The Riemann-Hurwitz formula
Publication date
2016
Authors
Oort, F.
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Part of book
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Abstract
Let ϕ : S → T be a surjective holomorphic map between compact Riemann surfaces. There is a formula relating the various invariants involved: the genus of S, the genus of T, the degree of ϕ and the amount of ramification. Riemann used this formula in case T has genus zero. Contemporaries referred to this general formula as ”Riemann’s theorem”. Proofs were given by Zeuthen and Hurwitz. We discuss this formula in its historical context, and in modern generalizations.
Keywords
Riemann surfaces, algebraic curves, coverings, ramification, Belyi’s theorem, Taverne
Citation
Oort, F 2016, The Riemann-Hurwitz formula. in The Legacy of Bernhard Riemann After One Hundred and Fifty Years. vol. II, Advanced Lectures in Mathematics, vol. 35.2, Higher Education Press and International Press, pp. 567-594.