Noam D. Elkies
Department of Mathematics, Harvard University, Cambridge, MA 02138
office: (617)4954625; fax: (617)4955132
email:
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.

Integers expressible in the form a^{4} + b^{4},
pages 2228 in Mathematical Buds Vol.3 (H.Ruderman, ed.;
Norman, Oklahoma: Mu Alpha Theta, 1984).

An improved lower bound on the greatest element of a sumdistinct
set of fixed order,
Jour. Comb. Theory A 41 (Jan. 1986), 8994.

The existence of infinitely many supersingular primes for every
elliptic curve over Q,
Invent. Math. 89 (1987), 561568.

On A^{4} + B^{4} + C^{4} = D^{4},
Math. of Comp. 51 (Oct. 1988), 825835.

Supersingular primes for elliptic curves over real number fields,
Compositio Math. 72 (1989), 165172.

The automorphism group of the modular curve X_{0}(63),
Compositio Math. 74 (1990), 203208.

Distribution of supersingular primes,
Astérisque 198199200
(1991; proceedings of Journées Arithmétiques 1989),
127132.

On the Hurwitz scheme and its monodromy
(with D. Eisenbud, J. Harris, and R. Speiser),
Compositio Math. 77 (1991), 95117.

On the packing densities of superballs and other bodies
(with A.M. Odlyzko and J.A. Rush),
Invent. Math. 105 (1991), 613639.

ABC implies Mordell,
International Math. Research Notices 1991 #7, 99109
[bound with Duke Math. J. 64 (1991)].

Alternating sign matrices and domino tilings I, II
(with G. Kuperberg, M. Larsen, and J. Propp),
Journal of Algebraic Combinatorics 1 (1992),
111132 and 219234.
math.CO/9201305
on the arXiv.

MordellWeil lattices in characteristic 2:
I. Construction and first properties,
International Math. Research Notices 1994 #8, 343361;
II. The Leech lattice as a MordellWeil lattice,
Invent. Math. 128 (1997), 18;
III. A MordellWeil lattice of rank 128,
Experimental Math. 10 (2001) #3, 467473.

Wiles minus epsilon implies Fermat,
pages 3840 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).

Heegner point computations,
Lecture Notes in Computer Science 877
(proceedings of ANTS1, 5/94; L.M. Adleman and M.D. Huang, eds.),
122133.

On numbers and endgames,
pages 135150 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.

A characterization of the Z^{n} lattice,
Math. Research Letters 2 (1995), 321326
(math.NT/9906019
on the arXiv).

Lattices and codes with long shadows,
Math. Research Letters 2 (1995), 643651
(math.NT/9906086
on the arXiv).

Local statistics for random domino tilings of the Aztec diamond
(with H. Cohn and J. Propp),
Duke Math. J. 85 #1 (Oct. 1996), 117166.

The exceptional cone and the Leech lattice (with B.H. Gross),
International Math. Research Notices 1996 #14, 665698.

Elliptic and modular curves over finite fields
and related computational issues, pages 2176 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).

Explicit modular towers,
pages 2332 in
Proceedings of the ThirtyFifth Annual Allerton Conference
on Communication, Control and Computing
(1997, T. Basar, A. Vardy, eds.),
Univ. of Illinois at UrbanaChampaign 1998
(math.NT/0103107
on the arXiv).

Embeddings into the Integral Octonions (with B.H. Gross),
Pacific J. Math., Dec. 1997
(Olga TausskyTodd Memorial Issue), 147158.

Shimura curve computations,
Lecture Notes in Computer Science 1423
(proceedings of ANTS3, 1998; J.P.Buhler, ed.), 147
(math.NT/0005160
on the arXiv).
corrigendum;
further corrections
mostly by David Jao (to be implemented soon)

The stillLife density problem and its generalizations,
pages 228253 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.

Linearized algebra and finite groups of Lie type.
I: Linear and symplectic groups, pages 77107 in
Applications of curves over finite fields (Seattle, 1997)
= Contemp. Math. 245, Providence: AMS, 1999.

The Klein quartic in number theory, pages 51102 in
The Eightfold Way: The Beauty of Klein’s Quartic Curve
(S.Levy, ed.; Cambridge Univ. Press, 1999; also online at the
MSRI Publications site)

Rational points near curves and small nonzero
x^{3}y^{2}
via lattice reduction,
Lecture Notes in Computer Science 1838
(proceedings of ANTS4, 2000; W.Bosma, ed.), 3363
(math.NT/0005139
on the arXiv).

Explicit towers of Drinfeld modular curves,
Progress in Mathematics 202 (2001), 189198
(Proceedings of the 3rd European Congress of Mathematics,
Barcelona, 7/2000: paper presented at the minisymposium on
“curves over finite fields and codes”;
math.NT/0005140
on the arXiv).

Lattices, Linear Codes, and Invariants (2part expository article),
Notices of the American Math. Society
47 (2000),
12381245
and
13821391.

Cubic rings and the exceptional Jordan algebra (with B.H. Gross),
Duke Math. J. 109 #2 (2001), 383409.

Excellent nonlinear codes from modular curves, pages 200208 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.

On finite sequences satisfying linear recursions,
New York J. Math. 8 (2002), 8597
= http://nyjm.albany.edu:8000/j/2002/85.html
(math.CO/0105007
on the arXiv).

Curves Dy^{2}=x^{3}x
of Odd Analytic Rank,
Lecture Notes in Computer Science 2369
(proceedings of ANTS5, 2002; C.Fieker and D.R.Kohel, eds.), 244251.
math.NT/0208056
on the arXiv.

Trinomials
ax^{7}+bx+c
and ax^{8}+bx+c
with Galois Groups of Order 168 and 8*168 (with Nils Bruin),
Lecture Notes in Computer Science 2369
(proceedings of ANTS5, 2002; C.Fieker and D.R.Kohel, eds.), 172188.

Appendix to "New Optimal Tame Towers of Function Fields
over Small Finite Fields" by WenChing 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 ANTS5, 2002; C.Fieker and D.R.Kohel, eds.), 384389.

Higher Nimbers in pawn endgames on large chessboards,
pages 6178 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.

The mathematical knight (with Richard Stanley),
Math. Intelligencer 25 #1 (2003), 2234.
[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.]

New upper bounds on sphere packings I (with H. Cohn),
Annals of Math. 157 (2003), 689714
(math.MG/0110009
on the arXiv).

On the Sums ,
Amer. Math. Monthly 110 #7
(Aug.Sep. 2003), 561573. Nearly isomorphic with
math.CA/0101168
on the arXiv.
Corrigenda: Amer. Math. Monthly 111 #5
(May 2004), 456.

On Elliptic Kcurves,
Progress in Mathematics 224 (2004), 8191
(Proceedings of the 7/2002 Barcelona Euroconference on
“Modular Curves and Abelian Varieties”,
ed. J.Cremona, J.C.Lario, J.Quer, and K.Ribet).

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), 399422
(math.NT/0208060
on the arXiv).

Elliptic Curves of Large Rank and Small Conductor (with M.Watkins),
Lecture Notes in Computer Science 3076
(proceedings of ANTS6, 2004; D.Buell, ed.), 4256.
math.NT/0403374
on the arXiv.

Elliptic Curves x^{3} + y^{3} = k of High Rank
(with N.F.Rogers),
Lecture Notes in Computer Science 3076
(proceedings of ANTS6, 2004; D.Buell, ed.), 184193.
math.NT/0403116
on the arXiv.

Gaps in Sqrt(n) mod 1 and ergodic theory (with C.T.McMullen),
Duke Math. J. 123 #1 (2004), 95139.

The conjugate dimension of algebraic numbers
(with N.Berry, A.Dubickas, B.Poonen, and C.Smyth),
Quart. J. Math. 55 (2004), 237252
(math.NT/0308069
on the arXiv).

New Directions in Enumerative Chess Problems,
Electronic J. of Combinatorics
11(2) (20042005) [Stanley60 Festschrift],
Article #4
(math.CO/0508645
on the arXiv).

Reduction of CM Elliptic Curves and Modular Function Congruences
(with K.Ono and T.Yang),
International Math. Research Notices 2005 #44, 26952707
(math.NT/0512350
on the arXiv).

SylvesterGallai Theorems for Complex Numbers and Quaternions
(with Lou M. Pretorius and Konrad J. Swanepoel),
Discrete and Computational Geometry
35 #3 (3/2006), 361373
(math.MG/0403023
on the arXiv).

The Mathieu group M_{12} and its pseudogroup extension M_{13}
(with John H. Conway and Jeremy L. Martin),
Experimental Math. 15 (2006) #2, 223236
(math.GR/0508630
on the arXiv).

Points of Low Height on Elliptic Curves and Surfaces
I: Elliptic surfaces over P^{1} with small d,
Lecture Notes in Computer Science 4076
(proceedings of ANTS7, 2006; F.Hess, S.Pauli, and M.Pohst, ed.), 287301.
math.AG/0608593
on the arXiv.

Shimura Curves for Level3 Subgroups of the (2,3,7) Triangle Group,
and Some Other Examples,
Lecture Notes in Computer Science 4076
(proceedings of ANTS7, 2006; F.Hess, S.Pauli, and M.Pohst, ed.), 302316.
math.NT/0409020
on the arXiv.

On some pointsandlines problems and configurations,
Periodica Mathematica Hungarica 53 #12 (2006), 133148.
math.MG/0612749
on the arXiv.

The D_{4} Root System Is Not Universally Optimal
(with Henry Cohn, John H. Conway, and Abhinav Kumar),
Experimental Math. 16 (2006) #3, 313320
(math.NT/0607447
on the arXiv).

Shimura Curve Computations Via K3 Surfaces of NéronSeveri Rank
at Least 19,
Lecture Notes in Computer Science 5011
(proceedings of ANTS8, 2008; A.J.van der Poorten and A.Stein, eds.),
196211.
arXiv:0802.1301v1
[math.NT].

About the cover: Rational curves on a K3 surface,
pages 14 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.

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), 105108.
arXiv:0905.4306v1
[math.NT].

On the Classification of Type II Codes of Length 24
(with Scott Duke Kominers),
SIAM J. Discrete Math. 23 #4 (2010),
21732177.
arXiv:0902.1942v2
[math.NT].
 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 00029939(10)102846;
138 #8 (August 2010), 28632872.
arXiv:0812.2579v2
[math.MG].

Weighted Generating Functions for Type II Lattices and Codes
(with Scott Duke Kominers), pages 63108 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

Minimal Suniversality criteria may vary in size
(with Daniel M. Kane and Scott Duke Kominers),
J. de Théorie de Nombres de Bordeaux
25 #3 (2013), 557564.
arXiv:1101.5662

Modular forms and K3 surfaces (with Matthias Schütt),
Advances in Math. 240 (2013), 106131.
arXiv:0809.0830
(2008, revised 2013).

Genus bounds for curves with fixed Frobenius eigenvalues
(with Everett W. Howe and Christophe Ritzenthaler),
Proc. Amer. Math. Soc. 142 (2014), 7184.
arXiv:1006.0822
(2010, revised 2012; officially posted to the journal site 18 September 2013).

K3 surfaces and equations for Hilbert modular surfaces (with Abhinav Kumar),
Algebra and Number Theory 8:10 (2014), 22972411.
(DOI: 10.2140/ant.2014.8.2297;
arXiv: 1209.3527)

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, 6398.
Copy on Andrew Granville’s page

Genus 1 Fibrations on the Supersingular K3 Surface
in Characteristic 2 with Artin Invariant 1
(with Matthias Schütt),
Asian J. Math. 19 #3 (2015), 555581.
arXiv:1207.1239

“Scrambling” georeferenced data to protect privacy
induces bias in distance estimation
(with Günther Fink, and Till Bärnighausen),
Population and Environment 37 #1
(Sep. 2015), 8398.

Moduli Spaces of Quadratic Rational Maps with
a Marked Periodic Point of Small Order
(with Jérémy Blanc and Jung Kyu Canci)
International Math. Research Notices 2015 #23, 1245912489
(arXiv:1305.1054;
DOI: 10.1093/imrn/rnv063)

Permutations that Destroy Arithmetic Progressions in Elementary
pGroups
(with Ashvin Swaminathan),
Electronic J. of Combinatorics
24 #1 (2017), Paper #P1.20
(1601.07541
[math.NT] on the arXiv).

Crossing numbers of complete graphs, pages 218249 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.
rest of mathematical
Curriculum Vitae
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