We use the "higher Hida theory" recently introduced by the second author to
p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4),
contributing to coherent cohomology of Siegel threefolds in positive degrees.
We apply this new method to construct p-adic L-functions associated to the
degree 4 (spin) L-function of automorphic representations of GSp(4), and the
degree 8 L-function of GSp(4) x GL(2).Comment: Updated with minor corrections. To appear in "Duke Math Journal" (see
https://projecteuclid.org/accepted/euclid.dmj
