
Andrew V. Sutherland
genus 1:
genus 2:
genus 3:
drew@math.mit.edu

I am a Principal Research Scientist here in the math department at MIT, focused on computational number theory and arithmetic geometry.
Here is a larger photograph and links to my arXiv, MathSciNet, and Google Scholar pages. My office is in room 2341 in the Simons Building (Building 2).
My work is supported by grants from the National Science Foundation and the Simons Foundation.
Latest News
We currently have job openings for Research Scientists to work with us on the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation.
Recent/Upcoming Events
Joint Meetings, San Diego Convention Center, Jan 1013, 2018.
Number Theory Seminar, UC Boulder, Jan 16, 2018.
Birational Geometry and Arithmetic, ICERM, May 1418, 2018.
SQuaRE on Galois representations, AIM, June 48, 2018.
Torsion groups and Galois representations of elliptic curves, University of Zagreb (Croatia), June 2529, 2018.
Building Bridges: 4th EU/US Summer School, Alfred Renyi Institute of Mathematics, Budapest (Hungary), July 913, 2018
ANTS XIII, University of Wisconsin, Madison, July 1620, 2018
Explicit Methods in Number Theory, Oberwolfach (Germany), July 2228, 2018.
Arithmetic of Curves, Baskerville Hall, UK, August 1317, 2018.
Arithmetic Geometry, Number Theory, and Computation, MIT, August 2024, 2018.
Silverberg 2018, University of California, Irvine, September 1721, 2018.
Current Teaching
Number Theory (18.785), Fall 2017.
Current Editorial/Board Positions
Mathematics of Computation, American Mathematical Society, 2014present.
Research in Number Theory, Springer, 2015present.
Number Theory Foundation, 2016present.
Lfunctions and Modular Forms Database, 2016present.
Publications (click title for arxiv version, journal/conference name for published version)
Torsion subgroups of rational elliptic curves over the compositum of all cubic fields, with Harris B. Daniels, Alvaro LozanoRobledo, and Filip Najman, Mathematics of Computation 87 (2018), 425458. (related magma scripts)
Modular curves of primepower level with infinitely many rational points, with David Zywina, Algebra and Number Theory 11 (2017), 11991229. (related magma scripts)
Torsion subgroups of elliptic curves over quintic and sextic number fields, with Maarten Derickx, Proceedings of the AMS, 145 (2017), 42334245. (related GitHub repository)
Finding elliptic curves with a subgroup of prescribed size, with Igor E. Shparlinski, International Journal of Number Theory 13 (2017), 133152.
Computing Lseries of geometrically hyperelliptic curves of genus three, with David Harvey and Maike Massierer, Algorithmic Number Theory 12th International Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 220234.
A database of genus 2 curves over the rational numbers, with Andrew R. Booker, Jeroen Sijsling, John Voight, and Dan Yasaki, Algorithmic Number Theory 12th International Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 235254.
A census of zeta functions of quartic K3 surfaces over F_2, with Kiran S. Kedlaya, Algorithmic Number Theory 12th International Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 111.
Computing images of Galois representations attached to elliptic curves, Forum of Mathematics, Sigma 4 (2016), e4 (79 pages) (related magma scripts).
SatoTate groups of y^{2}=x^{8}+c and y^{2}=x^{7}cx, with Francesc Fité, in Frobenius distributions: LangTrotter and SatoTate conjectures, Contemporary Mathematics 663 (2016), AMS, 103126.
Computing HasseWitt matrices of hyperelliptic curves in average polynomial time, II, with David Harvey, in Frobenius distributions: LangTrotter and SatoTate conjectures, Contemporary Mathematics 663 (2016), AMS, 127148.
SatoTate groups of some weight 3 motives, with Francesc Fité and Kiran S. Kedlaya, in Frobenius distributions: LangTrotter and SatoTate conjectures, Contemporary Mathematics 663 (2016), AMS, 57102.
Class polynomials for nonholomorphic modular functions, with Jan Bruinier and Ken Ono, Journal of Number Theory 161 (2016), 204229.
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), 14611483.
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) 308322.
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), 20672199.
Variants of the Selberg sieve, and bounded intervals containing many primes, D.H.J. Polymath, Research in the Mathematical Sciences 1 (2014).
Computing HasseWitt matrices of hyperelliptic curves in average polynomial time, with David Harvey, Algorithmic Number Theory 11th International Symposium (ANTS XI), LMS Journal of Computation and Mathematics 17 (2014), 257273.
SatoTate distributions of twists of y^{2}=x^{5}x and y^{2}=x^{6}+1, with Francesc Fité, Algebra and Number Theory 8 (2014), 543585.
On the distribution of Atkin and Elkies primes, with Igor E. Shparlinski, Foundations of Computational Mathematics 14 (2014), 285297.
Isogeny volcanoes, Algorithmic Number Theory 10th International Symposium (ANTS X), MSP Open Book Series 1 (2013), 507530.
On the evaluation of modular polynomials, Algorithmic Number Theory 10th International Symposium (ANTS X), MSP Open Book Series 1 (2013), 531555. (Selfridge Prize) (errata).
Deterministic elliptic curve primality proving for a special sequence of numbers, with Alexander Abatzoglou, Alice Silverberg, and Angela Wong, Algorithmic Number Theory 10th International Symposium (ANTS X), MSP Open Book Series 1 (2013), 120.
Identifying supersingular elliptic curves, LMS Journal of Computation and Mathematics 15 (2012), 317325.
Accelerating the CM method, LMS Journal of Computation and Mathematics 15 (2012), 172204.
SatoTate distributions and Galois endomorphism modules in genus 2, with Francesc Fité, Kiran S. Kedlaya, and Victor Rotger, Compositio Mathematica 148 (2012), 13901442 (errata).
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), 12351270.
A localglobal principle for rational isogenies of prime degree, Journal de Théorie des Nombres de Bordeaux 24 (2012), 475485.
A lowmemory algorithm for finding short product representations in finite groups, with Gaetan Bisson, Designs, Codes, and Cryptography 63 (2012), 113.
Constructing elliptic curves over finite fields with prescribed torsion, Mathematics of Computation 81 (2012), 11311147.
Modular polynomials via isogeny volcanoes, with Reinier Bröker and Kristin Lauter, Mathematics of Computation 81 (2012), 12011231.
Computing Hilbert class polynomials with the Chinese Remainder Theorem, Mathematics of Computation 80 (2011), 501538.
Structure computation and discrete logarithms in finite abelian pgroups, Mathematics of Computation 80 (2011), 477500. (related magma script)
Computing the endomorphism ring of an ordinary elliptic curve over a finite field, with Gaetan Bisson, Journal of Number Theory, 113 (2011), 815831.
Class invariants by the CRT method, with Andreas Enge, Algorithmic Number Theory 9th International Symposium (ANTS IX), LNCS 6197, Springer, 2010, 142156.
An explicit height bound for the classical modular polynomial, with Reinier Bröker, Ramanujan Journal 22 (2010), 293313.
On a theorem of Mestre and Schoof, with John E. Cremona, Journal de Théorie des Nombres de Bordeaux 22 (2010), 353358.
Hyperelliptic curves, Lpolynomials, and random matrices,
with Kiran S. Kedlaya, in Arithmetic, Geometry, Cryptography and Coding Theory (AGCT11, 2007), Contemporary Mathematics 487, AMS, 2009, 119162.
A generic approach to searching for Jacobians, Mathematics of Computation 78 (2009), 485507.
Computing Lseries of hyperelliptic curves,
with Kiran S. Kedlaya, Algorithmic Number Theory 8th International Symposium (ANTS VIII), LNCS 5011, Springer, 2008, 312326.
Order computations in generic groups, PhD thesis, Massachusetts Institute of Technology, 2007 (errata).
(George M. Sprowls Award for Outstanding Thesis in Computer Science)
Preprints
SatoTate distributions of twists of the Fermat and the Klein quartics, with Francesc Fite and Elisa Lorenzo Garcia, 2017 (related magma scripts).
Maps between curves and arithmetic obstructions, with Felipe Voloch, 2017.
Fast Jacobian arithmetic for hyperelliptic curves of genus 3, 2016. (related magma script)
SatoTate Distributions, 2016.
Lecture Notes
Elliptic Curves (18.783), Spring 2017 (OCW link).
Number Theory (18.785), Fall 2016 (OCW link ).
Primes is in P (18.095 lecture), January 2017.
Perfect Forward Secrecy (18.095 lecture), January 2016.
Arithmetic geometry(18.782), lecture notes for an introductory course on arithmetic geometry, 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.
Talks (selected)
Computing Lfunctions 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, FrankenAkademie Schloss Schney (Germany), July 2017
Computing Lseries of genus 3 curves, Workshop on Arithmetic Geometry and Computer Algebra, University of Oldenburg (Germany), June 2017
Strong arithmetic equivalence, AGCT16, CIRM Luminy (France), June 2017
Modular curves of primepower level with infinitely many rational points, Arithmetic Aspects of Explicit Moduli Problems, BIRS Banff (Canada), May 2017
SatoTate 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: Conference in Honour of John Cremona's 60th Birthday, Warwick University (UK), April 2016
Modular curves of primepower level with infinitely many rational points, AMS Southeast Sectional Meeting, University of Georgia, Athens, March 2016
SatoTate distributions, ColumbiaCUNYNYU Number Theory Seminar, CUNY Graduate Center, February 2016
SatoTate distributions of abelian varieties, Abelian Varieties MultiSite 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: Introduction (talk 1), A variational problem (talk 4), Narrow admissible tuples (talk 5), Oberwolfach (Germany), July 2015
Computing the image of Galois representations attached to elliptic curves, AGCT15, CIRM Luminy (France), May 2015
Computing the image of Galois, Dartmouth Mathematics Colloquium, October 2014
SatoTate groups of abelian varieties of dimension g ≤ 3, Workshop on statistics and number theory, CRM Montreal (Canada), September 2014
Computing HasseWitt 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 SatoTate conjecture, CNTA XIII (Canada), June 2014
The SatoTate conjecture for abelian varieties, Heilbronn Seminar, Bristol University (UK), February 2014 (also at Yale, Amherst, and Boston College March/April 2014)
Counting points on curves in average polynomial time, Frobenius Distributions on Curves, CIRM (France), February 2014
The SatoTate conjecture for abelian varieties, University of Chicago, January 2014
New bounds on gaps between primes, BrandeisHarvardMITNortheastern Joint Colloquium, October 2013
Isogeny volcanoes, Palmetto Number Theory Series (PANTS XX), September 2013
Computing Lseries of low genus curves, SIAM Conference on Applied Algebraic Geometry, August 2013
Computing the image of Galois, Workshop on Number Theory, Geometry, and Cryptography, University of Warwick (UK), June 2013
SatoTate 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 SatoTate conjecture, Brandeis, February 2013
Computing the image of Galois representations attached to elliptic curves, Joint Mathematics Meetings, January 2013
SatoTate distributions in genus 2, Princeton/IAS (video) and Quebec/Vermont Number Theory Seminar, November 2012
On the evaluation of modular polynomials, ECC 2012 October 2012
On the computation and evaluation of modular polynomials, Brown University, September 2012
On the evaluation of modular polynomials, ANTS X, July 2012
Isogeny volcanoes: a computational perspective, ANTS X, July 2012
Computing the image of Galois, CNTA XII (Canada), June 2012
Computing the modular equation, BarcelonaBostonTokyo Number Theory Seminar in Memory of Fumiyuki Momose, May 2012
Identifying supersingular elliptic curves, Joint Mathematics Meetings, January 2012
SatoTate distributions in genus 2, Boston University, November 2011
Telescopes for mathematicians, Computational Research in Boston and Beyond (MIT), September 2011
Hyperelliptic curves, Lpolynomials, and random matrices, MSRI and Emory University, February 2011
Genus 1 point counting in quadratic space and essentially quartic time, Joint Mathematics Meetings, January 2011
Genus 1 point counting in quadratic space and essentially quartic time, ColumbiaCUNYNYU, September 2010
Class invariants by the CRT method, ANTS IX (France), July 2010
A localglobal principle for rational isogenies of prime degree, CNTA XI (Canada), July 2010
Lpolynomial 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 (Canada), August 2009
Powered by volcanoes: Three new algorithms, Fields Institute (Canada), May 2009
Computing the endomorphism ring of an ordinary elliptic curve, CCR La Jolla, April 2009
SatoTate in genus 2, MIT, March 2009
Computing class polynomials with the Chinese Remainder Theorem, Microsoft Research, November 2008
Computing Hilbert class polynomials with the CRT method, ECC 2008 (Netherlands), September 2008
Computing Lseries of hyperelliptic curves, ANTS VIII (Canada), May 2008
Subexponential performance from generic group algorithms, MIT, April 2008
Thesis defense, MIT, May 2007
Beating the birthday paradox, MIT, April 2007
Data
Narrow admissible tuples database, part of the bounded gaps between primes polymath project.
Partition class polynomials, as described in Class polynomials for nonholomorphic modular functions
Elliptic curve pointcounting records
Table of factored norms of singular moduli
Record CM constructions of elliptic curves
Modular polynomials of all levels up to 300 for the jfunction
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
Pairingfriendly Edwards curves of nearprime order with embedding degree 6
Pairingfriendly curves of prime order with embedding degree 6
Pairingfriendly curves of prime order with embedding degree 10
Optimized equations for X_{1}(m,mn) for m ≥ 2 and mn^2 ≤ 120
Optimized equations for X_{1}(N) for N ≤ 100
(Updated and extended February 2014 based on joint work with Mark van Hoeij))
Alternative defining equations for X_{1}(N) for N ≤ 190
Defining equations for X_{1}(N) in raw form for N ≤ 101
SatoTate distributions in genus 1.
SatoTate distributions in genus 2.
SatoTate distributions in genus 3 (a few examples).
101 useful trace zero varieties
Gallery of large Jacobians
Software
classpoly_v1.0.2.tar, as described in Computing Hilbert class polynomials with the CRT method
and Class invariants by the CRT method. Requires the ff_poly_big library.
smoothrelation_v1.3.tar, as described in Computing the endomorphism ring of an ordinary elliptic curve over a finite field.
smalljac_v4.1.3.tar, as described in Computing Lseries of hyperelliptic curves. Requires the ff_poly library.
ff_poly_v1.2.7.tar, fast finite field arithmetic over word size prime fields (up to 61 bits).
ff_poly_big_v1.2.7.tar, fast finite field arithmetic over word size prime fields (up to 61 bits), uses 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 DMS1115455 and DMS1522526.