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Let $K$ be a number field. We consider a local-global principle for elliptic curves $E / K$ that admit (or do not admit) a rational isogeny of prime degree $ℓ$ . For suitable $K$ (including $K = ℚ$ ), we prove that this principle holds for all $ℓ \equiv 1 mod 4$ , and for $ℓ < 7$ , but find a counterexample when $ℓ = 7$ for an elliptic curve with $j$ -invariant $2268945 / 128$ . For $K = ℚ$ we show that, up to isomorphism, this is the only counterexample.

A moduli approach to quadratic $m a t h b b Q$ -curves realizing projective mod $p$ Galois representations.

J. A. Fernández (2008)

Revista Matemática Iberoamericana

A new record for the canonical height on an elliptic curve over $ℂ (t)$ .

Jain, Sonal (2010)

The New York Journal of Mathematics [electronic only]

A note on a product formula for the cubic Gauss sum

Hiroshi Ito (2012)

Acta Arithmetica

A note on arithmetic progressions on elliptic curves.

Campbell, Garikai (2003)

Journal of Integer Sequences [electronic only]

A note on arithmetic progressions on quartic elliptic curves.

Ulas, Maciej (2005)

Journal of Integer Sequences [electronic only]

A note on integral points on elliptic curves

Mark Watkins (2006)

Journal de Théorie des Nombres de Bordeaux

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...

A Note on squares in arithmetic progressions, II

Enrico Bombieri, Umberto Zannier (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We show that the number of squares in an arithmetic progression of length $N$ is at most $c_{1} N^{3 / 5} {(\log N)}^{c_{2}}$ , for certain absolute positive constants $c_{1}$ , $c_{2}$ . This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent $\frac{2}{3}$ in place of our $\frac{3}{5}$ . The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus $5$ as in [1].

A parametric family of elliptic curves

Andrej Dujella (2000)

Acta Arithmetica

A procedure to calcute torsion of elliptic curves over Q.

Darrin Doud (1998)

Manuscripta mathematica

À propos de la conjecture de Manin pour les courbes elliptiques modulaires

Ahmed Abbes, Emmanuel Ullmo (1996)

Compositio Mathematica

A quantitative primitive divisor result for points on elliptic curves

Patrick Ingram (2009)

Journal de Théorie des Nombres de Bordeaux

Let $E / K$ be an elliptic curve defined over a number field, and let $P \in E (K)$ be a point of infinite order. It is natural to ask how many integers $n \geq 1$ fail to occur as the order of $P$ modulo a prime of $K$ . For $K = ℚ$ , $E$ a quadratic twist of $y^{2} = x^{3} - x$ , and $P \in E (ℚ)$ as above, we show that there is at most one such $n \geq 3$ .

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

Joseph H. Silverman (1987)

Journal für die reine und angewandte Mathematik

A refined cojecture of Mazur-Tate type for Heegner points.

Henri Darmon (1992)

Inventiones mathematicae

A remark on certain simultaneous divisibility sequences

Stefan Barańczuk, Piotr Rzonsowski (2014)

Colloquium Mathematicae

We investigate possible orders of reductions of a point in the Mordell-Weil groups of certain abelian varieties and in direct products of the multiplicative group of a number field. We express the result obtained in terms of divisibility sequences.

A search for high rank congruent number elliptic curves.

Dujella, Andrej, Janfada, Ali S., Salami, Sajad (2009)

Journal of Integer Sequences [electronic only]

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