Universal periods of hyperelliptic curves and their applications

We construct universal power series for differential 1-forms and period integrals of Schottky–Mumford uniformized hyperelliptic curves over local fields. Using these universal 1-forms and periods, we characterize Siegel modular forms vanishing on the hyperelliptic Jacobian locus, and construct universal and p-adic solutions of the Korteweg–de Vries hierarchy.