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, focusing on computational number theory.
Here is a fairly recent photograph, and a link to my Google scholar page.
Recent
Admissible sequence of length 5,453 and diameter 56,152 (June 19, 2016). Subject to confirmation of the value k0=5,453,
this implies that there are infintiely many pairs of primes separated by a gap of at most 56,152. See the associated polymath project for details and the latest results.
Source code for generating dense admissible sequences (June 2013).
An introduction to arithmetic geometry (MIT course 18.782), will be offered in the Fall of 2013. See the course web page for details (May 2013).
Upcoming Talks/Workshops
Number Theory, Geometry, and Cryptography, University of Warwick, July 1-5, 2013
Explicit Methods in Number Theory, Oberwolfach, July 14-20, 2013
SIAM Conference on Applied Algebraic Geometry, Colorado State University, August 1-4, 2013
SAC 2013 -- Selected Areas in Cryptography, Simon Fraser University, August 14-16, 2013
Publications (click title for arxiv version, journal name for journal version)
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.
Isogeny volcanoes, Algorithmic Number Theory 10th International Symposium (ANTS X), 2012.
On the evaluation of modular polynomials, Algorithmic Number Theory 10th International Symposium (ANTS X), 2012. (Selfridge Prize)
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), 2012.
Sato-Tate distributions and Galois endomorphism modules in genus 2, with Francesc Fité, Kiran Kedlaya, and Victor Rotger, Compositio Mathematica 148 (2012), 1390-1442.
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.
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.
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, Algorithmic Number Theory 9th International Symposium (ANTS IX), LNCS 6197, Springer, 2010, pp. 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, Arithmetic, Geometry, Cryptography and Coding Theory (AGCT-11, 2007), Contemporary Mathematics 487, AMS, 2009, pp. 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, pp. 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
Class polynomials for nonholomorphic modular functions, with Jan Bruinier and Ken Ono, 2013.
Sato-Tate groups of some weight 3 motives, with Francesc Fité and Kiran S. Kedlaya, 2012.
Sato-Tate distributions of twists of y2=x5-x and y2=x6+1, with Francesc Fité, 2012.
On the distribution of Atkin and Elkies primes, with Igor E. Shparlinski, 2011.
Lecture Notes
Elliptic curves (18.783), lecture notes for an introductory MIT course on elliptic curves, Spring 2013.
Torsion subgroups of elliptic curves over number fields, extended notes from a lecture given at the Harvard Seminar on Mazur's torsion theorem, December 2012.
Talks
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, February 2013
Computing the image of Galois representations attached to elliptic curves, Joint Mathematics Meetings, January 2013
Sato-Tate distributions in genus 2, Princeton/IAS and Quebec/Vermont number theory seminars, 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, June 2012
Computing the modular equation, Barcelona-Boston-Tokyo Number Theory Seminar in Memory of Fumiyuki Momose, May 2012
Identifying supersingular elliptic curves, Joint Mathematics Meetings, 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, 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, Columbia-CUNY-NYU, September 2010
Class invariants by the CRT method, ANTS IX, July 2010
A local-global principle for rational isogenies of prime degree, CNTA XI, July 2010
L-polynomial distributions of genus 2 curves, ETH Zurich, May 2010
Genus 1 point counting in quadratic space and essentially quartic time, CRM Montreal, April 2010
Decomposing class polynomials with the CRT method, CRM Montreal, 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 August 2009
Powered by volcanoes: Three new algorithms, Fields Institute, 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, November 2008
Computing Hilbert class polynomials with the CRT method, ECC 2008, September 2008
Computing L-series of hyperelliptic curves, ANTS VIII, May 2008
Subexponential performance from generic group algorithms, MIT, April 2008
Thesis defense, MIT, spring 2007
Beating the birthday paradox, MIT, spring 2007
Data
Partition class polynomials, as described in Class polynomials for nonholomorphic modular functions
Elliptic curve point-counting records
Table of factored norms of singular moduli
Record CM constructions of elliptic curves
Modular polynomials for the Weber ƒ function
Modular polynomials for various modular functions used by classpoly
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
Defining equations for X1(N) in raw form for N ≤ 101
Alternative defining equations for X1(N) for N ≤ 190
Optimized equations for X1(N) for N ≤ 50
(Updated October 2012 to include optimized equation for X_1(24) derived from the recent result of Jeon, Kim, and Lee)
Sato-Tate distributions in genus 1.
Sato-Tate distributions in genus 2.
Standard L-polynomial coefficient distributions in genus 1, 2, and 3
101 useful trace zero varieties
Gallery of large Jacobians
Software
classpoly_v1.0.1.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_v1.2.tar, as described in Computing the endomorphism ring of an ordinary elliptic curve over a finite field.
smalljac_v4.0.23.tar, as described in Computing L-series of hyperelliptic curves. Requires the ff_poly library.
ff_poly_v1.2.0.tar, fast finite field arithmetic over prime fields of size up to 261.
ff_poly_big_v1.2.0.tar, fast finite field arithmetic over prime fields of size up to 261, uses David Harvey's zn_poly library to more efficiently handle polynomials of large degree.
galrep_v0.tar, as described in Computing the image of Galois representations attached to an elliptic curve, preliminary version.
galrep_ecdata_big.dat, auxiliary elliptic curve data for more extensive galrep computations (approx 104 MB)
galrep_gl2data_big.dat, auxiliary GL(2,Z/ellZ) conjugacy class data for more extensive galrep computations (approx 4 MB)
Acknowledgements
Many of the research products (publications/data/software) listed above were supported all or in part by NSF Grant DSM-1115455.
Genus 3:
