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 2-341 in the Simons Building (Building 2). For fall 2019 my office hours are M 2:30-3:30 and T 1:30-2:30.

My work is supported by grants from the National Science Foundation and the Simons Foundation; I am part of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation.

Latest News
     569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 (joint with Andrew Booker).
     (-385495523231271884)^3 + 383344975542639445^3 + 98422560467622814^3 (joint with Andrew Booker).
     (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3 (joint with Andrew Booker).
     (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 (joint with Andrew Booker).

Recent/Upcoming Events
    Computational Mathematics Colloquium, Universitry of Waterloo, Waterloo, Canada, November 7, 2019.
    Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting, Simons Foundation, New York City, January 9-10, 2020.
    JMM Special Session on Rational Points: Theory and Computation, Denver, Colorado, January 15-16, 2020.
    JMM Special Session on Algorithms, Experimentation, and Applications in Number Theory, Denver, Colorado, January 16, 2020.
    British Mathematical Colloquium (BMC), University of Glasgow (Scotland), April 6-9, 2020.
    Arithmetic, Algebra, and Algorithms -- Celebrating the mathematics of Hendrik Lenstra, ICMS, Edinburgh (Scotland), April 13-17, 2020.
    Arithmetic Geometry, Number Theory, and Computation III, ICERM, Providence, Rhode Island, June 1-5, 2020.
    FOCM 2020, Simon Fraser University, Vancouver (Canada), June 15-24, 2020.
    ANTS-XIV, University of Auckland, Auckland (New Zealand), June 30 - July 4, 2020.
    Number Theory Informed by Computation, PCMI, Park City, Utah, July 5-25, 2020.
    Modern Breakthroughs in Diophantine Problems, BIRS, Banff (Canada), August 30 - September 4, 2020.

Current Teaching
    Number Theory I (18.785), Fall 2019

Current Editorial/Board Positions
    Mathematics of Computation (associate editor), American Mathematical Society, 2014-present.
    Research in Number Theory (editor in chief), Springer, 2015-present.
    Number Theory Foundation (board member), 2016-present.
    L-functions and Modular Forms Database (managing editor), 2016-present.
    CoCalc (advisory board member), 2018-present.

Publications (click title for arxiv version, journal/conference name for published version)
    Maps between curves and arithmetic obstructions, with Felipe Voloch, 2017, in Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics 722, AMS, 2019, 167--175.
    A database of nonhyperelliptic genus 3 curves over Q, Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), Open Book Series 2 (2019), 443-459. (related data and magma scripts)
    Fast Jacobian arithmetic for hyperelliptic curves of genus 3, Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), Open Book Series 2 (2019), 425-442. (related magma script)
    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. (related magma scripts) (errata).
    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. (related magma scripts)
    Modular curves of prime-power level with infinitely many rational points, with David Zywina, Algebra and Number Theory 11 (2017), 1199-1229. (related magma scripts)
    Torsion subgroups of elliptic curves over quintic and sextic number fields, with Maarten Derickx, Proceedings of the AMS, 145 (2017), 4233-4245. (related GitHub repository)
    Finding elliptic curves with a subgroup of prescribed size, with Igor E. Shparlinski, International Journal of Number Theory 13 (2017), 133-152.
    Computing L-series of geometrically hyperelliptic curves of genus three, with David Harvey and Maike Massierer, Twelth Algorithmic Number Theory Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 220-234.
    A database of genus 2 curves over the rational numbers, with Andrew R. Booker, Jeroen Sijsling, John Voight, and Dan Yasaki, Twelth Algorithmic Number Theory Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 235-254.
    A census of zeta functions of quartic K3 surfaces over F2, with Kiran S. Kedlaya, Twelth Algorithmic Number Theory Symposium (ANTS XII), LMS Journal of Computation and Mathematics 19 (2016), 1-11.
    Computing images of Galois representations attached to elliptic curves, Forum of Mathematics, Sigma 4 (2016), e4, 79 pages. (related magma scripts).
    Sato-Tate groups of y2=x8+c and y2=x7-cx, with Francesc Fité, in Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics 663 (2016), AMS, 103-126.
    Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II, with David Harvey, in Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics 663 (2016), AMS, 127-148.
    Sato-Tate groups of some weight 3 motives, with Francesc Fité and Kiran S. Kedlaya, in Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics 663 (2016), AMS, 57-102.
    Class polynomials for nonholomorphic modular functions, with Jan Bruinier and Ken Ono, Journal of Number Theory 161 (2016), 204-229.
    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.
    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.
    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.
    Variants of the Selberg sieve, and bounded intervals containing many primes, D.H.J. Polymath, Research in the Mathematical Sciences 1 (2014).
    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.
    Sato-Tate distributions of twists of y2=x5-x and y2=x6+1, with Francesc Fité, Algebra and Number Theory 8 (2014), 543-585.
    On the distribution of Atkin and Elkies primes, with Igor E. Shparlinski, Foundations of Computational Mathematics 14 (2014), 285-297.
    Isogeny volcanoes, Tenth Algorithmic Number Theory Symposium (ANTS X), MSP Open Book Series 1 (2013), 507-530.
    On the evaluation of modular polynomials, Tenth Algorithmic Number Theory Symposium (ANTS X), MSP Open Book Series 1 (2013), 531-555. (Selfridge Prize) (errata).
    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), MSP Open Book Series 1 (2013), 1-20.
    Identifying supersingular elliptic curves, LMS Journal of Computation and Mathematics 15 (2012), 317-325.
    Accelerating the CM method, LMS Journal of Computation and Mathematics 15 (2012), 172-204.
    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).
    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.
    A local-global principle for rational isogenies of prime degree, Journal de Théorie des Nombres de Bordeaux 24 (2012), 475-485 (errata).
    A low-memory algorithm for finding short product representations in finite groups, with Gaetan Bisson, Designs, Codes, and Cryptography 63 (2012), 1-13.
    Constructing elliptic curves over finite fields with prescribed torsion, Mathematics of Computation 81 (2012), 1131-1147.
    Modular polynomials via isogeny volcanoes, with Reinier Bröker and Kristin Lauter, Mathematics of Computation 81 (2012), 1201-1231.
    Computing Hilbert class polynomials with the Chinese Remainder Theorem, Mathematics of Computation 80 (2011), 501-538.
    Structure computation and discrete logarithms in finite abelian p-groups, Mathematics of Computation 80 (2011), 477-500. (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), 815-831.
    Class invariants by the CRT method, with Andreas Enge, Ninth Algorithmic Number Theory Symposium (ANTS IX), LNCS 6197, Springer, 2010, 142-156.
    An explicit height bound for the classical modular polynomial, with Reinier Bröker, Ramanujan Journal 22 (2010), 293-313.
    On a theorem of Mestre and Schoof, with John E. Cremona, Journal de Théorie des Nombres de Bordeaux 22 (2010), 353-358.
    Hyperelliptic curves, L-polynomials, and random matrices, with Kiran S. Kedlaya, in Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics 487, AMS, 2009, 119-162.
    A generic approach to searching for Jacobians, Mathematics of Computation 78 (2009), 485-507.
    Computing L-series of hyperelliptic curves, with Kiran S. Kedlaya, Algorithmic Number Theory 8th International Symposium (ANTS VIII), LNCS 5011, Springer, 2008, 312-326.
    Order computations in generic groups, PhD thesis, Massachusetts Institute of Technology, 2007 (errata). (George M. Sprowls Award for Outstanding Thesis in Computer Science)

Preprints
    Sato-Tate groups of abelian threefolds: a preview of the classification, Francesc Fité and Kiran Kedlaya, 2019.
    Arithmetic invariants from Sato-Tate moments, with Edgar Costa and Francesc Fité, 2019.
    Sato-Tate Distributions, 2016.

Lecture Notes
    Elliptic Curves (18.783), Spring 2019. (OCW link).
    Number Theory I (18.785), Fall 2018 (OCW link 2017).
    Arithmetic Equvialence and Isospectrality, minicourse in Topics in Algebra (18.708), Spring 2018.
    Number Theory II (18.786), Spring 2018.
    Primes is in P (18.095 lecture), January 2017.
    Perfect Forward Secrecy (18.095 lecture), January 2016.
    Arithmetic geometry (18.782), (an introductory course for undergraduates), 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)
    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, Lecture 2, Lecture 3, Lecture 4).
    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.
    Winter school on Sato-Tate distributions, Lecture 1 (video), Lecture 2 (video), Lecture 3 (video), Lecture 4 (video) Arizona Winter School, Southwest Center for Arithmetic Geometry, March 2016.
    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: 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, 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 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.
    Winter school on Sato-Tate distributions of curves (joint with Francesc Fité), Lecture 1 (slides, video), Lecture 2 (video), Lecture 3 (video), Lecture 4 (video), Lecture 5 (slides, video), Lecture 6 (video), CIRM Luminy (France), February 2014.
    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.

Data
    Sato-Tate distributions in genus 1.
    Sato-Tate distributions in genus 2.
    Sato-Tate distributions in genus 3 (a few examples).

    genus 2 curves over of small discriminant over Q (these curves all appear in the LMFDB).
    genus 3 hyperelliptic curves of small discriminant over Q (format is D:N:[f(x),h(x)], for hyperelliptic curve y2 + h(x)y=f(x) with absolute discriminant D and conductor N).
    genus 3 non-hyperelliptic curves of small discriminant over Q.

    Modular polynomials of all levels up to 300 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 described in Class polynomials for nonholomorphic modular functions

    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
    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 L-series 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 DMS-1115455 and DMS-1522526, and Simons Foundation grant 550033. Computational support has been provided by Google and Charity Engine.