Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights

Workshop theme

It is a classical problem to understand the value of the Riemann zeta-function at n=2, 3, 4, ..... For n even this value is πⁿ times a non-zero rational number, but for n odd much less is known. In the 70s Borel proved that, for a number field k, and n≥2, up to some simple factors and a non-zero rational number, ζ_k(n) equals the regulator R_n(k) associated to the K-group K_2n−1(k) defined earlier by Quillen. Because ζ_Q(s)=ζ(s), and K₃(Q), K₇(Q), ... are finite whereas K₅(Q), K₉(Q), ... have rank one, this explains the behaviour of ζ(n).

Using the functional equation of ζ_k(n), Borel’s result gives the main ingredient of the first non-zero coefficient in the Taylor expansion at s=1−n of ζ_k(s), but the Lichtenbaum conjecture is more precise: up to sign and a power of 2, this coefficient should equal |K_2n−1(O_k)_tor|^-1|K_2n−2(O_k)| R_n(k), where O_k is the ring of algebraic integers in k. Nowadays there are various conjectures on special values of zeta-functions and their generalisations. Techniques to prove results have involved Iwasawa theory, as well as more explicit descriptions of the K-groups involved in terms of complexes of algebraic cycles or formal generators and relations.

One of those conjectures (predating Borel’s result and Lichtenbaum’s conjecture, and, in fact, one of the Millenium Prize Problems of the Clay Mathematical Institute) is the conjecture by Birch and Swinnerton-Dyer, on the behaviour at s=1 of L(E, s) for an elliptic curve E over a number field, where L(E, s) is the analogue of the zeta-function in this context. Its statement involves another K-group, K₀(E), as well as periods of 1-forms on the associated Riemann surface, a height pairing, and some more refined arithmetic invariants. Such periods tie up with Hodge theory, one of the other areas covered by this workshop.

Applications for this workshop are now closed

Fees

Registration Only

Registration Package: £208
Student Registration Package: £158

The Registration Package includes admission to all seminars, lunches and refreshments on the days that lectures take place (Monday - Friday), wine reception and formal dinner, but does not include other meals or accommodation.

Formal Dinner Only

Formal Dinner: £55

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event. For all remaining participants who would like to attend, such as programme participants, the above charge will apply.

Accommodation

Unfortunately we do not have any accommodation to offer so all successful applicants will need to source their own accommodation.

Please see the Hotels Combined website for a list of local hotels and guesthouses.

Meals

Lunch Lunch timings and location will be confirmed with timetable.

Evening Meal Participants are free to make their own arrangements for dinner.

Formal Dinner The Formal Dinner location and date is to be confirmed.

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event

The Institute kindly requests that any papers published as a result of this programme’s activities are credited as such. Please acknowledge the support of the Institute in your paper using the following text:

The author(s) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights, where work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1.

The Organisers would like to thank the following sponsors for their generous support of the event:

Timetable

Monday 20th June 2022
09:00 to 09:40	Registration	No Room Required
09:40 to 09:45	Christie Marr Isaac Newton Institute Welcome talk	Room 1
09:45 to 10:45	Minhyong Kim International Centre for Mathematical Sciences; University of Edinburgh Chair: James D. Lewis Path integrals and p-adic L-functions	Room 1
10:45 to 11:15	Group photo and Morning Coffee	No Room Required
11:15 to 12:15	Spencer Bloch University of Chicago Chair: James D. Lewis Height pairings for algebraic cycles. Biextensions, generalized cross-ratios, Tamagawa numbers, and the Birch and Swinnerton Dyer conjecture	Room 1
12:15 to 13:30	Lunch	No Room Required
13:30 to 14:30	Ana Caraiani Imperial College London Chair: James D. Lewis On the modularity of elliptic curves over imaginary quadratic fields	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Anna Cadoret Sorbonne Université Chair: James D. Lewis Degeneration loci of l-adic local systems	Room 1
16:00 to 17:00	Welcome Wine Reception at INI	No Room Required

Tuesday 21st June 2022
09:45 to 10:45	Francis Brown University of Oxford Chair: James D. Lewis Borel regulator, zeta-motives and moduli of abelian varieties (tropical or otherwise)	Room 1
10:45 to 11:15	Morning Coffee and Group Photo	No Room Required
11:15 to 12:15	Bruno Kahn Institut de Mathématiques de Jussieu Chair: James D. Lewis Universal Weil cohomology	Room 1
12:15 to 13:30	Lunch	No Room Required
13:30 to 14:30	Matthew Morrow Institut de Mathématiques de Jussieu Chair: James D. Lewis Motivic cohomology of singular varieties	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Matt Kerr Washington University in St. Louis Chair: James D. Lewis Differential equations and mixed Hodge structures	Room 1

Wednesday 22nd June 2022
09:45 to 10:45	Netan Dogra King's College London Chair: James D. Lewis p-adic integrals and rational points on families of curves	Room 1
10:45 to 11:15	Morning Coffee	No Room Required
11:15 to 12:15	Xi Chen University of Alberta Chair: James D. Lewis Rationally Inequivalent Points on Generic Hypersurfaces	Room 1
12:15 to 13:30	Lunch	No Room Required
13:30 to 17:00	Free Afternoon	No Room Required
19:30 to 22:00	Formal Dinner at Churchill College	No Room Required

Thursday 23rd June 2022
09:45 to 10:25	Tiago Jardim da Fonseca Universidade Estadual de Campinas Chair: James D. Lewis Elliptic KZB equations via the universal vector extension	Room 1
10:25 to 10:55	Morning Coffee	No Room Required
10:55 to 11:35	Steven Charlton Universität Hamburg Chair: James D. Lewis Zagier's polylogarithm conjecture on $\zeta_F(4)$ and an explicit 4-ratio	Room 1
11:35 to 12:15	Quentin Gazda Max Planck Institute for Mathematics Chair: James D. Lewis Regulators in function fields arithmetic	Room 1
12:15 to 13:30	Lunch	No Room Required
13:30 to 14:30	Gregory Pearlstein Texas A&M University; Università di Pisa Chair: James D. Lewis TBA	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Phillip Griffiths Institute for Advanced Study, Princeton; University of Miami Chair: James D. Lewis Extended Period Mapping - Clay Lecturer	Room 1
16:00 to 17:00	Clay Lecturer Reception	No Room Required

Friday 24th June 2022
09:45 to 10:45	Amalendu Krishna Indian Institute of Science Chair: James D. Lewis Tame class field theory over local fields	Room 1
10:45 to 11:15	Morning Coffee	No Room Required
11:15 to 12:15	Paulo Lima-Filho Texas A&M University Chair: James D. Lewis Multiplicative properties of higher regulators via current transforms	Room 1
12:15 to 13:30	Lunch	No Room Required
13:30 to 14:30	Yunqing Tang Université Paris Saclay; CNRS (Centre national de la recherche scientifique) Chair: James D. Lewis Applications of arithmetic holonomicity theorems	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Stephen Lichtenbaum Brown University Chair: James D. Lewis Special values of Zeta-functions of regular schemes projective over the integers	Room 1

K-theory, algebraic cycles and motivic homotopy theory

Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights

Organisers

Rob de Jeu Vrije Universiteit Amsterdam
James D. Lewis University of Alberta

Speakers

Spencer Bloch University of Chicago
Francis Brown University of Oxford
Anna Cadoret Sorbonne Université
Ana Caraiani Imperial College London
Steven Charlton Universität Hamburg
Xi Chen University of Alberta
Netan Dogra King's College London
Quentin Gazda Max Planck Institute for Mathematics
Phillip Griffiths Institute for Advanced Study, Princeton; University of Miami
Tiago Jardim da Fonseca Universidade Estadual de Campinas
Bruno Kahn Institut de Mathématiques de Jussieu
Matt Kerr Washington University in St. Louis
Minhyong Kim International Centre for Mathematical Sciences; University of Edinburgh
Amalendu Krishna Indian Institute of Science
Stephen Lichtenbaum Brown University
Paulo Lima-Filho Texas A&M University
Matthew Morrow Institut de Mathématiques de Jussieu
Gregory Pearlstein Texas A&M University; Università di Pisa
Yunqing Tang Université Paris Saclay; CNRS (Centre national de la recherche scientifique)

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Participants

Alexey Ananyevskiy None / Other; Steklov Mathematical Institute, Russian Academy of Sciences
Donu Arapura Purdue University
Masanori Asakura Hokkaido University
Ran Azouri Universität Zürich
Paul Balmer University of California, Los Angeles
Przemyslaw Baszihter Universiteit van Amsterdam
Federico Binda University of Milan; Università degli Studi di Milano
Spencer Bloch University of Chicago
Vasily Bolbachan Higher School of Economics, Moscow
Patrick Brosnan University of Maryland, College Park
Francis Brown University of Oxford
François Brunault ENS - Lyon
Viktor Burghardt Northwestern University
Anna Cadoret Sorbonne Université
Ana Caraiani Imperial College London
Steven Charlton Universität Hamburg
Xi Chen University of Alberta
David Corwin University of California, Berkeley
Rob de Jeu Vrije Universiteit Amsterdam
Pedro Del Ángel Centro de Investigación en Matemáticas
Netan Dogra King's College London
Charles Doran University of Alberta
Elden Elmanto Harvard University
Olivier Fouquet Université de Franche-Comté
Wojtek Gajda Adam Mickiewicz University
Herbert Gangl Durham University
Quentin Gazda Max Planck Institute for Mathematics
Phillip Griffiths Institute for Advanced Study, Princeton; University of Miami
Mark Gross University of Cambridge
Gabriela Guzman Polska Akademia Nauk
José Jaime Hernández Castillo CIMAT (Centro de Investigación en Matemáticas), Mexico
Tiago Jardim da Fonseca Universidade Estadual de Campinas
Roy Joshua Ohio State University
Bruno Kahn Institut de Mathématiques de Jussieu
Mahesh Kakde Indian Institute of Science
Elana Kalashnikov University of Waterloo
Matt Kerr Washington University in St. Louis
Adeel Khan Institute of Mathematics, Academia Sinica, Taipei, Taiwan
Minhyong Kim International Centre for Mathematical Sciences; University of Edinburgh
Lukas Kofler University of Cambridge
Kees Kok Universiteit van Amsterdam
Naomi Kraushar Stanford University
Amalendu Krishna Indian Institute of Science
Yeuk Hay Joshua Lam IHES
Jonghoo Lee Ohio State University
Marc Levine Universität Duisburg-Essen
Stephen Lichtenbaum Brown University
Joshua Lieber Max Planck Institute for Mathematics
Paulo Lima-Filho Texas A&M University
Ma Luo University of Oxford
Lorenzo Mantovani Johannes Gutenberg-Universität Mainz
Diego Marques Universidade de Brasilia
Naruki Masuda Johns Hopkins University
Mona Merling University of Pennsylvania
Ben Moore University of Warwick
Matthew Morrow Institut de Mathématiques de Jussieu
Owen Patashnick University of Bristol; King's College London
Gregory Pearlstein Texas A&M University; Università di Pisa
Riccardo Pengo ENS - Lyon
Mohammad Reza Rahmati Centro de Investigación en Matemáticas
Jitendra Rathore Tata Institute of Fundamental Research
Charanya Ravi Universität Regensburg
Gabriel Ribeiro École Polytechnique
Oliver Roendigs Universität Osnabrück
Anand Sawant Tata Institute of Fundamental Research
Marco Schlichting University of Warwick
Anthony Scholl University of Cambridge
Evgeny Shinder University of Sheffield
Bogdan Simeonov None / Other
Vasudevan Srinivas Tata Institute of Fundamental Research
Padmavathi Srinivasan University of Georgia
Jay Swar University of Oxford
Ashar Tafhim Bergische Universität Wuppertal
Yunqing Tang Université Paris Saclay; CNRS (Centre national de la recherche scientifique)
Burt Totaro University of California, Los Angeles
Charles Weibel Rutgers, The State University of New Jersey
Ursula Whitcher American Mathematical Society
Ferdinando Zanchetta University of Bologna