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We determine a method to find explicit defining equations for each compact Riemann surface which admits a cyclic group of automorphisms C p of prime order p such that the quotient space has genus 0.

Letν′ be the complementary in a compact Riemann surface ν of a point (or a finite set). In this paper are characterized the subfields, of the field of meromorphic functions inν′, containing sufficient functions to verify a factorization property, similar to that of the classical Weierstrass theorem. It is also seen that the field generated by the Baker functions is not of this type, and the problem is solved of determining the divisors, inν′, of the holomorphic functions admiting Weierstrass factorizations with Baker functions as factors. As an application, a theorem is obtained characterizing the infinite products, of meromorphic functions in ν with bounded degree, which converge normally inν′.

Let Γ g, n be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n

We give an elementary proof of continuity of the determinant in the parameter for a smooth family of laplacians (of the same nullity) on a smooth family of holomorphic vector bundles over a compact complex manifold. Families of unitary flat bundles over a compact Riemann surface are discussed, as an example.

We show that the vanishing of the Riemann theta function on some subvarieties of the Jacobi varieties of a compact Riemann surface determines the gap sequence at the base point of the Jacobi homomorphism. This gives other defining equations for the varieties of Weierstrass pointsW defined by Lax.

A degeneration of a compact, two-sheeted, Riemann surface in genus 2 is studied. Two branch points coalesce when the period matrix for the surface is degenerated. A Siegel modular transformation is applied to give the degenerating period matrix in the form of a corner matrix.

It is shown that a recent compactness theorem for Fuchsian groups, due to Mumford, remains valid for groups containing elliptic and parabolic elements.

A problem in the degeneration of a compact, two-sheeted, Riemann surface of genus 2 is studied, using theta function techniques. The three moveable branch points coalesce to the fourth branch point on the first limiting surface while the triangles formed by these points are all essentially similar. Applying a conformal map, we see that these points represent the finite branch points on the second of the limiting surfaces.