Cohomologie Automorphe et Compactifications partielles de certaines vari\'et\'es de Griffiths-Schmid

Abstract

We study some automorphic cohomology classes of degree one on the Griffiths-Schmid varieties attached to some unitary groups in 3 variables. Using partial compactifications of those varieties, constructed by K. Kato and S. Usui, we define for such a cohomology class some analogues of Fourier-Shimura coefficients, which are cohomology classes on certain elliptic curves. We show that a large space of such automorphic classes can be generated by those with rational "coefficients". More precisely, we consider those cohomology classes that come from Picard modular forms, via some Penrose-like transform studied in a previous article : we prove that the coefficients of the classes thus obtained can be computed from the coefficients of the Picardform by a similar transform defined at the level of the elliptic curve.