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Displaying similar documents to “On the average value of the canonical height in higher dimensional families of elliptic curves”

Some examples of 5 and 7 descent for elliptic curves over Q

Tom Fisher (2001)

Journal of the European Mathematical Society

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We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.

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...

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).

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...

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.

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.

On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

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We consider a random, uniformly elliptic coefficient field a on the lattice d . The distribution · of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function G ( t , x , y ) satisfy optimal annealed estimates which are L 2 and L 1 , respectively, in probability, i.e., they obtained bounds on | x G ( t , x , y ) | 2 1 / 2 and | x y G ( t , x , y ) | . In particular, the elliptic Green’s function G ( x , y ) satisfies optimal annealed bounds. In their recent work,...