Displaying similar documents to “Small discriminants of complex multiplication fields of elliptic curves over finite fields”

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...

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.

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,...

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

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Let f ( x ) be a cubic, monic and separable polynomial over a field of characteristic p 3 and let E be the elliptic curve given by y 2 = f ( x ) . In this paper we prove that the coefficient at x 1 2 p ( p - 1 ) in the p –th division polynomial of E equals the coefficient at x p - 1 in f ( x ) 1 2 ( p - 1 ) . For elliptic curves over a finite field of characteristic p , the first coefficient is zero if and only if E is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...

Rational points on X 0 + ( p r )

Yuri Bilu, Pierre Parent, Marusia Rebolledo (2013)

Annales de l’institut Fourier

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Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of X 0 + ( p r ) ( ) , for r > 1 and  p a prime number exceeding 2 · 10 11 . This includes the case of the curves X split ( p ) . We then prove, with the help of computer calculations, that the same holds true for  p in the range 11 p 10 14 , p 13 . The combination of those results completes the qualitative study of rational points on X 0 + ( p r ) undertook in our previous work, with the only exception of  p r = 13 2 .

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

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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...