Abstract
We construct an Euler system for Galois representations associated to cohomological cuspidal automorphic representations of $\operatorname{GSp}\_4$, using the pushforwards of Eisenstein classes for $\operatorname{GL}\_2 \times \operatorname{GL}\_2$.
Highlights
Using the theory of Λ-adic Eisenstein classes initiated by Kings, we show that these Euler system classes can be interpolated p-adically as the parameters vary
This leads to a definition of a “motivic p-adic L-function” for Π, which is a p-adic measure on Z×p interpolating the images of the Euler system classes under the Bloch–Kato logarithm and dual-exponential maps at p
Our motivic p-adic L-function should interpolate the critical values of the spin L-function of Π
Summary
We fix a prime and collect some definitions and results regarding smooth representations of the groups GL2(Q ), G(Q ), and H(Q ) on complex vector spaces. Let M : I(χ, ψ) → I(ψ, χ) be the normalised standard intertwining operator, defined by analytic continuation to s = 0 of the integral. If χ/ψ = |·|−1 we interpret the right-hand side as 0, so the elements fφ,χ,ψ all land in the 1-dimensional subrepresentation Let us evaluate these integrals explicitly for some specific choices of φ, assuming that χ and ψ are unramified characters. We let χ1 × χ2 ρ denote the representation of G afforded by the space of smooth functions f : G → C satisfying f a∗ ∗ ∗. |a2b| = |c|3/2 χ1(a)χ2(b)ρ(c)f (g), with G acting by right translation We refer to such representations as irreducible principal series. We consider the action of the spherical Hecke algebra H(KG \G /KG ) on σKG when σ is an unramified principal series representation.
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