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Infinite rank of elliptic curves over a b

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then E ( a b ) has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then E ( K a b ) has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over K a b .

Integral points on the elliptic curve y 2 = x 3 - 4 p 2 x

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E : y 2 = x 3 - 4 p 2 x . Further, let N ( p ) denote the number of pairs of integral points ( x , ± y ) on E with y > 0 . We prove that if p 17 , then N ( p ) 4 or 1 depending on whether p 1 ( mod 8 ) or p - 1 ( mod 8 ) .

Invariance of the parity conjecture for p -Selmer groups of elliptic curves in a D 2 p n -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

We show a p -parity result in a D 2 p n -extension of number fields L / K ( p 5 ) for the twist 1 η τ : W ( E / K , 1 η τ ) = ( - 1 ) 1 η τ , X p ( E / L ) , where E is an elliptic curve over K , η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal ( L / K ) = D 2 p n , and X p ( E / L ) is the p -Selmer group. The main novelty is that we use a congruence result between ε 0 -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p -parity conjecture (using...

Involutory elliptic curves over 𝔽 q ( T )

Andreas Schweizer (1998)

Journal de théorie des nombres de Bordeaux

For n 𝔽 q [ T ] let G be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve X 0 ( 𝔫 ) . We determine all 𝔫 and G for which the quotient curve G X 0 ( 𝔫 ) is rational or elliptic.

Iwasawa theory for elliptic curves over imaginary quadratic fields

Massimo Bertolini (2001)

Journal de théorie des nombres de Bordeaux

Let E be an elliptic curve over , let K be an imaginary quadratic field, and let K be a p -extension of K . Given a set Σ of primes of K , containing the primes above p , and the primes of bad reduction for E , write K Σ for the maximal algebraic extension of K which is unramified outside Σ . This paper is devoted to the study of the structure of the cohomology groups H i ( K Σ / K , E p ) for i = 1 , 2 , and of the p -primary Selmer group Sel p ( E / K ) , viewed as discrete modules over the Iwasawa algebra of K / K .

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