Compact Riemann Surfaces (Lectures in Mathematics. ETH by R. Narasimhan

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.

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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.

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