Eigenvarieties for reductive groups

Abstract

We develop the theory of overconvergent cohomology introduced by G. Stevens, and we use it to give a construction of eigenvarieties associated to any reductive group G over Q such that G(R) has discrete series. We prove that the so-called eigenvarieties are equidimensional and generically flat over the weight space.

On the $$\mathrm {GL}_n$$ GL n -eigenvariety and a conjecture of Venkatesh

“…This result is well-known to experts, and amounts to a special case of [43,Prop. 4.3.10]; for the convenience of the reader, we give a fairly detailed sketch of the proof.…”

“…3. The assumption dim T x = dim − l 0 , conjectured by Hida in the ordinary case [29] and Urban in the general finite-slope case [43], is a non-abelian analogue of the Leopoldt conjecture, and seems to be of equivalent difficulty [34]. However, when n = 3, 4, we have l 0 = 1, in which case this equality follows from [27, Theorem 4.5.1].…”

“…Coleman-Mazur [17] then systematized these families, constructing a geometric object called the eigencurve, whose points are in bijection with finite slope p-adic eigenforms and which locally are modelled on the families constructed by Coleman. Eigenvarieties for reductive groups other than GL 2 have since been constructed by a number of different authors; let us mention in particular here the work of Buzzard, Chenevier, Emerton, and Urban [9,12,21,43]. The eigenvarieties studied in this paper are those constructed by the first named author [27], following an earlier construction of Ash-Stevens [3].…”

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

“…Ash and Stevens developed this idea much further in [AS08]: as conceived there, overconvergent cohomology works for any connected reductive Q-group G split at p, and leads to natural candidates for quite general eigenvarieties. When the group G der (R) possesses discrete series representations, Urban [Urb11] used overconvergent cohomology to construct eigenvarieties interpolating classical forms occuring with nonzero Euler-Poincaré multiplicities, and showed that his construction yields spaces which are equidimensional of the same dimension as weight space. In this article, we develop the theory of eigenvarieties for a connected reductive group over a number field F , building on the ideas introduced in [AS08] and [Urb11], and we formulate precise conjectures relating these spaces with representations of the absolute Galois group Gal(F /F ).…”

A p-adic Labesse–Langlands transfer

“…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.…”