully some of exact control (to the extent I can prove by the time of the conference).

16:30-17:30 Christian Wuthrich - On a specific Galois cohomology group of elliptic curves

Abstract. Let p be a prime number, E an elliptic curve defined over Q and set K = Q(E[p]). When does the group H^1(K/Q,E[p]) vanish? Very often it does, but not always. After explaining the list of all cases, I will include one application of the non-vanishing.

18:30 Conference dinner

9:30-10:30 Romyar Sharifi - Modular symbols in Iwasawa theory

Abstract. I will explain a very explicit, conjectural relationship between first homology groups of modular curves modulo Eisenstein ideals and second K-groups of cyclotomic integer rings, in a form the mildly refines the published conjecture. Taken up the modular and cyclotomic towers, this conjecture can be viewed as a refinement of the main conjecture of Iwasawa theory. This Iwasawa-theoretic version of the original conjecture has been proven by Fukaya and Kato up to torsion when the relevant Kubota-Leopoldt p-adic L-function has no multiple zeros. I will discuss my work on a refinement of their method. Finally, I will very briefly hint at work of Fukaya, Kato, and myself on higher-dimensional analogues of the conjecture.

11:00-12:00 Jacques Tilouine - Big image of modular Galois representations associated to p-adic families of bounded slope

Abstract. In a recent work, Hida proved, under some mild assumptions, that the image of the Galois representation associated to a non CM p-adic family of p-ordinary Hecke eigenforms contains a congruence subgroup of non zero level; he also relates the best level to p-adic L-functions. In a joint work with A. Conti and A. Iovita, we generalise these results, under similar assumptions to the case of (non CM) finite slope p-adic families. The new ingredient replacing ordinarity is relative Sen theory.

Lunch break
13:15-14:15 Masato Kurihara - Iwasawa theory and zeta elements

Abstract. In this talk I first talk on the Galois module structure of ideal class groups and Weil etale cohomology groups, using Rubin-Stark elements, and on their relation with the zeta elements in the equivariant Tamagawa number conjecture for the Tate motive. Using these elements, I discuss Iwasawa theory over general number fields. This is a joint work with D. Burns and T. Sano. Aside from mathematics, I also plan to mention with concrete examples how deep John's understanding of Japanese culture is.

14:30-15:30 Mahesh Kakde - Congruences for p-adic L-function of a non-CM elliptic curve

Abstract. I will talk about congruences needed to construct p-adic L-functions for elliptic curves without complex multiplication.


Victor Abrashkin (Durham)      Toby Gee (Imperial College)      Otto Overkamp (Imperial College)
Adebisi Agboola (Santa Barbara)      Jean Gillibert (Toulouse)      Ekin Ozman (Istanbul)
Gülçin Akarsu (Ankara)      Krzysztof Górnisiewicz (Poznań)      Jae-Suk Park (Pohang)
Jennifer Balakrishnan (Oxford)      Andrew Granville (Montreal/UCL)      Jeehoon Park (Pohang)
Francesca Balestrieri (Oxford)      Benjamin Green (Oxford)      Vandita Patel (Warwick)
Amaro Barreal (Aalto U Helsinki)      Elena Griniari (Springer)      Bernadette Perrin-Riou (Paris)
Alex Bartel (Warwick)      Xuejun Guo (Nanjing)      Gautier Ponsinet (Quebec)
Boris Bartolome (Bordeaux/Göttingen)      Haruzo Hida (UCLA)      Hourong Qin (Nanjing)
Monique van Beek (Cambridge)      Camilla Hollanti (Aalto U Helsinki)      Mohammad Rahmati (Guanajuato)
Rebecca Bellovin (Berkeley)      Andreas Holmstróm (IHES)      Giovanni Rosso (Paris/Leuven)
Laurent Berger (Lyon)      Chen Huan (Paris)      Sandra Rozensztajn (Lyon)
Tobias Berger (Sheffield)      Canberk İrimağzi (Koç)      Karl Rubin (Irvine)
Massimo Bertolini (Duisburg-Essen)      Dimitar Jetchev (Lausanne)      Justin Scarfy (Vancouver)
Alex Best (Cambridge)      Rob de Jeu (Amsterdam)      Peter Schneider (Münster)
Alexander Betts (Oxford)      Henri Johnston (Exeter)      Tony Scholl (Cambridge)
Bryan Birch (Oxford)      Bruce Jordan (New York)      Isabella Scott (St Andrews)
Christopher Birkbeck (Warwick)      Mahesh Kakde (King's College)      Gianluigi Sechi (CRS4-Cagliari)
Matthew Bisatt (Warwick)      Berke Karagoz (Koç)      Ehud de Shalit (Jerusalem)
Chris Blake (Cambridge)      Alex Karrila (Aalto U Helsinki)      Romyar Sharifi (Arizona)
Ivan Blanco-Chacon (Aalto U Helsinki/Dublin)      Kazuya Kato (Chicago)      Jack Shotton (Imperial College)
Werner Bley (München)      Jukka Keranen (UCLA)      Samir Siksek (Warwick)
Ferdinand Blomqvist (Aalto U Helsinki)      Yukako Kezuka (Cambridge)      Vishal Solanki (King's College)
Thanasis Bouganis (Durham)      Chan-Ho Kim (Irvine)      David Spencer (Sheffield)
Lynn Brandon (Springer)      Minhyong Kim (Oxford)      Florian Sprung (Princeton/IAS)
Kazim Büyükboduk (Koç)      Guido Kings (Regensburg)      Jim Stankewicz (Bristol)
James Cann (UCL)      Oscar Kivinen (Aalto U Helsinki)      Peter Swinnerton-Dyer (Cambridge)
Marco Caselli (Warwick)      Masato Kurihara (Keio)      Tuomas Tajakka (Aalto U Helsinki)
Antonio Cauchi (UCL)      Robert Kurinczuk (Bristol)      Nao Takeshi (Tokyo)
Kęstutis Česnavičius (Berkeley)      Jack Lamplugh (UCL, waived)      Jack Thorne (Cambridge)
Zexiang Chen (Cambridge)      Emmanuel Lecouturier (Paris)      Ye Tian (Beijing)
Xiaoyun Cheng (Nanjing)      Junghwan Lim (Oxford)      Jacques Tilouine (Paris)
Rudolf Chow (Sheffield)      David Loeffler (Warwick)      Alex Torzewski (Warwick)
John Coates (Cambridge)      Matteo Longo (Padua)      Bach Tran (Edinburgh)
Andrea Conti (Paris)      Céline Maistret (Warwick)      George Turcas (Cambridge)
Jonathan Crawford (Durham)      Anton Mallasto (Aalto U Helsinki)      Niko Väisänen (Aalto U Helsinki)
John Cremona (Warwick)      Chloe Martindale (Leiden/Bordeaux)      Rodolfo Venerucci (Duisburg-Essen)
Andrzej Dąbrowski (Szczecin)      Jolanta Marzec (Bristol)      Otmar Venjakob (Heidelberg)
Ishai Dan-Cohen (Duisburg-Essen)      Marc Masdeu (Warwick)      Jan Vonk (Oxford)
Cosmin Davidescu (Cambridge)      Makiko Mase (Tokyo)      David Watson (Exeter)
Heline Deconinck (Warwick)      Barry Mazur (Harvard)      Victor Weenink (Microsoft)
Taiwang Deng (Paris)      William McCallum (Arizona)      Andrew Wiles (Oxford)
Fred Diamond (King's College)      Gary McConnell (Imperial College)      Chris Williams (Warwick)
Martin Dickson (Bristol)      Preda Mihăilescu (Göttingen)      Malte Witte (Paderborn/Heidelberg)
Netan Dogra (Oxford)      Arman Mimar (NYU Abu Dhabi)      Christian Wuthrich (Nottingham)
Tim Dokchitser (Bristol)      Adam Morgan (Bristol)      Donggeon Yhee (Sheffield)
Vladimir Dokchitser (Warwick)      Takayuki Morisawa (Tokyo)      Gergely Zábrádi (Budapest)
Cameron Fairweather (Durham)      Steffen Müller (Oldenburg)      Yasin Zaehringer (King's College)
Tom Fisher (Cambridge)      James Newton (Imperial College)      Sarah Zerbes (UCL)
Daniel Fretwell (Sheffield)      Rachel Newton (Bonn/IHES)      Shuai Zhai (Shandong/Cambridge)
Takako Fukaya (Chicago)      Christopher Nicholls (Oxford)      Yinan Zhang (Sydney)
Ildar Gaisin (Paris)      Andreas Nickel (Bielefeld)      Haiyan Zhou (Nanjing)
Wojciech Gajda (Poznań)      Jeanine Van Order (MPI Bonn)      Jialiang Zou (Leiden)


Conference photo (click to enlarge)

Pictures from the Kavli workshop, the Cambridge conference and the conference dinner.


We would like to thank Yukako Kezuka, Zexiang Chen, Julia Blackwell, Jack Lamplugh, Monique van Beek and David Stewart for their invaluable help, and all the speakers and the participants for making it into a very enjoyable conference.

--- The organisers.

Mill Lane Lecture Rooms Conference Cambridge
CMS/DPMMS Conference