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We solve the problem of embedding the universal cover of the Jacobian of any Riemann surface, X, into the “universal Jacobian” H1/2(S1), (see [7]). The latter is the Sobolev space of half-order differentiable functions on the unit circle. As the complex structure on X is deformed, we show how to define a (T (X)-parametrized) family of these natural embeddings, such that they provide a holomorphic homomorphism of a vector bundle over the Teichmüller space T (X), into the tautological vector bundle over the universal Siegel space which commutes with the universal period map. In sections 6 and 7 we study the above embeddings with reference to towers of Jacobians arising from towers of finite coverings of Riemann surfaces. We can pass to inductive limits and see the role of the commensurability automorphism groups acting on the limit objects.

Jacobians of Riemann surfaces and the Sobolev space $H^{1/2}$ on the circle