Corrigendum: On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields

The aim of this corrigendum is to correct an error in Corollary 10.7 to Theorem 10.6, one of the main results in the paper ‘On the cuspidal cohomology of $S$ -arithmetic subgroups of reductive groups over number fields’. This makes necessary a thorough investigation of the conditions under which a Cartan-type automorphism exists on $G_1=\mathrm {Res}_{\mathbb {C}/\mathbb {R}}G_0$ , where $G_0$ is a connected semisimple algebraic group defined over $\mathbb {R}$ .

Cuspidal cohomology of stacks of shtukas

Let $G$ be a connected split reductive group over a finite field $\mathbb{F}_{q}$ and $X$ a smooth projective geometrically connected curve over $\mathbb{F}_{q}$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

Analytic Tools

This chapter goes to the transcendental level, i.e., take an embedding ι‎ : E → ℂ, and extend the ground field to ℂ. The entirety of this chapter works over ℂ and therefore suppresses the subscript ℂ. It begins with the cuspidal parameters and the representation 𝔻λ‎ at infinity. Next, the chapter defines the square-integrable cohomology as well as the de Rham complex. Finally, cuspidal cohomology is addressed. Here, the chapter looks at the cohomological cuspidal spectrum and the consequence of multiplicity one and strong multiplicity one. It also shows the character of the component group I, before dropping the assumption that we are working over ℂ and go back to our coefficient system 𝓜̃λ‎,E defined over E.

ON IHARA'S LEMMA FOR DEGREE ONE AND TWO COHOMOLOGY OVER IMAGINARY QUADRATIC FIELDS

We prove a version of Ihara's Lemma for degree q = 1, 2 cuspidal cohomology of the symmetric space attached to automorphic forms of arbitrary weight (k ≥ 2) over an imaginary quadratic field with torsion (prime power) coefficients. This extends an earlier result of the author [Ihara's lemma for imaginary quadratic fields, J. Number Theory128(8) (2008) 2251–2262] which concerned the case k = 2, q = 1. Our method is different from [Ihara's lemma for imaginary quadratic fields, J. Number Theory128(8) (2008) 2251–2262] and uses results of Diamond [Congruence primes for cusp forms of weight k ≥ 2, Astérisque196–197 (1991) 205–213] and Blasius–Franke–Grunewald [Cohomology of S-arithmetic subgroups in the number field case, Invent. Math.116(1–3) (1994) 75–93]. We discuss the relationship of our main theorem to the problem of the existence of level-raising congruences.

THE AUTOMORPHIC COHOMOLOGY AND THE RESIDUAL SPECTRUM OF HERMITIAN GROUPS OF RANK ONE

Let G/ℚ be an inner form of Sp4/ℚ which does not split over ℝ. Consequently, it is not quasi-split. In this paper we determine completely the automorphic cohomology of G. That is, we describe the Eisenstein and the cuspidal cohomology of congruence subgroups Γ of G with respect to arbitrary coefficient systems. In particular we establish precise nonvanishing results for cuspidal cohomology. In addition, we calculate the residual spectrum [Formula: see text] of G.