Generalizations of Thomae's Formula for Zn Curves
This book provides a comprehensive overview of the theory of theta functions, as applied to compact Riemann surfaces, as well as the necessary background for understanding and proving the Thomae formulae.The exposition examines the properties of a particu
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Developments in Mathematics VOLUME 21 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California
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GENERALIZATIONS OF THOMAE’S FORMULA FOR Zn CURVES HERSHEL M. FARKAS The Hebrew University, Institute of Mathematics SHAUL ZEMEL The Hebrew University, Institute of Mathematics
Hershel M. Farkas Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond J. Safra Campus, Givat Ram Jerusalem, 91904, Israel [email protected]
Shaul Zemel Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond J. Safra Campus, Givat Ram Jerusalem, 91904, Israel [email protected]
ISSN 1389-2177 ISBN 978-1-4419-7846-2 e-ISBN 978-1-4419-7847-9 DOI 10.1007/978-1-4419-7847-9 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification (2010): 20F10, 14H42
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To our wives, Sara and Limor
Introduction
In this book we shall present the background necessary to understand and then prove Thomae’s formulae for Zn curves. It is not our intention to actually give the proofs of all the necessary results concerning compact Riemann surfaces and their associated theta functions, but rather to give the statements which are necessary and refer the reader to the book [FK] for many of the proofs. Sometimes we shall sketch some proofs for the benefit of the reader. We begin with the objects that we shall be studying. A (fully ramified) Zn curve is the compact Riemann surface associated to an equation t
wn = ∏(z − λi )αi , i=1
where λi ̸= λ j if i ̸= j and the powers αi , 1 ≤ i ≤ t satisfy 1 ≤ αi ≤ n − 1, are relatively prime to n, and their sum is either divisible by n or relatively prime to n. Thus a concrete representation of the curve as a compact Riemann surface is that of an n-sheeted branched cover of the sphere with full branching over each point λi . We shall generally assume that there is no branching over the point at ∞, which is equivalent to the statement that ∑ti=1 αi is divisible by n. The Riemann–Hurwitz formula will now give that 2(g + n − 1)
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