Displaying similar documents to “On the ranks of elliptic curves in families of quadratic twists over number fields”

Elliptic curves associated with simplest quartic fields

Sylvain Duquesne (2007)

Journal de Théorie des Nombres de Bordeaux

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We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much...

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Joseph H. Silverman (2012)

Journal de Théorie des Nombres de Bordeaux

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A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to  1 modulo  m . We prove a similar result for polynomials  f ( X ) that are divisible in  ( ℤ / m ℤ ) [ X ] by a polynomial of the form 1 + X + ⋯ + X n for some n ≥ ϵ deg ( f ) . We also formulate and prove an analogous statement for elliptic curves.

A quantitative primitive divisor result for points on elliptic curves

Patrick Ingram (2009)

Journal de Théorie des Nombres de Bordeaux

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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 .