Simons Collaboration: Arithmetic Geometry, Number Theory, and Computation

An $8M collaborative project between Boston University, Brown University, Dartmouth College, Harvard University, and MIT

Scientific objectives
Sato-Tate data: Andrew Sutherland

Our common perspective is that advances in computational techniques will drive research in arithmetic geometry and number theory, both as a source of data and examples, and as an impetus for effective results; and that tackling central mathematical problems requires a combination of different computational platforms and techniques.

The dynamic interplay between experiment, theory, and computation has played a pivotal role in the development of number theory. Systematic computations have driven progress in recent decades, from the Birch-Swinnerton-Dyer conjecture in the 1960s based on computations with curves of genus one to recent progress on the Sato-Tate conjecture in genus two.

The frontier of research is now studying curves of higher genus, abelian surfaces, and K3 surfaces. Here, however, the development and implementation of practical algorithms has lagged behind the theory, and we seek to correct this imbalance. Available computational resources have reached a point where algorithms are now technically feasible. In contrast to the situation with curves of low genus, brute force computation typically yields very little; to obtain practical algorithms one must exploit the theoretical infrastructure of modern arithmetic geometry.

Simons Collaboration Team - 2017

Principal Investigators

Photo Jennifer S. Balakrishnan is a number theorist working on explicit methods for curves. Her research is motivated by various aspects of the classical and p-adic Birch and Swinnerton-Dyer conjectures, as well as the problem of algorithmically finding rational points on curves. She is currently the Clare Boothe Luce Assistant Professor of Mathematics at Boston University. Previously, she was a Titchmarsh Research Fellow at the Mathematical Institute of the University of Oxford, a Junior Research Fellow of Balliol College, Oxford, and an NSF Postdoctoral Fellow at Harvard. Balakrishnan received an AB and AM from Harvard University and a PhD in Mathematics from MIT.
Photo Noam D. Elkies is professor of mathematics at Harvard University. He did undergraduate work at Columbia and obtained his doctorate at Harvard in 1987. He is known for exhibiting many integral solutions to the equation

A4 + B4 + C4 = D4

settling a 200-year-old question of Euler. Elkies also proved that elliptic curves over the rational numbers admit infinitely many supersingular primes. He is famous for `extreme examples' in number theory: elliptic curves with large rank, K3 surfaces with large Picard group, etc. Elkies has also contributed to algorithms for elliptic curves over finite fields, characterizations of efficient sphere packings, and tilings of Aztec dimanonds by dominoes. In 2017 he was elected to the National Academy of Sciences.
Photo Brendan Hassett is Professor of Mathematics and Director of the Institute for Computational and Experimental Research in Mathematics at Brown University. He received his BA in 1992 from Yale and his PhD from Harvard in 1996 under the supervision of Joseph Harris. From 1996 to 2000 he worked as a Dickson Instructor at the University of Chicago, partly supported by a National Science Foundation Postdoctoral Research Fellowship. He was a faculty member at Rice University from 2000 to 2015 and chaired its mathematics department from 2009 to 2014. He has held visiting positions at the Mittag Leffler Institute in Stockholm, the Chinese University of Hong Kong, and the University of Paris (Orsay).

Brendan’s research focus is algebraic geometry. He enjoys geometric questions inspired from number theory, problems natural from an arithmetic perspective but with surprising geometric implications. He also likes classification problems involving a wide range of techniques, from the classical theory of algebraic curves through Hodge theory and birational geometry. Brendan has written more than 50 research papers and has authored or co-edited six books. His work has been recognized with a Sloan Research Fellowship, a National Science Foundation CAREER award, and the Charles W. Duncan Award for Outstanding Faculty at Rice. He is a Fellow of the American Mathematical Society. Brendan has co-organized numerous conferences and workshops, including the 2006 Clay Summer School in Arithmetic Geometry, the Spring 2009 program in Algebraic Geometry at MSRI, and the 2015 AMS Summer Institute in Algebraic Geometry.
Photo Bjorn Poonen received an A.B. in Mathematics and Physics from Harvard in 1989, and a Ph.D. in Mathematics from U.C. Berkeley in 1994. In 2008, after positions at MSRI, Princeton, and U.C. Berkeley, he moved to MIT, where he is the Claude Shannon Professor of Mathematics.

Poonen is known for developing and analyzing algorithms aimed at determining the set of rational points of a given variety. But his theorems also demonstrate the limitations of known methods, and even show that certain related problems are undecidable. His role in the collaboration is to develop algorithms for varieties over number fields and finite fields, to explore new computational approaches to theoretical problems in the field, and to join Andrew Sutherland in leading the team of scientists working at MIT on the project.

Poonen has received the Guggenheim, Packard, Rosenbaum, Simons, and Sloan fellowships, as well as a Miller Professorship, the Chauvenet Prize, and the MIT School of Science Prize in Undergraduate Teaching. He is also a four-time Putnam Competition winner, a Simons Investigator, a fellow of the American Academy of Arts and Sciences and of the American Mathematical Society, and the founding managing editor of Algebra & Number Theory. He will deliver an invited address at the 2018 International Congress of Mathematicians.
Photo Andrew Sutherland received his S.B. in mathematics from MIT in 1990. As an undergraduate, Sutherland co-founded the software company Escher Group, specializing in high-performance distributed computing, and after completing an NSF Graduate Fellowship at MIT, served as the company's Chief Technology Officer for ten years. Sutherland returned to academia and completed his Ph.D. in mathematics at MIT in 2007, winning the George M. Sprowles Prize for his thesis. He joined the MIT mathematics department in 2009, and was promoted to Principal Research Scientist in 2012.

Sutherland's research focuses on computational number theory and arithmetic geometry. He is best known for his work on efficient algorithms to compute zeta functions, and Galois representations, objects that play a key role in the research program of the collaboration. Together with Bjorn Poonen, Sutherland will lead the MIT-based team of researchers working on the project.

Sutherland was awarded the Selfridge Prize in 2012, and currently serves on the boards of Mathematics of Computation, the L-functions and Modular Forms Database, and the Number Theory Foundation. He is a managing editor of Research in Number Theory and current record-holder for the largest cloud-based research computation (580,000 cores).
Photo John Voight is associate professor of mathematics at Dartmouth College. He received his Ph.D. in 2005 from the University of California, Berkeley, and has held positions at the University of Sydney, the Institute for Mathematics and its Applications (IMA) at the University of Minnesota, and the University of Vermont.

Voight's research interests are in arithmetic geometry and number theory, with a focus on algorithmic aspects. His current research concerns computational problems for moduli spaces and automorphic forms. He is the author of a textbook on quaternion algebras (under review) and received an NSF CAREER award. He was the recipient of the Selfridge Prize in 2010 and currently serves on the board for the L-functions and Modular Forms DataBase (LMFDB)

Affliated Scientists

Photo Edgar Costa Edgar Costa is an Instructor in Applied and Computational Mathematics at Dartmouth College and will join the team based at MIT this summer. He received his Ph.D. in 2015 from the Courant Institute of Mathematical Sciences at New York University and was a Postdoctoral Fellow at the Institute for Computational and Experimental Research in Mathematics at Brown University for the semester program "Computational Aspects of the Langlands Program."

Costa's research interests are centered around effective methods in arithmetic geometry, arithmetic statistics, and number theory. His current research is focused on the development and application of theoretical and computational techniques to study the interconnections predicted by the Langland's program.
Photo John Cremona has been a Professor at the University of Warwick, UK since 2007. After obtaining his DPhil under Birch at Oxford in 1981, he held positions at the University of Michigan and Dartmouth College in the US and at the universities of Exeter and Nottingham in the UK. John Cremona is best known for developing and implementing modular symbol algorithms and using these to compile tables of elliptic curves over the rationals.

Since his thesis, he has also developed such algorithms over imaginary quadratic fields, and recently they have been used to compile similar elliptic curve tables over these fields. All these collections of elliptic curves are now in the LMFDB. More generally, John Cremona has developed many algorithms concerning the arithmetic of elliptic curves which he has contributed to software packages Pari/GP, Magma and SageMath. Since 2013, he has been PI on a major grant "LMF: L-functions and modular forms" from the UK EPSRC which has part-funded the computing infrastructure and workshops which underpin the LMFDB.
Photo Netan Dogra obtained his doctorate from the University of Oxford in 2015 and is a postdoc at Imperial College London. His main research interests are Diophantine Geometry and arithmetic fundamental groups.
Photo David Harvey is a Senior Lecturer and Australian Research Council Future Fellow at the University of New South Wales in Sydney, Australia. He received his Ph.D. from Harvard in 2008, and subsequently held a three-year postdoctoral position at New York University.

Harvey's research interests include algorithmic number theory, especially computing zeta functions of varieties over finite fields, and symbolic computation, especially algorithms for efficient arithmetic on large integers and polynomials.
Photo Kiran Kedlaya is the Stefan E. Warschawski Professor of Mathematics at University of California, San Diego. He received his PhD from MIT in 2000. He is a recipient of the Presidential Early Career Award for Scientists and Engineers, a Sloan Research Fellowship, and a Guggenheim Fellowship.

Prof. Kedlaya's research covers a variety of topics in algebraic geometry and number theory, with some emphasis on the computational aspects of these areas. He has developed practical algorithms for computing L-functions of algebraic varieties using p-adic analysis, and for tabulating isogeny classes of abelian varieties over finite fields. He is a frequent contributor to Sage and the LMFDB.
Photo Steffen Müller is an assistant professor at the Unviersity of Groningen. He received his PhD from Bayreuth in 2010 and was then a member of the mathematics department in Hamburg and in Oldenburg. He is mainly interested in explicit methods in arithmetic geometry, in particular the use of archimedean and p-adic height functions to study rational points on curves and abelian varieties.
Photo David Roe will join the team in February as a researcher at MIT. His study of p-adic computation includes work on computing L-functions of varieties, p-adic modular forms and methods for tracking precision. He is also interested in the local Langlands correspondence and p-adic tori. He contributes frequently to Sage, and helped create the LMFDB database of isogeny classes of abelian varieties.

David received an S.B. in mathematics and literature from MIT in 2006, completed his Ph.D. at Harvard in 2011, then worked as a postdoctoral fellow at the Universities of Calgary, British Columbia and Pittsburgh before returning to MIT.
Photo Joseph H. Silverman is a Professor of Mathematics at Brown University, where he has been on the faculty since 1988. He is a recipient of a Sloan Research Fellowship, a Guggenheim Fellowship, and an AMS Steele Prize. His primary research interests are elliptic curves, arithmetic geometry, arithmetic dynamics, and cryptography, and he has also written a number of the standard textbooks in these areas.
Photo Jan Vonk obtained his doctorate from the University of Oxford in 2015 and is currently a postdoctoral fellow in Montréal at McGill University. His work focuses on p-adic aspects of arithmetic geometry and their application to explicit class field theory of real quadratic number fields. His main research interests include p-adic methods for studying the arithmetic of higher genus curves, as well as related computational aspects, such as the explicit determination of rational points.
ICERM welcomes all past and future participants of our semester programs, workshops and Summer@ICERM undergraduate programs to come together for its annual mixer. Those interested in the new Simons collaboration `Arithmetic Geometry, Number Theory, and Computation' are also invited. The ICERM Mixer will be held being held during the 2018 Joint Mathematics Meeting in San Diego, California. The event will be hosted by ICERM Director Brendan Hassett on Friday, January 12 from 4:00pm to 6:00pm in the Marriott Marquis San Diego Marina Hotel. RSVP at Refreshments will be served.
Simons collaboration conference Arithmetic Geometry, Number Theory, and Computation at MIT (August 20-24, 2018)
co-organized by Dr. Balakrishnan:
Sage Days 88: Opening workshop for a year of coding sprints at the IMA (August 21-25, 2017)
Women in Numbers 4 (August 14-18, 2017)
Sage Days 87: p-adics in Sage and the LMFDB (July 17-22, 2017)
co-organized by Dr. Voight:
LMFDB workshop (June 12-16, 2017)
We are hiring a number of mathematicians and programmers to support the work of the collaboration.
Recent Publications

  • Ariyan Javanpeykar and John Voight, The Belyi degree is computable, arXiv:1711.00125

  • Jennifer S. Balakrishnan and Jan Tuitman, Explicit Coleman integration for curves, arXiv:1710.01673

  • Andrew V. Sutherland and José Felipe Voloch, Maps between curves and arithmetic obstructions, arXiv:1709.05734v1

  • Bjorn Poonen, Using zeta functions to factor polynomials over finite fields, arXiv:1710.00970

We are grateful for the support of the Simons Foundation. See the Foundation's, and Brown University's press release.

View the Simons Founation's Collaboration page for more information about the foundation and its collaborations.