Overconvergent algebraic automorphic forms

Abstract

Abstract. I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GLn. This leads to some new phenomena, including the appearance of intermediate spaces of "semi-classical" automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules.

A p-adic Labesse–Langlands transfer

“…the subgroup of matrices that are congruent to an upper triangular matrix mod p, and by I the Iwahori subgroup of SL 2 (Q p ). Then I and I are admissible in the sense of Definition 2.2.3 of [18]. Fix Haar measures on GL 2 (Q p ) and SL 2 (Q p ) normalized such that meas(I) = meas( I) = 1.…”

“…In this section we briefly recall some details of the construction in [18], check some technical properties of the eigenvarieties and construct the two auxiliary eigenvarieties that we need for the p-adic transfer. We remark here that due to the generality Loeffler works in he has to impose certain arithmetical conditions at various places.…”

“…a 1 < a 2 ) and define I to be the monoid generated by I and Σ + . Let H + p ( G) be the Hecke algebra associated to I as in Proposition 3.2.2 of [18], i.e. H + p ( G) is the subalgebra of the full Hecke algebra of G of compactly supported, locally constant E-valued functions on G with support contained in I.…”

“…They have been constructed for many reductive groups (cf. [5], [6], [11], [18] and [25]). Given a construction of eigenvarieties for two groups G and H as above, one can ask, whether the classical transfer interpolates to a morphism between these rigid spaces.…”

“…Using Buzzard's machine Loeffler (in [18]) established the existence of eigenvarieties for a wide class of groups and in particular for the groups G and G. The Lgroups of G and G are GL 2 (Q) and PGL 2 (Q) respectively. There is a natural projection GL 2 (Q) → PGL 2 (Q) and the transfer corresponding to this L-homomorphism was proved by Labesse and Langlands in [16].…”