Conjugates of strongly equivariant maps

Abstract

Let T : X\ -• X 2 be a strongly equivariant holomorphic embedding of one bounded symmetric domain into another. We show that if σ is an automorphism of C, then τ σ : X° -» X% is also strongly equivariant.

A Characterization of Certain Shimura Curves in the Moduli Stack of Abelian Varieties

“…Choose again an order O in A, and let O 1 be the group of units in O of reduced norm 1. For any discrete torsion free subgroupΓ ⊂ P ρ 1…”

“…As a preparation for the proof of Proposition 0.3 we will show in Section 1, using Simpson's correspondence between Higgs bundles and local systems, that the maximality of the Higgs field enforces a presentation of the local systems R 1 f * C X 0 and End(R 1 f * C X 0 ) using direct sums and tensor products of one weight one complex variation of Hodge structures L of rank two and several unitary local systems. Proposition 0.3 relates families reaching the Arakelov bound to totally geodesic subvarieties of the moduli space of abelian varieties, as considered by Moonen in [16], or to the totally geodesic holomorphic embeddings, studied by Abdulali in [1] (see Remark 2.5,b). As in [21] one could use the classification of Shimura varieties due to Satake [22] to obtain a complete list of those families, and to characterize them in terms of properties of their variation of Hodge structures.…”

Spectral analysis on planar mixed automorphic forms

“…Define j α ; α ∈ R, to be the automorphic factor given by j α (g, z) = e 2iα z,g −1 ·0 , g ∈ G, z ∈ C, (2.1) where here and elsewhere z denotes the imaginary part of the complex number z and ·,· denotes the usual hermitian scalar product on C. Also, let (ρ, τ ) be an equivariant pair [18,14,1,16]. That is, ρ is a G-endomorphism and τ : C → C a smooth compatible mapping such that τ (g.z) = ρ(g) · τ (z) (2.2) for every g ∈ G and z ∈ C. By Γ we denote a uniform lattice of the additive group (C, +) that can be seen as a discrete subgroup of G by the identification…”

Rational equivariant holomorphic maps of symmetric domains

“…It is known that the conjugate of such a Kuga fiber variety by an element σ ∈ Aut(C) is again a Kuga fiber variety determined by a totally geodesic map (cf. [1], [7]). Thus Theorem 4.8 holds true for such Kuga fiber varieties.…”