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2-Cohomology of semi-simple simply connected group-schemes over curves defined over p -adic fields

Jean-Claude Douai (2013)

Journal de Théorie des Nombres de Bordeaux

Let X be a proper, smooth, geometrically connected curve over a p -adic field k . Lichtenbaum proved that there exists a perfect duality: Br ( X ) × Pic ( X ) / between the Brauer and the Picard group of X , from which he deduced the existence of an injection of Br ( X ) in P X Br ( k P ) where P X and k P denotes the residual field of the point P . The aim of this paper is to prove that if G = G ˜ is an X e t - scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results the neutrality of every X e t -gerb which...

A descent map for curves with totally degenerate semi-stable reduction

Shahed Sharif (2013)

Journal de Théorie des Nombres de Bordeaux

Let K be a local field of residue characteristic p . Let C be a curve over K whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to- p rational torsion subgroup on the Jacobian of C . We also determine divisibility of line bundles on C , including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of C .

A note on the ramification of torsion points lying on curves of genus at least two

Damian Rössler (2010)

Journal de Théorie des Nombres de Bordeaux

Let C be a curve of genus g 2 defined over the fraction field K of a complete discrete valuation ring R with algebraically closed residue field. Suppose that char ( K ) = 0 and that the characteristic p of the residue field is not 2 . Suppose that the Jacobian Jac ( C ) has semi-stable reduction over R . Embed C in Jac ( C ) using a K -rational point. We show that the coordinates of the torsion points lying on C lie in the unique tamely ramified quadratic extension of the field generated over K by the coordinates of the p -torsion...

Calcul du nombre de points sur une courbe elliptique dans un corps fini : aspects algorithmiques

François Morain (1995)

Journal de théorie des nombres de Bordeaux

Nous décrivons dans cet article les algorithmes nécessaires à une implantation efficace de la méthode de Schoof pour le calcul du nombre de points sur une courbe elliptique dans un corps fini. Nous tentons d’unifier pour cela les idées d’Atkin et d’Elkies. En particulier, nous décrivons le calcul d’équations pour X 0 ( ) , premier, ainsi que le calcul efficace de facteurs des polynômes de division d’une courbe elliptique.

Canonical integral structures on the de Rham cohomology of curves

Bryden Cais (2009)

Annales de l’institut Fourier

For a smooth and proper curve X K over the fraction field K of a discrete valuation ring R , we explain (under very mild hypotheses) how to equip the de Rham cohomology H dR 1 ( X K / K ) with a canonical integral structure: i.e., an R -lattice which is functorial in finite (generically étale) K -morphisms of X K and which is preserved by the cup-product auto-duality on H dR 1 ( X K / K ) . Our construction of this lattice uses a certain class of normal proper models of X K and relative dualizing sheaves. We show that our lattice naturally...

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

Consider the families of curves C n , A : y ² = x + A x and C n , A : y ² = x + A where A is a nonzero rational. Let J n , A and J n , A denote their respective Jacobian varieties. The torsion points of C 3 , A ( ) and C 3 , A ( ) are well known. We show that for any nonzero rational A the torsion subgroup of J 7 , A ( ) is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to J 7 , A ( ) [ 2 ] (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for J 3 , A (A ≠ 4) and J 5 , A . We also almost...

Computations with Witt vectors of length 3

Luís R. A. Finotti (2011)

Journal de Théorie des Nombres de Bordeaux

In this paper we describe how to perform computations with Witt vectors of length 3 in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the j -invariant of the canonical lifting as a function on the j -invariant of the ordinary elliptic curve in characteristic p .

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