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Universal Bound on the Performance of Lattice Codes
 IEEE TRANS. INFORM. THEORY
, 1999
"... We present a lower bound on the probability of symbol error for maximumlikelihood decoding of lattices and lattice codes on a Gaussian channel. The bound is tight for error probabilities and signaltonoise ratios of practical interest, as opposed to most existing bounds that become tight asymptoti ..."
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We present a lower bound on the probability of symbol error for maximumlikelihood decoding of lattices and lattice codes on a Gaussian channel. The bound is tight for error probabilities and signaltonoise ratios of practical interest, as opposed to most existing bounds that become tight asymptotically for high signaltonoise 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.
Height Difference Bounds For Elliptic Curves over Number Fields
"... 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 bound ..."
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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.
Invariants des courbes de FreyHellegouarch et grands groupes de TateShafarevich
, 1998
"... 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)). ..."
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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
SIMATH  a computer algebra system for number theoretic applications
"... this paper is to give a survey of the wide range of number theoretic applications of the computer algebra system SIMATH [42] ..."
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this paper is to give a survey of the wide range of number theoretic applications of the computer algebra system SIMATH [42]
Elliptic curves, Lfunctions, and CMpoints
, 2002
"... this paper. The first four sections of the paper provide the standard background about elliptic curves and Lseries. 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 disc ..."
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this paper. The first four sections of the paper provide the standard background about elliptic curves and Lseries. 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 Lseries of elliptic curves have good analytic properties as predicted by the generalized TaniyamaShimura 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 Lseries: the rank of the MordellWeil group is equal to the order of vanishing of Lseries at the center (by the Birch and SwinnertonDyer 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
Basic Algorithms for Elliptic Curves
"... 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 ..."
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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
A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO x 3 − y 2  VALUES
"... Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = x 3 −y 2  values. Seventeen new values of k
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Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = x 3 −y 2  values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k ̸ = 0, (1.1), known as Mordell’s equation, is an elliptic curve and has only finitely many solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of k  was found early by M. Hall [2, 7] by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27
CRITICAL LVALUES OF LEVEL p NEWFORMS (mod p)
"... Abstract. Suppose that p ≥ 5 is prime, that F(z) ∈ S2k(Γ0(p)) is a newform, that v is a prime above p in the field generated by the coefficients of F, and that D is a fundamental discriminant. We prove nonvanishing theorems modulo v for the twisted central critical values L(F ⊗χD, k). For example, ..."
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Abstract. Suppose that p ≥ 5 is prime, that F(z) ∈ S2k(Γ0(p)) is a newform, that v is a prime above p in the field generated by the coefficients of F, and that D is a fundamental discriminant. We prove nonvanishing theorems modulo v for the twisted central critical values L(F ⊗χD, k). For example, we show that if k is odd and not too large compared to p, then infinitely many of these twisted Lvalues are nonzero (mod v). We give applications for elliptic curves. For example, we prove that if E/Q is an elliptic curve of conductor p, where p is a sufficiently large prime, there there are infinitely many twists D with X(ED/Q)[p] = 0, assuming the Birch and SwinnertonDyer conjecture for curves of rank zero as well as a weak form of Hall’s conjecture. The results depend on a careful study of the coefficients of halfintegral weight newforms of level 4p, which is of independent interest. Let k ≥ 1 be an integer and let
Let
, 2008
"... Relating decision and search algorithms for rational points on curves of higher genus ..."
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Relating decision and search algorithms for rational points on curves of higher genus
Journal de Théorie des Nombres de Bordeaux 00 (XXXX), 000–000 A note on integral points on elliptic curves
, 2006
"... Abstract. We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cas ..."
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Abstract. We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gröbner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional padic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth