Recently V.Drinfeld proved that the scheme of maps from a formal disc to a (possibly singular) algebraic variety for which the image of the generic point of the disc is non-singular has essentially finite dimensional singularities (we shall explain Drinfeld's result in detail).
We shall present a conjecture about a possible application of the above result to the study of automorphic L-functions and discuss some specific examples. Part of this work is joint with M.Finkelberg, D.Gaitsgory and I.Mirkovic.
