A norm compatible system of Galois cohomology classes for $\mathrm{GSp}(4)$

Abstract

For a given de Rham $p$-adic Galois representation $M$, a conjecture of Perrin–Riou associates a $p$-adic $L$-function for $M$ to a norm compatible system of Galois cohomology classes in the projective limit $\underleftarrow{\lim }{}_n{\mathrm{H}}^1(Q({\zeta }_{p^n}),M)$. We construct such a norm compatible system for the symplectic group $\mathrm{G}{\overleftarrow{\mathrm{Sp}}}_4$. Our classes are cup-products of torsion sections of the large elliptic polylogarithm pro-sheaf; we rely on its norm compatibility and on some computations of weights in the cohomology of Siegel threefolds of our previous work.