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I am a Principal Research Scientist in the mathematics department at MIT, focused on computational number theory and arithmetic geometry.
Here is a larger photograph, my CV, and links to my arXiv, MathSciNet, zbMath, DBLP, Google Scholar and Wikipedia pages. My office is in room 2-341 in the Simons Building (Building 2).
My work is supported by grants from the National Science Foundation and the Simons Foundation; I am a Principal Investigator in the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation.
Recent/Upcoming Events
Mathematics and Machine Learning closing workshop, Harvard CMSA, October 28-30, 2024.
Murmurations in arithmetic geometry and related topics, Simons Center for Geometry and Physics, November 11-15, 2024.
Joint meetings of the NZMS, AustMS, and AMS, University of Auckland, December 9-13, 2024.
JMM 2025, Seattle, Jan 8-11.
Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting, Simons Foundation, January 15-16, 2025.
The legacy of John Tate, and beyond, Harvard, March 17-21, 2025.
Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), CIRM, Luminy, June 9-13, 2025.
Algebaric points on curves, ICERM, June 23-27, 2025.
Journées Arithmétiques, University of Luxembourg , June 30-July 4, 2025.
LMFDB, Computation, and Number Theory (LuCaNT 2025), ICERM, July 7-11, 2025.
René 25: Celebrating the research interests of René Schoof, University of French Polynesia, August 18-22, 2025.
Current Teaching and Seminars
VaNTAGe (co-organized with Rachel Pries).
MIT number theory seminar (co-organized with Bjoon Poonen and Wei Zhang).
BC-MIT number theory seminar (co-organized with Solomon Friedberg, Ben Howard, Dubi Kelmer, Spencer Leslie, Keerthi Madapusi Pera, Bjorn Poonen, and Wei Zhang).
Current Editorial/Board/Organizational Positions
Mathematics of Computation (Associate Editor), 2014-present.
L-functions and Modular Forms Database (Managing Editor), 2016-present.
Research in Number Theory (Editor in Chief), Springer, 2017-present.
The Number Theory Foundation (President), 2019-present.
Algorithmic Number Theory Symposia (Vice Chair), 2019-present.
researchseminars.org (Administrator), 2020-present.
psetpartners.mit.edu (Administrator), 2020-present.
SageMath Inc (Board Member), 2023-present.
Institute for Computational and Experimental Research in Mathematics (Scientific Advisory Board Member), 2024-present.
Research Publications (click title for arxiv version, journal/conference name for published version)
Sato-Tate groups of abelian threefolds, with Francesc Fité and Kiran S. Kedlaya,
to appear in Memoirs of the American Mathematical Society. (code)
Computing the endomorphism ring of an elliptic curve over a number field, with John Cremona,
LuCaNT: LMFDB, Computation, and Number Theory, Contemporary Mathematics 796 (2024), 75-102. (code)
[MR 4732684, DOI 10.1090/conm/796/15998]
Counting points on smooth plane quartics, with Edgar Costa and David Harvey,
Fifteenth Algorithmic Number Theory Symposium (ANTS XV), Research in Number Theory 9:1 (2023), 32 pages. (code)
[MR 4514545, Zbl 7622906, DOI 10.1007/s40993-022-00397-8]
Appendix to From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture
, by Seoyoung Kim and M. Ram Murty,
Mathematics of Computation 92 (2023), 385-408.
[MR 4496969, Zbl 7603635, DOI 10.1090/mcom/3773]
ℓ-adic images of Galois for elliptic curves over ℚ, with Jeremy Rouse and David Zureick-Brown, and an appendix with John Voight,
Forum of Mathematics, Sigma 10 (2022), 62 pages. (code and data, errata)
[MR 4468989, Zbl 07577487, DOI 10.1017/fms.2022.38]
Appendix to Computing L-Polynomials of Picard curves from Cartier-Manin matrices, by Sualeh Asif, Francesc Fité, and Dylan Pentland,
Mathematics of Computation 91 (2022), 943-971. (offprint, code)
[MR 4379983, Zbl 074473351, DOI 10.1090/mcom/3675]
Computing classical modular forms, with A.J. Best, J. Bober, A.R. Booker, E. Costa, J. Cremona, M. Derickx, M. Lee, D. Lowry-Duda, D. Roe, and J. Voight,
Arithmetic Geometry, Number Theory, and Computation, Simons Symposia (2021), 123-213. (offprint)
[MR 4427962, Zbl 7912218, DOI 10.1007/978-3-030-80914-0_4]
Stronger arithemtic equivalence
Discrete Analysis 2021, Paper No. 23, 23 pp.
[MR 4341956, Zbl 7471813, DOI 10.19086/da.29452]
On a question of Mordell, with Andrew R. Booker,
Proceedings of the National Academy of Sciences 118 (2021), no. 11., e2022377118 (code, commentary)
[MR 4279690, DOI 10.1073/pnas.2022377118]
Sato-Tate groups of abelian threefolds: a preview of the classification, with Francesc Fité and Kiran S. Kedlaya,
Arithmetic Geometry, Cryptography, and Coding Theory, Contemporary Mathematics 770 (2021), 103-129. (offprint)
[MR 4280389, Zbl 1497.11158, DOI 10.1090/conm/770/15432]
Counting points on superelliptic curves in average polynomial time,
Fourteenth Algorithmic Number Theory Symposium (ANTS XIV), The Open Book Series 4 (2020), 403-422.
[MR 4235126, Zbl 1472.11322, DOI 10.2140/obs.2020.4.403]
Arithmetic invariants from Sato-Tate moments, with Edgar Costa and Francesc Fité,
Comptes Rendus Mathematique 357 (2019), 823-826.
[MR 4038255, Zbl 1444.11127, DOI 10.1016/j.crma.2019.11.008]
Sato-Tate distributions,
Analytic Methods in Arithmetic Geometry, Contemporary Mathematics 740 (2019), 197-248. (offprint)
[MR 4033732, Zbl 1440.11176, DOI 10.1090/conm/740/14904]
A database of nonhyperelliptic genus 3 curves over ℚ,
Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), The Open Book Series 2 (2019), 443-459. (code and data)
[MR 3952027, Zbl 1516.14058, DOI 10.2140/obs.2019.2.443]
Fast Jacobian arithmetic for hyperelliptic curves of genus 3,
Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), The Open Book Series 2 (2019), 425-442. (code)
[MR 3952026, Zbl 1516.14061, DOI 10.2140/obs.2019.2.425]
Maps between curves and arithmetic obstructions, with Felipe Voloch,
Arithmetic Geometry: Computations and Applications, Contemporary Mathematics 722, AMS, 2019, 167--175. (offprint)
[MR 3896855, Zbl 1464.11129, DOI 10.1090/conm/722/14532]
Sato-Tate distributions of twists of the Fermat and the Klein quartics, with Francesc Fite and Elisa Lorenzo Garcia,
Research in the Mathematical Sciences 5 (2018), article 41. (code, errata)
[MR 3864839, Zbl 1451.11050, DOI 10.1007/s40687-018-0162-0]
Torsion subgroups of rational elliptic curves over the compositum of all cubic fields, with Harris B. Daniels, Alvaro Lozano-Robledo, and Filip Najman,
Mathematics of Computation 87 (2018), 425-458. (offprint, code, errata)
[MR 3716201, Zbl 1422.11132, DOI 10.1090/mcom/3213]
Torsion subgroups of elliptic curves over quintic and sextic number fields, with Maarten Derickx,
Proceedings of the AMS 145 (2017), 4233-4245. (offprint, code)
[MR 3690609, Zbl 1421.11049, DOI 10.1090/proc/13605]
Modular curves of prime-power level with infinitely many rational points, with David Zywina,
Algebra and Number Theory 11 (2017), 1199-1229. (code, errata)
[MR 3671434, Zbl 1374.14022, DOI 10.2140/ant.2017.11.1199]
Computing L-series of geometrically hyperelliptic curves of genus three, with David Harvey and Maike Massierer,
Twelfth Algorithmic Number Theory Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 220-234.
[MR 3540957, Zbl 1404.11143, DOI 10.1112/S1461157016000383]
Finding elliptic curves with a subgroup of prescribed size, with Igor E. Shparlinski,
International Journal of Number Theory 13 (2017), 133-152.
[MR 3573417, Zbl 1377.11074, DOI 10.1142/S1793042117500099]
A database of genus 2 curves over the rational numbers, with Andrew R. Booker, Jeroen Sijsling, John Voight, and Dan Yasaki,
Twelfth Algorithmic Number Theory Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 235-254. (data)
[MR 3540942, Zbl 1391.14075, DOI 10.1112/S146115701600019X]
A census of zeta functions of quartic K3 surfaces over F2, with Kiran S. Kedlaya,
Twelfth Algorithmic Number Theory Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 1-11.
[MR 3540942, Zbl 1391.14075, DOI 10.1112/S1461157016000140]
Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II, with David Harvey,
Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics 663 (2016), 127-148. (code, offprint)
[MR 3502941, Zbl 1417.11121, DOI 10.1090/conm/663/13352]
Sato-Tate groups of y2=x8+c and y2=x7-cx, with Francesc Fité,
Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics 663 (2016), 103-126. (offprint)
[MR 3502940, Zbl 1411.11090, DOI 10.1090/conm/663/13351]
Sato-Tate groups of some weight 3 motives, with Francesc Fité and Kiran S. Kedlaya,
Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics 663 (2016), 57-102. (offprint)
[MR 3502939, Zbl 1411.11089, DOI 10.1090/conm/663/13350]
Computing images of Galois representations attached to elliptic curves,
Forum of Mathematics, Sigma 4 (2016), e4, 79 pages. (code)
[MR 3482279, Zbl 1361.11040, DOI 10.1017/fms.2015.33]
A framework for deterministic primality proving using elliptic curves with complex multiplication, with Alexander Abatzoglou, Alice Silverberg, and Angela Wong,
Mathematics of Computation 85 (2016), 1461-1483.
[MR 3454371, Zbl 1345.11088, DOI 10.1090/mcom/3001]
Class polynomials for nonholomorphic modular functions, with Jan Bruinier and Ken Ono,
Journal of Number Theory 161 (2016), 204-229.
[MR 3435725, Zbl 1332.11047, DOI 10.1016/j.jnt.2015.07.002]
Variants of the Selberg sieve, and bounded intervals containing many primes, as part of D.H.J. Polymath,
Research in the Mathematical Sciences 1 (2014).
[MR 3373710, Zbl 1400.11164, DOI 10.1186/s40687-014-0012-7]
On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average, with Igor Shparlinski,
LMS Journal of Computation and Mathematics 18 (2015) 308-322.
[MR 3349320, Zbl 1400.11164, DOI 10.1112/S1461157015000017]
New equidistribution estimates of Zhang type, with W. Castryck, E. Fouvry, G. Harcos, E. Kowalski, P. Michel, P. Nelson, E. Paldi, J. Pintz, T. Tao, and X.-F. Xie,
Algebra and Number Theory 8 (2014), 2067-2199.
[MR 3294387, Zbl 1307.11097, DOI 10.2140/ant.2014.8.2067]
Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, with David Harvey,
Eleventh Algorithmic Number Theory Symposium (ANTS XI), LMS Journal of Computation and Mathematics 17 (2014), 257-273. (code)
[MR 3240808, Zbl 1296.11076, DOI 10.1112/S1461157014000187]
Sato-Tate distributions of twists of y2=x5-x and y2=x6+1, with Francesc Fité,
Algebra and Number Theory 8 (2014), 543-585.
[MR 3218802, Zbl 1303.14051, DOI 10.2140/ant.2014.8.543]
On the distribution of Atkin and Elkies primes, with Igor E. Shparlinski,
Foundations of Computational Mathematics 14 (2014), 285-297.
[MR 3179585, Zbl 1312.11048, DOI 10.1007/s10208-013-9181-9]
Isogeny volcanoes,
Tenth Algorithmic Number Theory Symposium (ANTS X), The Open Book Series 1 (2013), 507-530.
[MR 3207429, Zbl 1345.11044, DOI 10.2140/obs.2013.1.507]
On the evaluation of modular polynomials,
Tenth Algorithmic Number Theory Symposium (ANTS X), The Open Book Series 1 (2013), 531-555. (errata, award)
[MR 3207430, Zbl 1344.11087, DOI 10.2140/obs.2013.1.531]
Deterministic elliptic curve primality proving for a special sequence of numbers, with Alexander Abatzoglou, Alice Silverberg, and Angela Wong,
Tenth Algorithmic Number Theory Symposium (ANTS X), The Open Book Series 1 (2013), 1-20.
[MR 3207405, Zbl 1344.11082, DOI 10.2140/obs.2013.1.1]
Identifying supersingular elliptic curves,
LMS Journal of Computation and Mathematics 15 (2012), 317-325.
[MR 2988819, Zbl 1307.11072, DOI 10.1112/S1461157012001106]
Sato-Tate distributions and Galois endomorphism modules in genus 2, with Francesc Fité, Kiran S. Kedlaya, and Victor Rotger,
Compositio Mathematica 148 (2012), 1390-1442 (errata).
[MR 2982436, Zbl 1269.11094, DOI 10.1112/S0010437X12000279]
Accelerating the CM method,
LMS Journal of Computation and Mathematics 15 (2012), 172-204.
[MR 2970725, Zbl 1343.11098, DOI 10.1112/S1461157012001015]
The probability that the number of points on the Jacobian of a genus 2 curve is prime, with Wouter Castryck, Amanda Folsom, and Hendrik Hubrechts,
Proceedings of the London Mathematical Society 104 (2012), 1235-1270.
[MR 2946086, Zbl 1333.11059, DOI 10.1112/plms/pdr063]
A local-global principle for rational isogenies of prime degree,
Journal de Théorie des Nombres de Bordeaux 24 (2012), 475-485 (errata).
[MR 2950703, Zbl 1276.11095, DOI 10.5802/jtnb.807]
A low-memory algorithm for finding short product representations in finite groups, with Gaetan Bisson,
Designs, Codes, and Cryptography 63 (2012), 1-13.
[MR 2890318, Zbl 1246.20031, DOI 10.1007/s10623-011-9527-8]
Modular polynomials via isogeny volcanoes, with Reinier Bröker and Kristin Lauter,
Mathematics of Computation 81 (2012), 1201-1231.
[MR 2869057, Zbl 1267.11125, DOI 10.1090/S0025-5718-2011-02508-1]
Constructing elliptic curves over finite fields with prescribed torsion,
Mathematics of Computation 81 (2012), 1131-1147.
[MR 2869053, Zbl 1267.11074, DOI 10.1090/S0025-5718-2011-02538-X]
Computing the endomorphism ring of an ordinary elliptic curve over a finite field, with Gaetan Bisson,
Journal of Number Theory 113 (2011), 815-831. (code)
[MR 2772473, Zbl 1225.11085, DOI 10.1016/j.jnt.2009.11.003]
Computing Hilbert class polynomials with the Chinese Remainder Theorem,
Mathematics of Computation 80 (2011), 501-538. (code)
[MR 2728992, Zbl 1231.11144, DOI 10.1090/S0025-5718-2010-02373-7]
Structure computation and discrete logarithms in finite abelian p-groups,
Mathematics of Computation 80 (2011), 477-500. (code)
[MR 2728991, Zbl 1225.11163, DOI 10.1090/S0025-5718-10-02356-2]
Class invariants by the CRT method, with Andreas Enge,
Ninth Algorithmic Number Theory Symposium (ANTS IX), Lecture Notes in Computer Science 6197 (2010), 142-156. (code)
[MR 2721418, Zbl 1260.11083, DOI 10.1007/978-3-642-14518-6_14]
An explicit height bound for the classical modular polynomial, with Reinier Bröker,
Ramanujan Journal 22 (2010), 293-313.
[MR 2670978, Zbl 1245.11079, DOI 10.1007/s11139-010-9231-8]
On a theorem of Mestre and Schoof, with John E. Cremona,
Journal de Théorie des Nombres de Bordeaux 22 (2010), 353-358.
[MR 2769066, Zbl 1223.11072, DOI 10.5802/jtnb.719]
Hyperelliptic curves, L-polynomials, and random matrices,
with Kiran S. Kedlaya,
Arithmetic, Geometry, Cryptography, and Coding Theory, Contemporary Mathematics 487 (2009), 119-162.
[MR 2555991, Zbl 1233.11074, DOI 10.1090/conm/487/09529]
A generic approach to searching for Jacobians,
Mathematics of Computation 78 (2009), 485-507.
[MR 2448717, Zbl 1208.14020, DOI 10.1090/S0025-5718-08-02143-1]
Computing L-series of hyperelliptic curves,
with Kiran S. Kedlaya,
Eighth Algorithmic Number Theory Symposium (ANTS VIII), Lecture Notes in Computer Science 5011 (2008), 312-326. (code)
[MR 2467855, Zbl 1232.11078, DOI 10.1007/978-3-540-79456-1_21]
Order computations in generic groups,
PhD thesis, Massachusetts Institute of Technology, 2007 (errata, award)
[MR 2717420]
Preprints
Doubly isogenous curves of genus two with a rational action of D6, with Jeremy Booher, Everett W. Howe, José Felipe Voloch, 2024. (code)
Computing Euler factors of genus 2 curves at odd primes of almost good reduction, with Céline Maistret, 2024. (code)
Sorting and labelling integral ideals in a number field, wih John Cremona and Aurel Page, 2020. (code)
Letters, lecture notes, and expository articles
Letter to Michael Rubinstein and Peter Sarnak, August 30, 2022.
Pset Partners, Notices of the AMS 68 (2021), no. 11, 1919-1923.
The L-Functions and Modular Forms Database, with John E. Cremona, John W. Jones, and John Voight, Notices of the AMS 68 (2021) no. 9, 1520-1522.
Research Seminars: A New Hope, with Edgar Costa, Bjorn Poonen, and David Roe, Mathematics Online First Collections, Springer, 2020.
Number Theory II (18.786), Spring 2024.
Elliptic Curves (18.783), Fall 2023. (2021 version available on OCW).
Number Theory I (18.785), Fall 2021. (also available on OCW)
Arithmetic Equivalence and Isospectrality, minicourse in Topics in Algebra (18.708), Spring 2018.
Primes is in P (18.095 lecture), January 2017.
Perfect Forward Secrecy (18.095 lecture), January 2016.
The "bounded gaps between primes" Polymath project - a retrospective, Newsletter of the European Mathematical Society 94 (2014), 13-23.
Arithmetic geometry (18.782), Fall 2013 (OCW link).
Torsion subgroups of elliptic curves over number fields, notes from a lecture given at the Harvard seminar on Mazur's torsion theorem, December 2012.
The mathematics of Spinpossible, expository article about the game Spinpossible, 2011.
Talks (selected)
Genus2 curves over Q of small conductor, CAVARET, Barcelona, June 2024
The fine art of point counting, Beeger Lecture, Nederlands Mathematisch Congres, April 2024
L-functions from nothing, Number Theory Seminar, Harvard, December, 2023.
Murmurations: A computational perspective, Computational algebra and Magma, University of Sydney (Australia), December, 2023.
Building a database of modular curves, Computational algebra and Magma, University of Sydney (Australia), November, 2023.
Breaking post-quantum cryptography with arithmetic geometry, Undergraduate Mathematics Association, MIT, October, 2023 (updated version of August 2022 talk at PCMI).
Murmurations of arithmetic L-functions, Number Theory Web Seminar, September, 2023 (also at ICERM in July 2023). (animations, video)
Murmurations of arithmetic L-functions, Arithmetic statistics, CIRM (France), May, 2023 (also at Princeton/IAS in April 2023). (video)
A database of modular curves, Arithmetic, Algebra, and Algorithms -- Celebrating the Mathematics of Hendrik Lenstra, ICMS (Scotland), April, 2023.
Sato-Tate groups of abelian threefolds, Algebraic geometry seminar, METU (Turkey), October 2022.
Diophantine computations, Arf Lecture, METU (Turkey), October 2022. (video)
On a question of Mordell, Mordell 2022, August 2022.
Enumerating mathematical objects in the cloud, Big Data in Pure Mathematics workshop, online, May 2022.
Number theory and the LMFDB, Harvard CMSA, April 2022.
Abelian surfaces and their L-functions, Simons Foundation (online), January 2022.
ℓ-adic images of Galois for elliptic curves over ℚ, Upstate Number Theory Conference, Union College, Schenectady, NY, October, 2021.
Stronger arithmetic equivalence, Around Frobenius Distributions II (online), June 2021.
Sato-Tate groups of abelian threefolds, Front Range Number Theory Day (online), September 2020. (video)
The L-functions and modular forms database, ICMS, July 2020. (video)
Counting points on superelliptic curves in average polynomial time, ANTS XIV, July, 2020. (video)
Mathematics research online: Hosting virtual events, online panel discussion co-organized with Bianca Viray, May, 2020. (video)
Computing L-functions of modular curves, MAGIC seminar, May, 2020.
Sums of three cubes, Number Theory Web seminar, May 2020 (also at Simons Foundation, JMM, and MIT in January 2020). (video)
Arithmetic L-functions and their Sato-Tate distributions, VaNTAGe semianr, April 2020. (video)
Sums of three cubes, Computational Mathematics Colloquium, University of Waterloo (Canada), November 2019.
Computing zeta functions and L-functions of curves, CMI-HMIR Summer School, University of Bristol (UK), June 2019.
Lecture 1 slides; Lecture 2 slides; Lecture 3 slides, Lecture 4 slides.
Counting points on modular curves, AGCT 2019, CIRM Luminy (France), June 2019.
Stronger arithmetic equivalence, Princeton/IAS Number Theory Seminar, Princeton, March 2019.
Building Telescopes for Mathematicians, Simons Lecture (video), Simons Foundation (NYC), January 2019.
Elliptic curve cryptography in a post-quantum world, Undergraduate Mathematics Association, MIT, November 2018.
A database of genus 3 curves over ℚ, ANTS XIII, University of Wisonsin, Madison, July 2018.
Fast Jacobian arithmetic for hyperelliptic curves of genus 3, ANTS XIII, University of Wisonsin, Madison, July 2018.
Computation in supersingular isogeny graphs, CTNT 2018, University of Connecticut, June 2018.
A database of genus 3 curves over ℚ, Birational Geometry and Arithmetic, ICERM (Providence), May 2018.
Strong arithmetic equivalence, University of Colorado and Brown University, February 2018.
Computing L-functions of hyperelliptic curves, Workshop on the Arithmetic of Hyperelliptic Curves, ICTP Trieste (Italy), September 2017.
Computing zeta functions in average polynomial time, AMMCS 2017, Waterloo (Canada), August 2017.
A database of genus 3 curves over ℚ, Rational Points 2017, Franken-Akademie Schloss Schney (Germany), July 2017.
Computing L-series of genus 3 curves, Workshop on Arithmetic Geometry and Computer Algebra, University of Oldenburg (Germany), June 2017.
Strong arithmetic equivalence, AGCT-16, CIRM Luminy (France), June 2017.
Modular curves of prime-power level with infinitely many rational points, Arithmetic Aspects of Explicit Moduli Problems, BIRS Banff (Canada), May 2017.
Sato-Tate in dimension 3, Harvard Number Theory Seminar, Harvard, December 2016.
Torsion subgroups of rational elliptic curves over the compositum of all cubic fields, Explicit Methods in Number Theory (Cremona 60), Warwick University (UK), April 2016.
Modular curves of prime-power level with infinitely many rational points, AMS Southeast Sectional Meeting, University of Georgia, Athens, March 2016.
Sato-Tate distributions, Arizona Winter School, Southwest Center for Arithmetic Geometry, March 2016.
Lecture 1 video; Lecture 2 video; Lecture 3 video; Lecture 4 video.
Sato-Tate distributions, Columbia-CUNY-NYU Number Theory Seminar, CUNY Graduate Center, February 2016.
Sato-Tate distributions of abelian varieties, Abelian Varieties Multi-Site Seminar Series, University of Washington, January 2016.
Computing the image of Galois representations attached to elliptic curves, Computational representation theory in number theory, Oregon State University, July 2015.
Sieve theory and small gaps between primes, Oberwolfach (Germany), July 2015.
Lecture 1 slides; Lecture 2 slides; Lecture 5 slides.
Computing the image of Galois representations attached to elliptic curves, AGCT 2015 (video), CIRM Luminy (France), May 2015.
Computing the image of Galois, Dartmouth Mathematics Colloquium, October 2014.
Sato-Tate groups of abelian varieties of dimension g ≤ 3, Workshop on statistics and number theory, CRM Montreal (Canada), September 2014.
Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, ANTS XI (Korea), August 2014.
Telescopes for mathematicians, Conference on the Impact of Computation in Number Theory, NCTS (Taiwan), July 2014.
The refined Sato-Tate conjecture, CNTA XIII (Ottawa, Canada), June 2014.
The Sato-Tate conjecture for abelian varieties, Heilbronn Seminar, Bristol University (UK), February 2014 (also Yale, Amherst, Boston College March/April 2014).
Counting points on curves in average polynomial time, Frobenius Distributions on Curves, CIRM (France), February 2014.
Winter school on Sato-Tate distributions of curves (joint with Francesc Fité), CIRM Luminy (France), February 2014.
Lecture 1 slides, video; Lecture 2 video; Lecture 3 video; Lecture 4 video; Lecture 5 slides, video; Lecture 6 video.
The Sato-Tate conjecture for abelian varieties, University of Chicago, January 2014.
New bounds on gaps between primes, Brandeis-Harvard-MIT-Northeastern Joint Colloquium, October 2013.
Isogeny volcanoes, Palmetto Number Theory Series (PANTS XX), Davidson College, September 2013.
Computing L-series of low genus curves, SIAM Conference on Applied Algebraic Geometry, Colorado State University, August 2013.
Computing the image of Galois, Workshop on Number Theory, Geometry, and Cryptography, University of Warwick (UK), June 2013.
Sato-Tate distributions, Emory University and CEDAR Workshop at UIC, May 2013.
Computing the image of Galois representations attached to elliptic curves, University of Connecticut and AMS East Sectional Meeting, April 2013.
The generalized Sato-Tate conjecture, Brandeis University, February 2013.
Computing the image of Galois representations attached to elliptic curves, JMM, San Diego, January 2013.
Sato-Tate distributions in genus 2, Princeton/IAS (video) and Quebec/Vermont Number Theory Seminar, November 2012.
On the evaluation of modular polynomials, ECC 2012, Querétaro (Mexico) October 2012.
On the computation and evaluation of modular polynomials, Brown University, September 2012.
On the evaluation of modular polynomials, ANTS X, University of California, San Diego, July 2012.
Isogeny volcanoes: a computational perspective, ANTS X, University of California, San Diego, July 2012.
Computing the image of Galois, CNTA XII (Canada), June 2012.
Computing the modular equation, Barcelona-Boston-Tokyo Number Theory Seminar in Memory of Fumiyuki Momose, May 2012.
Identifying supersingular elliptic curves, JMM, Boston, January 2012.
Sato-Tate distributions in genus 2, Boston University, November 2011.
Telescopes for mathematicians, Computational Research in Boston and Beyond (MIT), September 2011.
Hyperelliptic curves, L-polynomials, and random matrices, MSRI and Emory University, February 2011.
Genus 1 point counting in quadratic space and essentially quartic time, JMM, New Orleans, January 2011.
Genus 1 point counting in quadratic space and essentially quartic time, Columbia-CUNY-NYU, September 2010.
Class invariants by the CRT method, ANTS IX (Nancy, France), July 2010.
A local-global principle for rational isogenies of prime degree, CNTA XI, Acadia University (Canada), July 2010.
L-polynomial distributions of genus 2 curves, Rational Points, ETH (Switzerland), May 2010.
Genus 1 point counting in quadratic space and essentially quartic time, CRM Montreal (Canada), April 2010.
Decomposing class polynomials with the CRT method, CRM Montreal (Canada), April 2010.
Modular polynomials via isogeny volcanoes, CCR Princeton, February 2010.
Computing the image of Galois representations attached to an elliptic curve, Clay Mathematics Institute, December 2009.
Computing modular polynomials with the Chinese Remainder Theorem, ECC 2009 (Calgary, Canada), August 2009.
Powered by volcanoes: Three new algorithms, Fields Institute (Toronto, Canada), May 2009.
Computing the endomorphism ring of an ordinary elliptic curve, CCR La Jolla, April 2009.
Sato-Tate in genus 2, MIT, March 2009.
Computing class polynomials with the Chinese Remainder Theorem, Microsoft Research, Seattle, November 2008.
Computing Hilbert class polynomials with the CRT method, ECC 2008 (Leiden, Netherlands), September 2008.
Computing L-series of hyperelliptic curves, ANTS VIII (BIRS, Canada), May 2008.
Subexponential performance from generic group algorithms, MIT, April 2008.
Thesis defense, MIT, May 2007.
Beating the birthday paradox, MIT, April 2007.
Press (selected)
Elliptic curve "murmurations" found with AI take flight, Quanta, March 5, 2024.
After cracking the sum of cubes puzzle for 42, mathematicians discover a new solution for 3, MIT News, March 11, 2021.
For math fans: A hitchhiker's guide to the number 42., Scientific American, September 21, 2020.
A Ticketmaster for science seminars, MIT News, May 26, 2020.
Online directory makes free science talks easier to find in coronavirus era
, Nature, June 4, 2020.
Why the sum of three cubes is a hard math problem, Quanta Magazine, November 5, 2019.
Mathematicians find a completely new way to write the number 3, New Scientist, September 18, 2019.
Mathematicians crack elusive puzzle involving the number 42, New Scientist, September 6, 2019.
The answer to life, the universe and everything, MIT News, September 10, 2019.
Befreundete Kurven, Der Spiegel, May 14, 2016 (in German).
International team launches vast atlas of mathematical objects, MIT News, May 10, 2016.
The polyface of Polymath, Notices of the American Mathematical Society, June/July 2015.
Together and alone, closing the prime gap, Quanta Magazine, September 19, 2013.
Data
Sums of cubes
Sato-Tate distributions in genus 1.
Sato-Tate distributions in genus 2.
Sato-Tate distributions in genus 3.
genus 2 curves over ℚ.
genus 3 curves over ℚ.
Modular polynomials of all levels up to 400 and prime levels up to 1000 for the j-function
Modular polynomials of prime level up to 5000 for the Weber ƒ function
Modular polynomials of prime level up to 200 for various modular functions used by classpoly
Optimized equations for X1(m,mn) for m ≥ 2 and mn^2 ≤ 120
Optimized equations for X1(N) for N ≤ 100
Alternative defining equations for X1(N) for N ≤ 190
Defining equations for X1(N) in raw form for N ≤ 101
Table of factored norms of singular moduli
Partition class polynomials, as defined in Class polynomials for nonholomorphic modular functions, for n ≤ 750.
Elliptic curve point-counting records
Record CM constructions of elliptic curves
Pairing-friendly Edwards curves of near-prime order with embedding degree 6
Pairing-friendly curves of prime order with embedding degree 6
Pairing-friendly curves of prime order with embedding degree 10
101 useful trace zero varieties
Gallery of large Jacobians
Narrow admissible tuples database (part of the bounded gaps between primes polymath project).
Software
Genus2Euler, as decribed in Computing Euler factors of genus 2 curves over Q at primes of almost good reduction (with Cé'line Maistret).
EndECNF, as decribed in Computing the endomorphism ring of an elliptic curve over a number field (with John Cremona).
zcubes, as described in On a question of Mordell (with Andrew R. Booker).
galrep, as described in Computing images of Galois representations attached to elliptic curves.
rforest, as described in Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial-time (with David Harvey).
classpoly, as described in Computing Hilbert class polynomials with the CRT method
and Class invariants by the CRT method.
smoothrelation, as described in Computing the endomorphism ring of an ordinary elliptic curve over a finite field (with Gaetan Bisson).
smalljac, as described in Computing L-series of hyperelliptic curves.
ffpoly, fast finite field arithmetic over word size prime fields (up to 61 bits), using David Harvey's zn_poly library to more efficiently handle polynomials of large degree.
Acknowledgements
Many of the research products (publications/data/software) listed above were supported by NSF grants DMS-1115455 and DMS-1522526, and Simons Foundation grant 550033.
Computational support has been provided by Google Cloud and Charity Engine.
Collaborative messaging hosted by Zulip have facilitated collaboration within and among many of the projects and conferences I have been involved with over the years.
Accessibility
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genus 2: 
genus 3: