By R. Narasimhan

Those notes shape the contents of a Nachdiplomvorlesung given on the Forschungs institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. ok. Chandrasekharan and Prof. Jurgen Moser have inspired me to write down them up for inclusion within the sequence, released by way of Birkhiiuser, of notes of those classes on the ETH. Dr. Albert Stadler produced exact notes of the 1st a part of this direction, and intensely intelligible class-room notes of the remainder. with out this paintings of Dr. Stadler, those notes shouldn't have been written. whereas i've got replaced a few issues (such because the facts of the Serre duality theorem, right here performed solely within the spirit of Serre's unique paper), the current notes stick to Dr. Stadler's rather heavily. My unique target in giving the path was once twofold. i needed to offer the elemental theorems concerning the Jacobian from Riemann's personal standpoint. Given the Riemann-Roch theorem, if Riemann's equipment are expressed in sleek language, they fluctuate little or no (if in any respect) from the paintings of recent authors.

**Read or Download Compact Riemann Surfaces (Lectures in Mathematics. ETH Zurich) PDF**

**Similar calculus books**

**Plane Waves and Spherical Means: Applied to Partial Differential Equations**

The writer want to recognize his legal responsibility to all his (;Olleagues and neighbors on the Institute of Mathematical Sciences of recent York college for his or her stimulation and feedback that have contributed to the writing of this tract. the writer additionally needs to thank Aughtum S. Howard for permission to incorporate effects from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for his or her cooperation and aid, and especially Lipman Bers, who recommended the e-book in its current shape.

**A Friendly Introduction to Analysis**

This booklet is designed to be an simply readable, intimidation-free consultant to complicated calculus. rules and strategies of evidence construct upon one another and are defined completely. this is often the 1st e-book to hide either unmarried and multivariable research in this kind of transparent, reader-friendly surroundings. bankruptcy themes disguise sequences, limits of features, continuity, differentiation, integration, limitless sequence, sequences and sequence of features, vector calculus, features of 2 variables, and a number of integration.

This ebook, meant as a realistic operating advisor for calculus scholars, contains 450 workouts. it's designed for undergraduate scholars in Engineering, arithmetic, Physics, or the other box the place rigorous calculus is required, and should vastly profit an individual looking a problem-solving method of calculus.

- Spectral Theory of Self-Adjoint Operators in Hilbert Space
- The Continuum. A constructive approach to basic concepts of real analysis
- Perturbation methods
- Repetitorium Funktionentheorie: Mit über 200 ausführlich bearbeiteten Prüfungsaufgaben

**Extra info for Compact Riemann Surfaces (Lectures in Mathematics. ETH Zurich)**

**Example text**

If C is the set of critical points of f, and B = f(C), we have B C 1P'1 - {oo} = C Moreover, orda(j) = 2 if a E C, so that f-l(b) consists of exactly one point if bE B. ) = a for a E C) and we have 1'2 = identity. It is called the hyperelliptic involution of X. By the Riemann-Hurwitz formula, if X has genus g, we have since 1P'1 has genus 0, 2g-2=-4+ L(orda(f)-l), (~EC so that (since orda(j) = 2 for a E C), the number of branch points of each of these is a Weierstrass point of X). 2 E iC(z) so that (writing z for f(x)) we have u being holomorphic at both :r and r(x).

In fact, if X is hyperelliptic and f is of degree 2, we can take for D the divisor of poles of f [hO(D) 2': 2 since (1) 2': -D, (1) 2': -D]. Conversely, if hO(D) 2': 2, there is a non-constant f with (j) 2': - D. k=l and hl(Dm) we have = 0 for m 2': 2g - 1, while hO(Do) = hO(Dm) = m - 9 1, hl(Do) = g. Thus, for m 2': 2g - 1, + 1. Moreover, hO(Dm) -1 = 'L,{, (hO(Dk) - hO(Dk-l)) is the number of k :::;m which occur as the order of pole at P of an f E qX) holomorphic on X - P. Thus, the number "gaps" is m - (hO(Dm) - 1) = g.

Wg)' as I runs over HI (X, Z). We have Aak = (0, ... , 1, 0) = ek, the vector in reg with 1 in the k-th place and 0 elsewhere, and Ab> = (Ibk WI, , Ibk wg) (= Bk say) consists of the columns of the matrix B = (Bjk), Bjk = Ibjwk. Since Im(B) is positive definite, the vectors {el, ... ,eg,Bl, ... ,Bg} are linearly independent over JR. Since {ai, bj} generate HI (X, Z), we also have A = Zel + ... + Zeg + ZBl + ... + ZBg. These remarks imply that A is a lattice in lC9 with a compact quotient ABEL'S THEOREM.