Withdrawn: Beilinson's Theorem on Modular Curves

Version explicite du théorème de Beilinson pour la courbe modulaire X1(N)

“…Le théorème de Beilinson sur les courbes modulaires [1, §5] (voir également [2,12]) exprime la valeur en s = 2 de la fonction L d'une forme modulaire de poids 2 en termes d'un régulateur. Compte-tenu des applications potentielles de ce théorème [4,7,11], il est naturel d'en chercher une version explicite.…”

“…Un théorème profond de Beilinson [1,2,12] exprime le produit de valeurs spéciales L(f, 2)L(f, χ, 1), où χ est un caractère pair, en termes de De plus, le symbole de Milnor {uχ , u ψχ } appartient au sous-espace K 2 (X 1 (N )) ⊗ C.…”

“…For any Dirichlet character χ we define L(f, χ, s) = ∞ n=1 a n χ(n)n −s . Profound results of Beilinson [1,2,12] express L(f, 2)L(f, χ, 1), where χ is an even Dirichlet character, in terms of r N (γ χ ), ω f , with γ χ ∈ K 2 (X 1 (N )) ⊗ C and ω f = 2iπf (z) dz. However, Beilinson's formula is not explicit in the following sense:…”

The equivariant Tamagawa number conjecture: a survey

“…In the noncritical range n ≥ k one has results towards Conjecture 2 (the Beilinson conjecture), involving the construction of a subspace of H 1 f (M ) whose image under the regulator is related to L(f, n) (see [68] for k = 2 and n ≥ 2. The case n ≥ k > 2 has been announced in [30] but has not yet appeared in print).…”

Zagier's conjecture on L ( E ,2)

“…The function J q (z) is invariant under the shift z −→ qz and satisfies J q (z) = −J q (z −1 ). It follows from the main result of Beilinson in [B2], see also [SS2], that for a modular elliptic curve E over Q there always exists an element in K 2 (E) Z whose regulator gives (up to a standard nonzero factor) L(E, 2). So we get the formula…”