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.
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.
| Original language | English |
|---|---|
| Title of host publication | The Legacy of Bernhard Riemann After One Hundred and Fifty Years |
| Publisher | Higher Education Press and International Press |
| Pages | 567-594 |
| Volume | II |
| Publication status | Published - 2016 |
Publication series
| Name | Advanced Lectures in Mathematics |
|---|---|
| Publisher | Higher Education Press and International Press |
| Volume | 35.2 |
Keywords
- Riemann surfaces
- algebraic curves
- coverings
- ramification
- Belyi’s theorem