Groupes de Selmer et accouplements

Details Groupes de Selmer et accouplements Groupes de Selmer et accouplements

2002-11-14

arxiv.org/abs/math/0211469v1

Mathematics

Number Theory

Abstract added in migration

Scientific paper

Nekov\'a\v{r} vient de d\'emontrer que le rang de $E(\Q)$ pour une courbe elliptique $E$d\'efinie sur $\Q$ est de m\^eme parit\'e que la multiplicit\'e du z\'ero en $s=1$ de la fonction $L_{E}$ complexe associe\'e \`a $E/\Q$, lorsque le groupe de Tate-Shafarevich est fini. La clef de la d\'emonstration est le construction d'une forme altern\'ee et non d\'eg\'en\'er\'ee sur le quotient de $S(K)$ par sa partie divisible. Pour construire le forme altern\'ee, Nekov\'a\v{r} reprend compl\`etement la th\'eorie des groupes de Selmer en utilisant la formalisme des complexes. Il obtient ainsi d'autres applicationsen th\'eorie de Hida et autres. Nous allons faire ici cette construction en allant au plus court et de replacer ensuite ces r\'esultats dans un contexte plus g\'en\'eral. ----- Nekov\'a\v{r} recently proved that the rank of $E(\Q)$ for an elliptic curve $E$ defined over $\Q$ has the same parity as the zero of the $L$-function $L_{E}$ at $s=1$, when the Tate-Shafarevitch group is finite, in agreement with the conjecture of Birch and Swinnerton-Dyer. The key to the proof is the construction of a non-degenerate alternating form on the quotient of the Selmer group of $E$ by its divisible part. In order to construct this form, Nekov\'a\v{r} completely redoes the theory of Selmer groups, using the formalism of complexes. He thereby obtains other applications in the theory due to Hida and others. Here we will simplify this construction and place these results in a more general context.

Affiliated with

Also associated with

No associations

Rating

Groupes de Selmer et accouplements does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Groupes de Selmer et accouplements, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Groupes de Selmer et accouplements will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-473761