Displaying similar documents to “Some examples of 5 and 7 descent for elliptic curves over Q

On the average value of the canonical height in higher dimensional families of elliptic curves

Wei Pin Wong (2014)

Acta Arithmetica

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Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height h ̂ E ω of the specialized elliptic curve E ω with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient ( h ̂ E ω ( P ω ) ) / h ( ω ) over all nontorsion P ∈ E(K).

On invariants of elliptic curves on average

Amir Akbary, Adam Tyler Felix (2015)

Acta Arithmetica

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We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let e E ( p ) be the exponent of the group of rational points of the reduction modulo p of E over the finite field p . Let be the family of elliptic curves E a , b : y 2 = x 3 + a x + b , where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, 1 / | | E p x e E k ( p ) = C k l i ( x k + 1 ) + O ( ( x k + 1 ) / ( l o g x ) c ) as x → ∞, as long...

Infinite rank of elliptic curves over a b

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

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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 .

Elliptic curves with ( [ 3 ] ) = ( ζ 3 ) and counterexamples to local-global divisibility by 9

Laura Paladino (2010)

Journal de Théorie des Nombres de Bordeaux

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We give a family h , β of elliptic curves, depending on two nonzero rational parameters β and h , such that the following statement holds: let be an elliptic curve and let [ 3 ] be its 3-torsion subgroup. This group verifies ( [ 3 ] ) = ( ζ 3 ) if and only if belongs to h , β . Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer...

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski (2015)

Czechoslovak Mathematical Journal

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We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field 𝔽 p of p elements, such that the discriminant D ( E ) of the quadratic number field containing the endomorphism ring of E over 𝔽 p is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

A local-global principle for rational isogenies of prime degree

Andrew V. Sutherland (2012)

Journal de Théorie des Nombres de Bordeaux

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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 . For suitable K (including K = ), we prove that this principle holds for all 1 mod 4 , and for < 7 , but find a counterexample when = 7 for an elliptic curve with j -invariant 2268945 / 128 . For K = we show that, up to isomorphism, this is the only counterexample.

The Mordell-Weil bases for the elliptic curve y 2 = x 3 - m 2 x + m 2

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

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Let D m be an elliptic curve over of the form y 2 = x 3 - m 2 x + m 2 , where m is an integer. In this paper we prove that the two points P - 1 = ( - m , m ) and P 0 = ( 0 , m ) on D m can be extended to a basis for D m ( ) under certain conditions described explicitly.