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11G05, Elliptic curves over global fields — 4 results found.

      
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On Dependence of Rational Points on Elliptic Curves
C. R. Math. Rep. Acad. Sci. Canada Vol. 38 (2) 2016, pp. 75-84
Mohammad Sadek (Received: 2015/04/15, Revised: 2015/08/18)

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Let \(E\) be an elliptic curve defined over \(Q\). Let \(\Gamma\) be a subgroup of \(E(Q)\) and \(P\in E(Q)\). In \cite{Arithmetic}, it was proved that if \(E\) has no nontrivial rational torsion points, then \(P\in\Gamma\) if and only if \(P\in \Gamma\) mod \(p\) for finitely many primes \(p\). In this note, assuming the General Riemann Hypothesis, we provide an explicit upper bound on these primes when \(E\) does not have complex multiplication and either \(E\) is a semistable curve or \(E\) has no exceptional prime.

Soit \(E\) une courbe elliptique définie sur \(Q\). Soit \( \Gamma\) un sous-groupe de \( E(Q) \) et \( P \in E (Q) \). Dans \cite{Arithmetic}, il on a prouvé que si \( E \) n’a pas de points de torsion rationels non trivials, alors \( P \in \Gamma \) si et seulement si \( P \in \Gamma \) mod \( p \) pour un nombre fini de nombres premiers \( p \). Dans cette note, supposant l’hypothèse général de Riemann, nous fournissons une borne-supérieure explicite sur ces nombres premiers quand \( E \) n’a pas de multiplication complexe et soit \( E \) est une courbe semi-stable soit \( E \) n’a aucun nombre premier exceptionnel.

Further Remarks on Rational Albime Triangles
C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (2) 2017, pp. 67-76
Jasbir S. Chahal; Josselin Kooij; Jaap Top (Received: 2015/07/26, Revised: 2016/11/14)

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In this note we present further number theoretic properties of the rational albime triangles, in particular, the distribution of acute vs. obtuse rational albime triangles. The notion of albime triangle is extended to include the case of external angle bisector. The proportion of internal vs. external rational albime triangles is also computed.

Dans cette note, nous présentons des propriétés supplémentaires (concernant la théorie des nombres) des triangles rationnels ‘albimes’; en particulier, la distribution des triangles rationnels albimes aigus contre obtus. La notion de triangle albime est développé pour comprendre le cas d’extérieur bissectrice. On calcule aussi la proportion des triangles rationnels albimes internes contre externes.

Variation in the number of points on elliptic curves and applications to excess rank
C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (4), 2005 pp. 111–120
Steven J. Miller (Received: 2005/07/19)

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Michel proved that for a one-parameter family of elliptic curves over \(\mathbb{Q}(T)\) with non-constant \(j(T)\) that the second moment of the number of solutions modulo \(p\) is \(p^2 + O(p^{3/2})\). We show this bound is sharp by studying \(y^2 = x^3 + Tx^2 + 1\). Lower order terms for such moments in a family are related to lower order terms in the \(n\)-level densities of Katz and Sarnak, which describe the behavior of the zeros near the central point of the associated \(L\)-functions. We conclude by investigating similar families and show how the lower order terms in the second moment may affect the expected bounds for the average rank of families in numerical investigations.

Michel a démontré que pour une famille de courbes élliptiques à un paramètre sur \(\mathbb{Q}(T)\) avec \(j(T)\) non-constant, le second moment du nombre de solutions modulo \(p\) est \(p^2 + O(p^{3/2})\). Nous montrons que cette limite est précise en étudiant \(y^2 = x^3 + Tx^2 + 1\). Pour de tels moments dans une famille, les termes d’ordre inférieur sont liés aux termes dans les \(n\)-niveaux de densité de Katz et Sarnak, qui decrivent le comportement des zéros près du point central des \(L\)-fonctions associées. Nous concluons en recherchant des familles semblables et en montrant comment les termes d’ordre inférieur dans le second moment peuvent affecter les bornes pour le rang moyen de familles dans des simulations numériques.

Bounding the torsion in CM elliptic curves
C. R. Math. Rep. Acad. Sci. Canada Vol. 23 (1) 2001, pp. 1–5
D. Prasad / C.S. Yogananda (Received: 2000/01/21)

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

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05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F11 11F66 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K14 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F25 39B72 42C05 43A07 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 83C05

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