Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups
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483 Robert D. M. Accola
ETHICS ETH-HB *00100000135731*
II II U~IUII II II UlIII II MII III Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups m
Springer-Verlag
Berlin. Heidelberg. NewYork 1975
Author Prof. Robert D. M. Accola Department of Mathematics Brown University Providence, R.I. 02912 USA
Library of Congress Cataloging in Publication Data
Accola, Robert D M 1929Riemann sufaoes, theta functions, and abelian aut omorphism groups. (Lecture notes in mathematics ; 483) Bibliography: p. Includes index. i. Riemann surfaces. 2. Functions, Theta. 5. Aut omorphisms. 4. Abe lian groups. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 483. QA3.L28 no. 483 [QA333] 510'.8 [515'.223] 75-25928
AMS Subject Classifications (1970): 14 H40, 30A46 ISBN 3-540-07398-1 ISBN 0-387-07398-1
Springer-Verlag Berlin Heidelberg 9 New 9 York Springer-Verlag New York Heidelberg 9 Berlin 9
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Contents
Part I I Introduction
I
2 Remarks on general coverings
3 Resum~ of the Riemamnvamishlmg theorem RAmified normal coverings
7 8
5 Abeliancovers
12
6 Main ~esults
19
Part II i Introduction
32
2 Completely ramified abelian covers
@0
3 Two-sheeted covers
50
@ Other applications
56
5 Closing remarks
63
Part III i Introduction
66
2 Castelnuovo's method and P0-hyperellipticity
70
3 Extensions
7@
The p - 2 conjecture for p = 5
79
5 Elliptic-hyperelliptic surfaces of genus five
81
6 Elliptic-hyperelliptic surfaces of genus three
88
7 Cyclic groups of order three for genus two
9@
8 Some local characterizations
95
9 Closing remarks
98
References
100
Index
102
PA2~ X x)
i.
Introduction.
Torelli's
type of a Riemann surface
theorem states
is determined
class of) one of its period matrices. some property
If a Riemann surface has
by some property
a property which
period matrix at hand. characterizations problem,
by {the equivalence
not shared by all Riemann surfaces
should be characterized hopefully
that the conformal
then this fact
of the period matrix,
is independent
of the particular
The main tool for effecting
is Riemann's
solution
often called Riemann's
such
to the .~cobi inversion
vanishing
theorem.
Riemann's
theorem relates vanishing
properties
of the theta function
the. Jacobian of a surface
to certain
linear series on the surface.
Since special properties existence
of special
will be reflected, properties
on a Riemann
linear series,
via Riemann's
for
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