Period relations for cusp forms of GSp(4)

Let F be a totally real number field and let pi be a cuspidal automorphic representation of GSp(4)(A F), which contributes irreducibly to coherent cohomology. If pi has a Bessel model, we may attach a period p(pi) to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods p(pi) and all of their twists p(pi circle times xi) under arbitrary algebraic Hecke characters xi. In the appendix, we show that (g, K)-cohomological cusp forms of GSp(4)(A F) all qualify to be of the above type - providing a large source of examples. We expect that these period relations for GSp(4)(A F) will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, whichwe hope to report on in Part II of this paper