We present a lower bound on the probability of symbol error for maximum-likelihood decoding of lattices and lattice codes on a Gaussian channel. The bound is tight for error probabilities and signal-to-noise ratios of practical interest, as opposed to most existing bounds that become tight asymptotically for high signal-to-noise ratios. The bound is also universal; it provides a limit on the highest possible coding gain that may be achieved, at specific symbol error probabilities, using any lattice or lattice code in n dimensions. In particular, it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains. The asymptotic (as n !1) behavior of the new bound is shown to coincide with the Shannon limit for Gaussian channels.

Introduction Soit E une courbe elliptique d'efinie sur Q, et X son groupe de TateShafarevich, d'efinie par X = ker / H 1 (Q; E) \Gamma! Y v H 1 (Q v ; E) ! ; o`u v d'ecrit l'ensemble des places finies et 1 de Q (voir [30] pour une d'efinition des groupes H 1 (Q; E) et H 1 (Q v ; E)). Ce groupe est ab'elien, et sa finitude pour toute courbe elliptique est encore conjecturale, mais, en 1987, Rubin [29] a donn'e un exemple d'une famille infinie de courbes elliptiques pour lesquelles X est fini. Ces courbes sont `a multiplication complexe, et d'efinies sur un corps quadratique imaginair

this paper is to give a survey of the wide range of number theoretic applications of the computer algebra system SIMATH [42]

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and ˆ h be the canonical height on E. Bounds for the difference h − ˆ h are of tremendous theoretical and practical importance. It is possible to decompose h − ˆ h as a weighted sum of continuous bounded functions Ψυ: E(Kυ) → R over the set of places υ of K. A standard method for bounding h − ˆ h, (due to Lang, and previously employed by Silverman) is to bound each function Ψυ and sum these local ‘contributions’. In this paper we give simple formulae for the extreme values of Ψυ for nonarchimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ. For real archimedean υ a method for sharply bounding Ψυ was previously given by Siksek (1990). We complement this by giving two methods for sharply bounding Ψυ for complex archimedean υ.

this paper. The first four sections of the paper provide the standard background about elliptic curves and L-series. We will start with elliptic curves defined by Weierstrass equations, and address two arithmetic questions: to compute the Mordell Weil group (Lang's conjecture) and to bound the discriminant in terms of the conductor (Szpiro's conjecture). Then we assume that the L-series of elliptic curves have good analytic properties as predicted by the generalized Taniyama-Shimura conjecture. In other words, we always work on modular elliptic curves (or more generally, abelian varieties of GL 2 -type). Of course, over Q, such a conjecture has been proved recently by Wiles and completed by Taylor, Diamond, Conrad, and Brueil. Both arithmetic questions addressed earlier have their relation with L-series: the rank of the Mordell-Weil group is equal to the order of vanishing of L-series at the center (by the Birch and Swinnerton-Dyer conjecture); the discriminant is essentially the degree of the strong modular parameterization. The theory of complex multiplications then provides many examples of modular elliptic curves and abelian varieties of GL 2 -type, and the foundation for the theory of Shimura varieties

b 2 b 4 216b 6 : The discriminant is then = b 2 2 b 8 8b 3 4 27b 2 6 + 9b 2 b 4 b 6 and the modular invariant j = c 3 4 : However, if E is dened over a number eld K, we shall use the short Weierstrass form (1:2) E : Y 2 = X 3 + aX + b (a; b 2 K) with discriminant = 16(4a 3 + 27b 2 ) = 16 0 and modular invariant j = 12 3 4a 3<F12.2