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A remark on Tate's algorithm and Kodaira types

Tim Dokchitser, Vladimir Dokchitser (2013)

Acta Arithmetica

We remark that Tate’s algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of v ( a i ) / i . As an application, we deduce the behaviour of Kodaira types in tame extensions of local fields.

An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Lucien Bénéteau, M. Abou Hashish (2004)

Commentationes Mathematicae Universitatis Carolinae

The Generalized Elliptic Curves ( GECs ) are pairs ( Q , T ) , where T is a family of triples ( x , y , z ) of “points” from the set Q characterized by equalities of the form x . y = z , where the law x . y makes Q into a totally symmetric quasigroup. Isotopic loops arise by setting x * y = u . ( x . y ) . When ( x . y ) . ( a . b ) = ( x . a ) . ( y . b ) , identically ( Q , T ) is an entropic GEC and ( Q , * ) is an abelian group. Similarly, a terentropic GEC may be characterized by x 2 . ( a . b ) = ( x . a ) ( x . b ) and ( Q , * ) is then a Commutative Moufang Loop ( CML ) . If in addition x 2 = x , we have Hall GECs and ( Q , * ) is an exponent 3

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

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 zero loci...

Dynamical systems arising from elliptic curves

P. D'Ambros, G. Everest, R. Miles, T. Ward (2000)

Colloquium Mathematicae

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose...

Formes de jacobi et formule de Weber p -adique

Abdelmejid Bayad (1999)

Journal de théorie des nombres de Bordeaux

Dans ce texte, on construit sur un corps local de caractéristique strictement positive, un analogue p -adique aux formes de Jacobi méromorphes complexes D L ( z ; ϕ ) , étudiées dans [3] et [4]. Le théorème principal établit que les formes de Jacobi p -adiques obtenues satisfont deux relations de distribution et d’inversion additives. L’analogue p -adique à une formule de Weber généralisée est prouvé comme corollaire du théorème principal.

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...

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