Noam D. Elkies

Department of Mathematics, Harvard University, Cambridge, MA 02138
office: (617)495-4625;  fax: (617)495-5132
e-mail: elkies@math.harvard.edu
mathematical PUBLICATIONS
Complete* through 2009.
Also, see here for all my arXiv articles, including some not yet published and thus not listed below.
  1. Integers expressible in the form a4 + b4, pages 22-28 in Mathematical Buds Vol.3 (H.Ruderman, ed.; Norman, Oklahoma: Mu Alpha Theta, 1984).
  2. An improved lower bound on the greatest element of a sum-distinct set of fixed order, Jour. Comb. Theory A 41 (Jan. 1986), 89-94.
  3. The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), 561-568.
  4. On A4 + B4 + C4 = D4, Math. of Comp. 51 (Oct. 1988), 825-835.
  5. Supersingular primes for elliptic curves over real number fields, Compositio Math. 72 (1989), 165-172.
  6. The automorphism group of the modular curve X0(63), Compositio Math. 74 (1990), 203-208.
  7. Distribution of supersingular primes, Astérisque 198-199-200 (1991; proceedings of Journées Arithmétiques 1989), 127-132.
  8. On the Hurwitz scheme and its monodromy (with D. Eisenbud, J. Harris, and R. Speiser), Compositio Math. 77 (1991), 95-117.
  9. On the packing densities of superballs and other bodies (with A.M. Odlyzko and J.A. Rush), Invent. Math. 105 (1991), 613-639.
  10. ABC implies Mordell, International Math. Research Notices 1991 #7, 99-109 [bound with Duke Math. J. 64 (1991)].
  11. Alternating sign matrices and domino tilings I, II (with G. Kuperberg, M. Larsen, and J. Propp), Journal of Algebraic Combinatorics 1 (1992), 111-132 and 219-234. math.CO/9201305 on the arXiv.
  12. Mordell-Weil lattices in characteristic 2:
    I. Construction and first properties, International Math. Research Notices 1994 #8, 343-361;
    II. The Leech lattice as a Mordell-Weil lattice, Invent. Math. 128 (1997), 1-8;
    III. A Mordell-Weil lattice of rank 128, Experimental Math. 10 (2001) #3, 467-473.
  13. Wiles minus epsilon implies Fermat, pages 38-40 in Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (J.Coates and S.-T.Yau, eds.; Boston: International Press, 1995; proceedings of the 12/93 conference on elliptic curves and modular forms at the Chinese University of Hong Kong).
  14. Heegner point computations, Lecture Notes in Computer Science 877 (proceedings of ANTS-1, 5/94; L.M. Adleman and M.-D. Huang, eds.), 122-133.
  15. On numbers and endgames, pages 135-150 in Games of No Chance (R.J.Nowakowski, ed.; MSRI Publ. #29, 1996 via Cambridge Univ. Press; proceedings of the 7/94 MSRI conference on combinatorial games). math.CO/9905198 on the arXiv.
  16. A characterization of the Zn lattice, Math. Research Letters 2 (1995), 321-326 (math.NT/9906019 on the arXiv).
  17. Lattices and codes with long shadows, Math. Research Letters 2 (1995), 643-651 (math.NT/9906086 on the arXiv).
  18. Local statistics for random domino tilings of the Aztec diamond (with H. Cohn and J. Propp), Duke Math. J. 85 #1 (Oct. 1996), 117-166.
  19. The exceptional cone and the Leech lattice (with B.H. Gross), International Math. Research Notices 1996 #14, 665-698.
  20. Elliptic and modular curves over finite fields and related computational issues, pages 21-76 in Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin (D.A. Buell and J.T. Teitelbaum, eds.; AMS/International Press, 1998).
  21. Explicit modular towers, pages 23-32 in Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (1997, T. Basar, A. Vardy, eds.), Univ. of Illinois at Urbana-Champaign 1998 (math.NT/0103107 on the arXiv).
  22. Embeddings into the Integral Octonions (with B.H. Gross), Pacific J. Math., Dec. 1997 (Olga Taussky-Todd Memorial Issue), 147-158.
  23. Shimura curve computations, Lecture Notes in Computer Science 1423 (proceedings of ANTS-3, 1998; J.P.Buhler, ed.), 1-47 (math.NT/0005160 on the arXiv). corrigendum; further corrections mostly by David Jao (to be implemented soon)
  24. The still-Life density problem and its generalizations, pages 228-253 in Voronoï’s Impact on Modern Science, Book I (P. Engel, H. Syta, eds.; Institute of Math., Kyiv 1998 = Vol.21 of Proc. Inst. Math. Nat. Acad. Sci. Ukraine). math.CO/9905194 on the arXiv.
  25. Linearized algebra and finite groups of Lie type. I: Linear and symplectic groups, pages 77-107 in Applications of curves over finite fields (Seattle, 1997) = Contemp. Math. 245, Providence: AMS, 1999.
  26. The Klein quartic in number theory, pages 51-102 in The Eightfold Way: The Beauty of Klein’s Quartic Curve (S.Levy, ed.; Cambridge Univ. Press, 1999; also on-line at the MSRI Publications site)
  27. Rational points near curves and small nonzero |x3-y2| via lattice reduction, Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63 (math.NT/0005139 on the arXiv).
  28. Explicit towers of Drinfeld modular curves, Progress in Mathematics 202 (2001), 189-198 (Proceedings of the 3rd European Congress of Mathematics, Barcelona, 7/2000: paper presented at the mini-symposium on “curves over finite fields and codes”; math.NT/0005140 on the arXiv).
  29. Lattices, Linear Codes, and Invariants (2-part expository article), Notices of the American Math. Society 47 (2000), 1238-1245 and 1382-1391.
  30. Cubic rings and the exceptional Jordan algebra (with B.H. Gross), Duke Math. J. 109 #2 (2001), 383-409.
  31. Excellent nonlinear codes from modular curves, pages 200-208 in STOC‘01: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Hersonissos, Crete, Greece. Isomorphic with math.NT/0104115 on the arXiv.
  32. On finite sequences satisfying linear recursions, New York J. Math. 8 (2002), 85-97 = http://nyjm.albany.edu:8000/j/2002/8-5.html (math.CO/0105007 on the arXiv).
  33. Curves Dy2=x3-x of Odd Analytic Rank, Lecture Notes in Computer Science 2369 (proceedings of ANTS-5, 2002; C.Fieker and D.R.Kohel, eds.), 244-251. math.NT/0208056 on the arXiv.
  34. Trinomials ax7+bx+c and ax8+bx+c with Galois Groups of Order 168 and 8*168 (with Nils Bruin), Lecture Notes in Computer Science 2369 (proceedings of ANTS-5, 2002; C.Fieker and D.R.Kohel, eds.), 172-188.
  35. Appendix to "New Optimal Tame Towers of Function Fields over Small Finite Fields" by Wen-Ching W. Li, Hiren Maharaj, and Henning Stichtenoth [identifying each of their four towers with a tower of classical modular curves], Lecture Notes in Computer Science 2369 (proceedings of ANTS-5, 2002; C.Fieker and D.R.Kohel, eds.), 384-389.
  36. Higher Nimbers in pawn endgames on large chessboards, pages 61-78 in More Games of No Chance (R.J.Nowakowski, ed.; MSRI Publ. #42, 2002 via Cambridge Univ. Press; proceedings of the 7/00 MSRI workshop on combinatorial games). math.CO/0011253 on the arXiv.
  37. The mathematical knight (with Richard Stanley), Math. Intelligencer 25 #1 (2003), 22-34. [Diagram 11 is misprinted: it should have White and Black Kings on d1 and h8 respectively, White Knight on b2, and Black Pawn on a3.]
  38. New upper bounds on sphere packings I (with H. Cohn), Annals of Math. 157 (2003), 689-714 (math.MG/0110009 on the arXiv).
  39. On the Sums ,   Amer. Math. Monthly 110 #7 (Aug.-Sep. 2003), 561-573. Nearly isomorphic with math.CA/0101168 on the arXiv. Corrigenda: Amer. Math. Monthly 111 #5 (May 2004), 456.
  40. On Elliptic K-curves, Progress in Mathematics 224 (2004), 81-91 (Proceedings of the 7/2002 Barcelona Euroconference on “Modular Curves and Abelian Varieties”, ed. J.Cremona, J.-C.Lario, J.Quer, and K.Ribet).
  41. Curves of every genus with many points, II: Asymptotically good families (with E.W.Howe, A.Kresch, B.Poonen, J.L.Wetherell, and M.E.Zieve), Duke Math. J. 122 #2 (2004), 399-422 (math.NT/0208060 on the arXiv).
  42. Elliptic Curves of Large Rank and Small Conductor (with M.Watkins), Lecture Notes in Computer Science 3076 (proceedings of ANTS-6, 2004; D.Buell, ed.), 42-56. math.NT/0403374 on the arXiv.
  43. Elliptic Curves x3 + y3 = k of High Rank (with N.F.Rogers), Lecture Notes in Computer Science 3076 (proceedings of ANTS-6, 2004; D.Buell, ed.), 184-193. math.NT/0403116 on the arXiv.
  44. Gaps in Sqrt(n) mod 1 and ergodic theory (with C.T.McMullen), Duke Math. J. 123 #1 (2004), 95-139.
  45. The conjugate dimension of algebraic numbers (with N.Berry, A.Dubickas, B.Poonen, and C.Smyth), Quart. J. Math. 55 (2004), 237-252 (math.NT/0308069 on the arXiv).
  46. New Directions in Enumerative Chess Problems, Electronic J. of Combinatorics 11(2) (2004-2005) [Stanley-60 Festschrift], Article #4 (math.CO/0508645 on the arXiv).
  47. Reduction of CM Elliptic Curves and Modular Function Congruences (with K.Ono and T.Yang), International Math. Research Notices 2005 #44, 2695-2707 (math.NT/0512350 on the arXiv).
  48. Sylvester-Gallai Theorems for Complex Numbers and Quaternions (with Lou M. Pretorius and Konrad J. Swanepoel), Discrete and Computational Geometry 35 #3 (3/2006), 361-373 (math.MG/0403023 on the arXiv).
  49. The Mathieu group M12 and its pseudogroup extension M13 (with John H. Conway and Jeremy L. Martin), Experimental Math. 15 (2006) #2, 223-236 (math.GR/0508630 on the arXiv).
  50. Points of Low Height on Elliptic Curves and Surfaces I: Elliptic surfaces over P1 with small d, Lecture Notes in Computer Science 4076 (proceedings of ANTS-7, 2006; F.Hess, S.Pauli, and M.Pohst, ed.), 287-301. math.AG/0608593 on the arXiv.
  51. Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group, and Some Other Examples, Lecture Notes in Computer Science 4076 (proceedings of ANTS-7, 2006; F.Hess, S.Pauli, and M.Pohst, ed.), 302-316. math.NT/0409020 on the arXiv.
  52. On some points-and-lines problems and configurations, Periodica Mathematica Hungarica 53 #1-2 (2006), 133-148. math.MG/0612749 on the arXiv.
  53. The D4 Root System Is Not Universally Optimal (with Henry Cohn, John H. Conway, and Abhinav Kumar), Experimental Math. 16 (2006) #3, 313-320 (math.NT/0607447 on the arXiv).
  54. Shimura Curve Computations Via K3 Surfaces of Néron-Severi Rank at Least 19, Lecture Notes in Computer Science 5011 (proceedings of ANTS-8, 2008; A.J.van der Poorten and A.Stein, eds.), 196-211. arXiv:0802.1301v1 [math.NT].
  55. About the cover: Rational curves on a K3 surface, pages 1-4 of Arithmetic Geometry: Proceedings of the Clay Mathematics Institute, Göttingen, 17 July – 11 August, 2006 (Henri Darmon, David Alexandre Ellwood, Brendan Hassett, and Yuri Tschinkel, eds.), Clay Math. Proceedings 8, 2009.
  56. Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80 (with Scott Duke Kominers), Proceedings of the American Math. Society 138 #1 (2010), 105-108. arXiv:0905.4306v1 [math.NT].
  57. On the Classification of Type II Codes of Length 24 (with Scott Duke Kominers), SIAM J. Discrete Math. 23 #4 (2010), 2173-2177. arXiv:0902.1942v2 [math.NT].
  58. Point configurations that are asymmetric yet balanced (with Henry Cohn, Abhinav Kumar, and Achill Schürmann), Proceedings of the American Math. Society, posted on March 23, 2010, PII S 0002-9939(10)10284-6; 138 #8 (August 2010), 2863-2872. arXiv:0812.2579v2 [math.MG].
  59. Weighted Generating Functions for Type II Lattices and Codes (with Scott Duke Kominers), pages 63-108 in Quadratic and Higher Degree Forms (Krishnaswami Alladi, Manjul Bhargava, David Savitt, and Pham Huu Tiep, eds.), Developments in Mathematics 31, 2013 (New York: Springer). arXiv:1111.2392
  60. Minimal S-universality criteria may vary in size (with Daniel M. Kane and Scott Duke Kominers), J. de Théorie de Nombres de Bordeaux 25 #3 (2013), 557-564. arXiv:1101.5662
  61. Modular forms and K3 surfaces (with Matthias Schütt), Advances in Math. 240 (2013), 106-131. arXiv:0809.0830 (2008, revised 2013).
  62. Genus bounds for curves with fixed Frobenius eigenvalues (with Everett W. Howe and Christophe Ritzenthaler), Proc. Amer. Math. Soc. 142 (2014), 71-84. arXiv:1006.0822 (2010, revised 2012; officially posted to the journal site 18 September 2013).
  63. K3 surfaces and equations for Hilbert modular surfaces (with Abhinav Kumar), Algebra and Number Theory 8:10 (2014), 2297-2411. (DOI: 10.2140/ant.2014.8.2297; arXiv: 1209.3527)
  64. Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, and Nicholas F. Rogers: Ranks of quadratic twists of elliptic curves, Publ. Math. Besançon 2014/2, 63-98. Copy on Andrew Granville’s page
  65. Genus 1 Fibrations on the Supersingular K3 Surface in Characteristic 2 with Artin Invariant 1 (with Matthias Schütt), Asian J. Math. 19 #3 (2015), 555-581. arXiv:1207.1239
  66. “Scrambling” geo-referenced data to protect privacy induces bias in distance estimation (with Günther Fink, and Till Bärnighausen), Population and Environment 37 #1 (Sep. 2015), 83-98.
  67. Permutations that Destroy Arithmetic Progressions in Elementary p-Groups (with Ashvin Swaminathan), Electronic J. of Combinatorics 24 #1 (2017), Paper #P1.20 (1601.07541 [math.NT] on the arXiv).
  68. Crossing numbers of complete graphs, pages 218--249 in The Mathematics of Various Entertaining Subjects, Volume 2: Research in Games, Graphs, Counting, and Complexity (Jennifer Beineke and Jason Rosenhouse, eds.), Princeton University Press 2017.
* Except for “boxed fillers” or problem proposals and solutions in the American Math. Monthly etc.
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