I need to learn the basic theory of Riemann surfaces and would like to pick a book which is most relevant to algebraic number theory. I have a good understanding of all underdraduate algebra and the basic theory of one complex variable. I do not have any prior exposure to algebraic geometry however. Can anyone recommend a book which would be a good match?
Many thanks!
The basic theory of Riemannian surfaces can be found in many books, see for example the literature given here. An important application to number theory is also the question on the spectrum of the Laplacian in Riemannian Geometry. This has a lot of number theory involved, see Selberg's eigenvalue conjecture. For literature, see also here; also Peter Sarnak's article Selberg's eigenvalue conjecture gives a nice survey, with "pictures" of the Riemannian surfaces $\mathbb{H}/\Gamma(N)$ involved. These surfaces are central from many points of view, for example, in the formulation of the (former) Shimura-Taniyama conjectures and the work of Andrew Wiles et al.
