## Riemann Surfaces

#### A brief history of the subject:

Riemann surfaces were ﬁrst deﬁned in Riemann’s 1851 PhD dissertation  where he laid down the geometric foundations of the theory of functions of one complex variable. The theory was advanced much further in Riemann’s 1857 epoch-making paper on the theory of abelian functions . It took mathematicians many years to fully grasp, develop, and extend Riemann’s ideas to its modern status. Today, the theory of Riemann surfaces occupies a central place in modern mathematics and mathematical physics, specially in areas such as geometric analysis, potential theory and PDE’s, number theory, algebraic geometry, string theory, and conformal ﬁeld theory.

One should however not get the impression that Riemann surface theory started in Riemann’s thesis! Riemann was on the shoulders of some giants like Abel and Jacobi, not to mention Cauchy’s work in complex analysis, and the works of Gauss, Legendre, and Euler on elliptic integrals and elliptic functions. In fact as the names of some of the most important theorems and concepts in the subject would suggest, (e.g. Abel’s addition theorem, the Abel-Jacobi map, Jacobian of a Riemann surface, etc.), many deep results, at least in their germinal forms, were obtained by Abel and Jacobi more than twenty years prior to Riemann’s thesis. It is fair to say that Riemann surfaces were invented (or discovered!) by Riemann in order to put the whole theory of algebraic functions in one variable, as well as the theory of abelian functions, on a ﬁrm geometric and analytic basis and to forge ahead equipped with this new geometric insight.

This subject has somehow two distinct ﬂavors: algebraic geometric and complex manifold theoretic. One of Riemann’s results that we shall prove in this course states that any compact Riemann surface is algebraic, i.e. is the set of solutions of a homogeneous polynomial equation. Thus the two approaches are equivalent as long as we work over C. With the appearance of Weyl’s 1913 classic , Riemann’s approach and his results found its deﬁni- tive form. For more on the history of the subject and its development one can check [6, 2] and references therein. Remmert’s article carefully follows the evolution of the subject from Riemann to present times from the point of view of complex manifold theory, while Dieudonn´ treats it as part of algebraic geometry.

#### Outlines and Lecture notes:

Here is a tentative list of topics I plan to cover.

• Riemann surfaces and holomorphic maps between them, examples: alge- braic curves, Riemann surfaces from analytic continuation, diﬀerential equa- tions, conformal structures. Problem of moduli.
• Covering spaces and monodromy, Riemann’s existence theorem.
• Differential forms, de Rham and Dolbeault cohomology, Laplace operators, harmonic forms, The Dirichlet principle and Hodge theory à la Riemann.
• Elliptic integrals and functions, doubly periodic function, Weierstrass ℘- function, The ﬁeld of meromorphic functions, Theta functions, classiﬁcation of Riemann surfaces of genus one, the modular curve.
• The Riemann-Hurwitz formula, the degree-genus formula. Field of mero- morphic functions, birational equivalence, connections with algebraic number theory. Hyperbolic surfaces, Gauss-Bonnet theorem.
• The main theorem for compact Riemann surfaces and its consequences: The Riemann-Roch formula, the uniformization theorem, automorphisms of Riemann surfaces, Weierstrass points.
• Divisors, line bundles, sheaf cohomology, Jacobi’s inversion problem, Abel- Jacobi map, Jacobian of Riemann surfaces. Abelian varieties and abelian functions.
• Moduli spaces of Riemann surfaces, Beltrami diﬀerentials, compactiﬁcation of moduli spaces.
• Belyi’s theorem, Grothendieck's dessins dènfant.

#### Textbook: Lectures on Riemann Surfaces (Graduate Texts in Mathematics) by Otto Forster

Here are some other good texts on the subject (the list is by no means exhaustive):

• J.-B. Bost,Introduction to compact Riemann surfaces, Jacobians, and abelian varieties, in From number theory to physics (Les Houches, 1989), Springer,

Berlin.1992, pp. 64–211. This is a particularly inspiring text full of his- torical background and references to original works (check Abel’s amazing 2-page proof of his addition theorem, or his discovery of what later came to be known as the genus of a curve).

• S. Donaldson, Riemann Surfaces, Oxford Graduate Texts in Mathematics.
• H. M. Farkas, I. Kra, Riemann Surfaces.
• J. Jost, Compact Riemann Surfaces.
• F. Kirwan, Complex Algebraic Curves.

This one is more on the algebraic geometry side and at a more elementary level. Very well written and suitable for an advanced undergraduate course.

• H. McKean, V. Moll, Elliptic curves: function theory, geometry, arithmetic.

A good place to start learning about connections between Riemann surfaces and arithmetic. Covers Kronecker-Weber and Mordel-Weil theorems. Very close to the style of originals.

#### References:

 J.-B. Bost, , Introduction to compact Riemann surfaces, Jacobians, and abelian varieties , in From number theory to physics (Les Houches, 1989), Springer, Berlin.1992, pp. 64–211.

 J. Dieudonne´, History of Algebraic geometry.

 S. Donaldson, Riemann Surfaces Oxford Graduate Texts in Mathematics 22, 2011.

 H. M. Farkas, I. Kra, , Riemann Surfaces (2nd ed.), Berlin, New York: Springer-Verlag.

 J. Jost, Compact Riemann Surfaces, Berlin, New York: Springer-Verlag.

 R. Remmert, From Riemann Surfaces to Complex Spaces.

 B. Riemann, GGrundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse. Inauguraldissertation, Göttingen 1851;Werke, pp. 3-48.

 B. Riemann, Theorie der Abelschen Functionen, J. Reine Angew. Math., 54 (1857), pp. 115155; Werke, pp. 88-142.

 H. Weyl, Die Idee der Riemannschen Fläche. Teubner, Leipzig, 1913. Annotated reedition, 1997. The concept of a Riemann surface (3rd ed.), New York: Dover Publications.