Seminar of Algebra

The adelic point of view on modular forms leads to the study of (smooth) complex representations of GL_2(Q_p) and SL_2(Q_p), for a prime number p. The more and more important rôle of congruences of modular forms (notably in the proof of Fermat’s Last Theorem) imposes the study of representations of the same groups on vector spaces over finite fields or their algebraic closure. When the characteristic l of the coefficient field C is not p, the irreducible (smooth) C-representations of GL_2(Q_p) have been determined by Vignéras quite some time ago, together with a Langlands correspondence with C-representations of the absolute Galois group of Q_p. In recent work, Vignéras and I give analogous, but more subtle, results for SL_2(Q_p). All results are valid for finite extensions of Q_p as well.