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3-Selmer groups for curves y 2 = x 3 + a

Andrea Bandini (2008)

Czechoslovak Mathematical Journal

We explicitly perform some steps of a 3-descent algorithm for the curves y 2 = x 3 + a , a a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.

A local-global principle for rational isogenies of prime degree

Andrew V. Sutherland (2012)

Journal de Théorie des Nombres de Bordeaux

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 ≡ 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 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 2 3 in place of our 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 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 ∈ E ( K ) be a point of infinite order. It is natural to ask how many integers n ≥ 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 ∈ E ( ℚ ) as above, we show that there is at most one such n ≥ 3 .

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