Elliptic curves and the Weil-Deligne group

Root numbers of abelian varieties

“…2(i)), which implies that the image by κ l of I is finite, hence κ l is trivial on an open subgroup of I (cf. [6], p. 148).…”

“…We are interested in the representation σ = (σ, N ) of W (K/K) associated to σ l by the standard procedure (see e.g., [6], §4).…”

“…is a nilpotent endomorphism (see [6], p. 130, §4 for more details). A priori, σ depends on the choice of l and ı, but by abuse of notation we write σ instead of σ l,ı .…”

Deux théorèmes de non-annulation de valeurs spéciales de fonctions L

“…Ce calcul peutégalementêtre effectué du côté de Galois en utilisant la correspondance de Langlands locale; il suffit alors de citer, par exemple, [Roh,Cor. page 146].…”

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“…The result stated in Main Theorem can be extended, if the parity conjecture holds true for E [Rohrlich 1994]. Let (E/‫,ޑ‬ s) be the complete L-series of the elliptic curve over ‫.ޑ‬ Denote by w(E) ∈ {±1} the root number in the functional equation…”