Most transcendence proofs rely on a collection of statements of a purely geometric nature: the ``zero estimates" (or ``vanishing lemmas"), which ensure injectivity (or surjectivity) of an evaluation map on sections of a vector bundle at zero-dimensional subschemes of the base. We shall present (and time permitting, prove) them in a unified way, covering two of the main areas of transcendence: periods of one-motives (Baker-Wustholz theory), and values of solutions of differential equations (Siegel-Shidlovsky theorem).
A new proof of the latter theorem will also be discussed, as well as a $q$-difference analogue of the corresponding ``vanishing lemma".
In each case, the vanishing lemmas exhibit a geometric obstruction which is the key not only to the trancendence results themselves, but also to their conjectural generalizations (Schanuel, Grothendieck, Andr\'e).
D. Bertrand: Le th\'eor\`eme de Siegel-Shidlovsky revisit\'e, Pr\'epublication de l'Institut de Math\'ematiques de Jussieu, No 390, Mai 2004.