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abelian extension abelian group Accola Algebraic automorphism groups canonical divisor canonical homology basis characterizations closed Riemann surface component Corollary 9 corresponding cyclic group defined denote differentials divisor of degree Edited elliptic elliptic surfaces elliptic-hyperelliptic involutions equation Farkas fixed points fixed-point-free involution formula 21 four four-group genus five admitting group of automorphisms group of order half-canonical series half-periods homomorphism hyperelliptic involution hyperelliptic surfaces integral divisor invariant under G isomorphic J(WQ Krazer Lemma lift linear series meromorphic function order eight order theta function period characteristic period matrix Proceedings 1974 proof quarter-periods quotient genus ramified abelian cover Riemann sphere Riemann surface Riemann-Hurwitz formula Riemann-Roch theorem subgroup Suppose surface of genus surfaces admitting T-families Theory theta function theta function vanishes theta-characteristics two-sheeted cover unramified vanishes to order Weierstrass points z2 x z2 zero