Browsing by Author "Elkies, Noam"
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The ABC's of Number Theory
Elkies, Noam (Harvard University, 2007)The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry ... 
Curves of Every Genus with Many Points, II: Asymptotically Good Families
Zieve, Michael E.; Wetherell, Joseph L.; Poonen, Bjorn; Kresch, Andrew; Howe, Everett W.; Elkies, Noam David (Duke University Press, 2004)We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following ... 
The \(D_4\) Root System is Not Universally Optimal
Cohn, Henry; Conway, John H.; Elkies, Noam; Kumar, Abhinav (AK Peters, 2007)We prove that the \(D_4\) root system (equivalently, the set of vertices of the regular 24cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 ... 
Elliptic Curves of Large Rank and Small Conductor
Elkies, Noam; Watkins, Mark (Springer Verlag, 2004)For \(r = 6, 7, . . . , 11\) we find an elliptic curve \(E/Q\) of rank at least \(r\) and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for \(r = 6)\) to over 100 (for \(r ... 
Explicit Towers of Drinfeld Modular Curves
Elkies, Noam (Springer Verlag, 2001)We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. 
Gaps in \(\sqrt{n}mod 1\) and Ergodic Theory
Elkies, Noam David; McMullen, Curtis T. (Duke University Press, 2004)Cut the unit circle \(S^1 = \mathbb{R}/\mathbb{Z}\) at the points \(\{\sqrt{1}\}, \{\sqrt{2}\}, . . ., \{\sqrt{N}\}\), where \(\{x\} = x mod 1\), and let \(J_1, . . . , J_N\) denote the complementary intervals, or gaps, ... 
Higher nimbers in pawn endgames on large chessboards
Elkies, Noam (Cambridge University Press, 2002)We answer a question posed in [Elkies 1996] by constructing a class of pawn endgames on mXn boards that show the Nimbers <i>*k</i> for many large <k>k</k>. We do this by modifying and generalizing T.R. Dawson’s “pawns game” ... 
The Klein Quartic in Number Theory
Elkies, Noam (Cambridge University Press, 1999)We describe the Klein quartic <i>X</i> and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing ... 
Lattices and codes with long shadows
Elkies, Noam (International Press, 1995)In an earlier paper we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n8. [A "characteristic vector" of L is a vector w in L such that ... 
The Mathieu group M12 and its pseudogroup extension M13
Conway, John H.; Elkies, Noam; Martin, Jeremy L. (AK Peters, 2006)We study a construction of the Mathieu group M12 using a game reminiscent of Loyd's "15puzzle." The elements of M12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There ... 
New directions in enumerative chess problems
Elkies, Noam (2005)Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves ... 
New Upper Bounds on Sphere Packings I
Cohn, Henry; Elkies, Noam (Princeton University, 2003)We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for ... 
On some pointsandlines problems and configurations
Elkies, Noam (Springer Verlag, 2006)We apply an old method for constructing pointsandlines configurations in the plane to study some recent questions in incidence geometry. 
Point Configurations That Are Asymmetric Yet Balanced
Cohn, Henry; Kumar, Abhinav; Elkies, Noam David; Schürmann, Achill (American Mathematical Society, 2010)A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to ... 
Points of low height on elliptic curves and surfaces I: Elliptic surfaces over P1 with small d
Elkies, Noam (Springer Verlag, 2006)For each of n = 1, 2, 3 we find the minimal height ˆh(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d = 12n (equivalently, of arithmetic genus n), and exhibit all (E, P) attaining this ... 
Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory
Kaplan, Nathan (20130930)The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed ... 
Reduction of CM elliptic curves and modular function congruences
Elkies, Noam; Ono, Ken; Yang, Tonghai (2005)We study congruences of the form F(j(z))  U(p) = G(j(z)) mod p, where U(p) is the pth Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. ... 
Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80
Elkies, Noam David; Kominers, Scott (American Mathematical Society, 2010)We show that, if \(L\) is an extremal Type II lattice of rank 40 or 80, then \(L\) is generated by its vectors of norm \(min(L)+2\). This sharpens earlier results of Ozeki, and the second author and Abel, which showed that ... 
Shimura curve computations via K3 surfaces of NeronSeveri rank at least 19
Elkies, Noam (Springer Verlag, 2008)It is known that K3 surfaces S whose Picard number rho (= rank of the NeronSeveri group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated ... 
Shimura Curves for Level3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples
Elkies, Noam (Springer Verlag, 2006)The (2,3,7) triangle group is known to be associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(cos2π/7) and unramified at all other places of K. This triangle group and its congruence ...