Drinfeld type automorphic forms and special values of L-functions over function fields
(Academia Sinica & IHÉS)
Amphitéâtre Léon Motchane (IHES)
Amphitéâtre Léon Motchane
Le Bois Marie
35, route de Chartres
By a function field K, we mean a field extension over a finite field with transcendence degree one. In the function field world, by the work of Deligne, Drinfeld, Jacquet-Langlands, Weil, and Zarhin, the "Drinfeld modular parametrization" always exists for every "non-isotrivial" elliptic curve E over K. Suppose E has split multiplicative reduction at a place ∞. Then there exists a unique "Drinfeld type" (with respect to ∞) automorphic cusp form fE such that its L-function coincides with the Hasse-Weil L-function of E over K. These forms can be viewed as analogue of classical weight 2 modular forms. In this talk, we will start with basic properties of Drinfeld type automorphic forms, and use them as tools to obtain explicit formulas for special values of the L-functions coming from non-isotrivial elliptic curves.