Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

Abstract

The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of G-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if G is an abelian group then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian G, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of G-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.

Hecke operators and analytic Langlands correspondence for curves over local fields

“…In his letters to us (2019), he conjectured compactness and the Hilbert-Schmidt property of averages of the Hecke operators over sufficiently many points. The idea that Hecke operators over C could be used to construct an analogue of the Langlands correspondence was suggested by Langlands [24], who sought to construct them in the case when G D GL 2 , X is an elliptic curve, and S D ; (however, for an elliptic curve X we can only define Hecke operators if S ¤ ;; see [14,Section 3]).…”

Class field theory, its three main generalisations, and applications

“…It is still not clear, even conjecturally, what happens when one passes from a curve to its finite étale cover, and the ramified case remains substantially open too. Frenkel [15] discusses possible analytic approached to automorphic fundings on complex algebraic curves; see also [3].…”

“…2dAAG = 2d adelic analysis and geometry studies properties of the zeta function of E by involving 2d analytic adeles and 2d zeta integrals. 15 Recall that the (Hasse) zeta function of E is E .s/ D Y x 1 jk.x/j s 1 ;…”

“…See also Remark 2.3.3 of [56, Part IV] and Section 4.2 of [55] 15. See Sections I, J, K of httpsW//www.maths.nottingham.ac.uk/personal/ibf/mp.html.…”

Automorphic forms for PGL(3) over elliptic function fields. Part 1: Graphs of Hecke operators

“…The geometric Langlands program (e.g., cf. [Fre20,FGV02,Gai15,Lau87]) is rooted in the following observation that is usually attributed to Weil (e.g., cf. [Fre04, Lemma 3.1]): the domain of unramified automorphic forms, modulo the right-action of the standard maximal compact subgroup K of PGL n (A), is naturally identified with the set PBun n X of isomorphism classes of P n−1 -bundles 1 on the smooth, projective and geometrically irreducible curve X with function field F.…”