Abstract
We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of {{,mathrm{GL},}}_3 over imaginary quadratic fields, using the cohomology of Shimura varieties for {text {GU}}(2, 1).
Highlights
1.1 Overview of the resultsEuler systems – families of global cohomology classes satisfying norm-compatibility relations – are among the most powerful tools available for studying the arithmetic of global Galois representations
Euler systems come in two flavours: full Euler systems, in which we have classes over almost all of the ray class fields E[m], where E is some fixed number field; or anticyclotomic Euler systems, where E is a CM field, and we restrict to ring class fields
ArtE : E×\(E× ⊗ a category of G (Af) )× →∼ Gal(E /E)ab is the Artin reciprocity map of class field theory, normalized so that geometric Frobenius elements are mapped to uniformizers, the map π0(YG ) ∼= E×\(E× ⊗ Af )× is Gal(E /E)-equivariant if we let σ ∈ Gal(E /E) act on E×\(E× ⊗ Af )× as multiplication by ArtE (σ )−1
Summary
Euler systems – families of global cohomology classes satisfying norm-compatibility relations – are among the most powerful tools available for studying the arithmetic of global Galois representations. Applying the étale regulator map and projecting to a cuspidal Hecke eigenspace, we obtain Euler systems in the conventional sense – as families of elements in Galois cohomology – associated to cohomological automorphic representations of G(A). Combining this with known theorems relating automorphic representations of G and of GL3 /E, we obtain the following: Theorem B Let be a RAECSDC1 automorphic representation of GL3 /E which is unramified and ordinary at the primes p | p. In the present paper we shall focus solely on the construction of the Euler system classes
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