arXiv subjects

Noam D. Elkies

Publications and source records attributed to Noam D. Elkies.

50 records · Page 3Linked to original sources

arXiv2002

Curves D y^2 = x^3 - x of odd analytic rank

For nonzero rational D, which may be taken to be a squarefree integer, let E_D be the elliptic curve Dy^2=x^3-x over Q arising in the "congruent number" problem. It is known that the L-function of E_D has sign -1, and thus odd analytic rank, if and only if |D| is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and Swinnerton-Dyer that the arithmetic rank of each of these curves E_D is odd, and therefore positive. We prove that E_D has positive rank for each D such that |D| is in one of the above congruence classes mod 8 and also satisfies |D|<10^6. Our proof is computational: we use the modular parametrization of E_1 or E_2 to construct a rational point P_D on each E_D from CM points on modular curves, and compute P_D to enough accuracy to usually distinguish it from any of the rational torsion points on E_D. In the 1375 cases in which we cannot numerically distinguish P_D from a torsion point of E_D, we surmise that P_D is in fact a torsion point but that E_D has rank 3, and prove that the rank is positive by searching for and finding a non-torsion rational point. We also report on the conjectural extension to |D|<10^7 of the list of curves E_D whose analytic rank is odd and greater than 1, which raises several new questions.

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arXiv2002

On finite sequences satisfying linear recursions

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which there exist a_0,...,a_m in k, not all zero, such that sum(a_i x_{i+j}, i=0..m) = 0 holds for each j=0,1,...,n-m. Equivalently, H_m is the set of (x_0,...,x_n) such that the (m+1)-by-(n-m+1) matrix with (i,j) entry x_{i+j} (i=0..m, j=0..n-m) has rank at most m. We use elementary linear and polynomial algebra to study these sets H_m. In particular, when k is a finite field of q elements, we write the characteristic function of H_m as a linear combination of characteristic functions of linear subspaces of dimensions m and m+1 in W_n. We deduce a formula for the discrete Fourier transform (DFT) of this characteristic function, and obtain some consequences. For instance, if the 2m+1 entries of a square Hankel matrix of order m+1 are chosen independently from a fixed but not necessarily uniform distribution mu on k, then as m->infty the matrix is singular with probability approaching 1/q provided the DFT of mu has l_1 norm less than sqrt(q). This bound sqrt(q) is best possible if q is a square.

math.CO↗

arXiv2001

Excellent nonlinear codes from modular curves

We introduce a new construction of error-correcting codes from algebraic curves over finite fields. Modular curves of genus g -> infty over a field of size q0^2 yield nonlinear codes more efficient than the linear Goppa codes obtained from the same curves. These new codes now have the highest asymptotic transmission rates known for certain ranges of alphabet size and error rate. Both the theory and possible practical use of these new record codes require the development of new tools. On the theoretical side, establishing the transmission rate depends on an error estimate for a theorem of Schanuel applied to the function field of an asymptotically optimal curve. On the computational side, actual use of the codes will hinge on the solution of new problems in the computational algebraic geometry of curves.

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arXiv2001

Explicit modular towers

We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X_0(l^n): n=1,2,3,...}. We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form, and speculate that perhaps every tower of this form that attains the Drinfeld-Vladut bound is modular.

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arXiv2000

Higher Nimbers in pawn endgames on large chessboards

We answer a question posed in [Elkies 1996] (math.CO/9905198) by constructing a class of pawn endgames on m-by-n boards that show the Nimbers *k for large k. We do this by modifying and generalizing T.R. Dawson's ``pawns game'' [Berlekamp et al. 1982] (Winning Ways I). Our construction works for m>8 and n sufficiently large; on the basis of computational evidence we conjecture, but cannot yet prove, that the construction yields *k for all integers k.

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arXiv2000

Mordell-Weil lattices in characteristic 2, III: A Mordell-Weil lattice of rank 128

In the first paper of this series, we constructed a family of lattices in dimensions 2^{n+1} for positive integers n, and proved that the associated lattice packings of spheres equal or exceed the previous records for several values of n. In this paper we analyze the case n=6, which is now the first case where the resulting lattice MW128 yields a previously unknown lattice of record density. We prove that the elliptic curve used to construct this lattice has trivial Tate-Shafarevich group, and exhibit on that curve a rational point whose canonical height attains our lower bound on the minimal norm of MW128. We thus obtain the normalized center density of the lattice. We then report on a computation that determined all the rational points of minimal norm, thus obtaining the kissing number of MW128. Like the packing density, the kissing number of MW128 is by a considerable factor the largest known kissing number of a lattice in this dimension.

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arXiv2000

Explicit towers of Drinfeld modular curves

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.

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arXiv2000

Shimura curve computations

We give some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. We then illustrate these methods by working out several examples in varying degrees of detail. For instance, we compute coordinates for all the rational CM points on the curves "X(1)" associated with the quaternion algebras over Q ramified at {2,3}, {2,5}, {2,7}, and {3,5}. We conclude with a list of open questions that may point the way to further computational investigation of these curves.

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arXiv2000

Rational points near curves and small nonzero |x^3-y^2| via lattice reduction

We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat cubic, we find all integer solutions of |x^3+y^3-z^3| >N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M<N^3), the computational costs are essentially as low as possible. Moreover the algorithm readily parallelizes. It not only yields new numerical examples but leads to theoretical results, difficult open questions, and natural generalizations. We also adapt our algorithm to investigate Hall's conjecture: we find all integer solutions of 0<|x^3-y^2|<<x^(1/2) with x<X in time O(X^(1/2) L(X)). By implementing this algorithm with X=10^18 we shattered the previous record for x^(1/2)/|x^3-y^2|. The O(X^{1/2} L(X)) bound is rigorous; its proof also yields new estimates on the distribution mod 1 of sqrt(cx^3) for any positive rational c.

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arXiv1999

Lattices and codes with long shadows

In an earlier paper (math.NT/9906019) 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 n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissing number of unimodular lattices with minimal norm 2. We then state and prove analogues of these results for self-dual codes, and relate them directly to the lattice problems via "Construction A".

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arXiv1999

A characterization of the Z^n lattice

We use theta series and modular forms to prove that Z^n is the only integral unimodular lattice of rank n without characteristic vectors of norm <n, i.e. the only integral unimodular lattice not containing a vector w such that (w,w)<n and 2|(v,v+w) for all lattice vectors v. By the work of Kronheimer and others on the Seiberg-Witten equation this yields an alternative proof of a theorem of Donaldson on the geometry of 4-manifolds.

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arXiv1999

The still-Life density problem and its generalizations

A "still Life" is a subset S of the square lattice Z^2 fixed under the transition rule of Conway's Game of Life, i.e. a subset satisfying the following three conditions: 1. No element of Z^2-S has exactly three neighbors in S; 2. Every element of S has at least two neighbors in S; 3. Every element of S has at most three neighbors in S. Here a ``neighbor'' of any x \in Z^2 is one of the eight lattice points closest to x other than x itself. The "still-Life conjecture" is the assertion that a still Life cannot have density greater than 1/2 (a bound easily attained, for instance by {(x,y): x is even}). We prove this conjecture, showing that in fact condition 3 alone ensures that S has density at most 1/2. We then consider variations of the problem such as changing the number of allowed neighbors or the definition of neighborhoods; using a variety of methods we find some partial results and many new open problems and conjectures.

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arXiv1999

The identification of three moduli spaces

It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G yield two G-covers of the moduli space of configurations of six points on the projective line modulo PGL_2, via the 3- and 2-torsion of the Jacobians of the double and triple cyclic covers of P^1 branched at those six points. Remarkably these two covers are isomorphic. This was proved over C by transcendental methods by Hunt and Weintraub. We give an algebraic proof valid over any field not of characteristic 2 or 3 that contains the cube roots of unity. We then explore the connection between this $G$-cover and the elliptic surface $y^2 = x^3 + sextic(t), whose Mordell-Weil lattice is E_8 with automorphisms by a central extension of G.

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arXiv1999

On numbers and endgames: Combinatorial game theory in chess endgames

In an investigation of the applications of Combinatorial Game Theory to chess, we construct novel mutual Zugzwang positions, explain an otherwise mysterious pawn endgame from "A Guide to Chess Endings" (Euwe and Hooper), show positions containing non-integer values (fractions, switches, tinies, and loopy games), and pose open problems concerning the values that may be realized by positions on either standard or nonstandard chessboards.

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