Curves with prescribed symmetry and associated representations of mapping class groups

Abstract

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra QG to the algebra of Q-endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation acts Q-irreducibly in a G-isogeny space of H 1 (C; Q) and with image often a Q-almost simple group.

A remark on the homology of finite coverings of a surface

“…Note that the image by Φ of H 1 (Γ(C σ ); Q) in H 1 (S; Q) does not depend on the choice of the splittings. Moreover, since irreducible QG-modules are self-dual, the image by Φ of Since, by definition, the subspace H σssc 1 (S; Q) of H 1 (S; Q) is preserved by the group G and by the centralizer Mod(S) G of G in the mapping class group Mod(S), Corollary 3.5 in [1] then implies that H σssc 1 (S; Q) contains the isotypic components of all irreducible QG-modules afforded by H 1 (Γ(C σ ); Q). In particular, it contains the image by Φ of its dual H 1 (Γ(C σ ); Q).…”

Linear representations of hyperelliptic mapping class groups

“…More precisely, there is, for g ≥ 2, a natural Z/2-gerbe H g → M 0,[2g+2] defined by assigning to a genus g hyperelliptic curve C, the genus zero curve C/ι, where ι is the hyperelliptic involution of C, labeled by the branch points of the cover C → C/ι. In the genus 1 case, there is a Z/2-gerbe M 1,1 → M 0, [3]+1 , where by the notation " [3] + 1" we mean that there are 4 labels, one distinguished and the others unordered. There are natural isomorphisms H g ι ∼ = M 0,[2g+2] and M 1,1 ι ∼ = M 0, [3]+1 , where by we denote the operation of erasing the generic group of automorphisms from an algebraic stack.…”

“…In the genus 1 case, there is a Z/2-gerbe M 1,1 → M 0, [3]+1 , where by the notation " [3] + 1" we mean that there are 4 labels, one distinguished and the others unordered. There are natural isomorphisms H g ι ∼ = M 0,[2g+2] and M 1,1 ι ∼ = M 0, [3]+1 , where by we denote the operation of erasing the generic group of automorphisms from an algebraic stack.…”

“…This short exact sequence has the following geometric meaning (cf. Section 3 in [3]). Let S G B ∼ = S g,n , where B is the set of inertia points of S G .…”

“…On the other hand, the natural inclusion φ : G → Mod(S) determines the moduli stack M φ of complex G-curves of topological type φ (cf. Section 1 in [3]) whose fundamental group is isomorphic to Mod(S) G . There is a natural morphism M φ → M[K] which is a Z(G)gerbe and whose associated short exact sequence of fundamental groups identifies with the sequence (1).…”