Zeta elements and modular forms
(Un exposé de Tony Scholl au séminaire de théorie
des nombres de Chevaleret le 3 février 2003)
Résumé :
Let $f$ be a newform on $\Gamma_1(N)$ of weight $k\ge 2$. We show how to
construct `zeta elements', belonging to a suitable motivic cohomology
group attached to the motive of $f$, which are related to special
$L$-values in two ways: via the archimedean regulator map, to the
(non-critical) $L$-values at $s=k$ of $f$ and its twists; and via the
$p$-adic dual exponential map, to the (critical) values at $s=k-1$.
These generalise Kato's $K_2$ elements for $k=2$.