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By Pierre Deligne

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If p + and p _ are distinct points on a marked compact Riemann surface M and z+ and z_ are any points of the universai covering space M lying over p + and p _ respectively, then associated to the divisor b = 1· p + -1' p _ is a complex analytic line bundle over M represented by the factor of automorphy Pr' where tEcr9 has the coordinates ti=wlz+)-wi(z_) in terms of the canonical Abelian integrals wi for M. Proof. Any factor of automorphy representing the line bundIe associated to the divisor b must have zero characteristic elass as a consequence of Theorem 2a; and it then follows from the weak form of Abel's theorem and Theorem 4 that this factor of automorphy is analytically equivalent to a flat factor of automorphy Pr for some tE cr g• The factor of automorphy Pr admits a meromorphic relatively automorphic function f having divisor b(f) = b on M.

Having found this function, the mapping h: M -+CC* defined by h(z) = exp 2ni f(z) is continuous and satisfies h(Tz) h(z) - 1 = i;(T, z) for all elements TET and points zEM; and therefore the factor of automorphy i; is continuously equivalent to the trivial factor of automorphy, and the proof of the theorem is coneluded. The same argument of course shows that two factors of automorphy are CD equivalent precisely when they have the same characteristic elass, upon using Ca) rather than merely continuous partitions of unity.

Thus the poles can be specified quite arbitrarily; and the argument extends quite easily to arbitrary singularities of higher order. Now let ef; be a meromorphic Abelian differential on M having nontrivial simple poles at two distinct points p + and p _; and after multiplying by a suitable constant, assume that ef; has residues + 1 at the point p + and -1 at the point p _ on M. The most general such differential is obtained from ef; by adding a complex analytic Abelian differential to ef;; and a canonical such differential can be specified by imposing some period restrictions, but some care must be taken in describing the periods of ef; since that differential form has singularities at which it is not locally exact.

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