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 2006.
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, ed.), 196-211. arXiv:0802.1301v1 [math.NT].
* Except for ``boxed fillers'' or problem proposals and solutions in the American Math. Monthly etc.
rest of mathematical Curriculum Vitae
back to Mathematics page