By Hershel M. Farkas
This booklet presents a accomplished review of the speculation of theta features, as utilized to compact Riemann surfaces, in addition to the mandatory historical past for figuring out and proving the Thomae formulae.
The exposition examines the homes of a specific type of compact Riemann surfaces, i.e., the Zn curves, and thereafter specializes in the right way to end up the Thomae formulae, which offer a relation among the algebraic parameters of the Zn curve, and the theta constants linked to the Zn curve.
Graduate scholars in arithmetic will enjoy the classical fabric, connecting Riemann surfaces, algebraic curves, and theta capabilities, whereas younger researchers, whose pursuits are concerning complicated research, algebraic geometry, and quantity thought, will locate new wealthy parts to discover. Mathematical physicists and physicists with pursuits additionally in conformal box thought would certainly savor the wonderful thing about this subject.
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Extra resources for Generalizations of Thomae's Formula for Zn Curves
This increases r(1/∆ ) from 1 to at least g, as needed by Riemann–Roch to show that ∆ is canonical. Therefore for every canonical divisor ∆ (integral or not) we have φP0 (∆ ) = −2KP0 , a fact which will be very useful in this book. We have seen that for an integral divisor ∆ of degree g on a Riemann surface there is a significant distinction between the cases i(∆ ) = 0 and i(∆ ) ≥ 1. 9 that the condition i(∆ ) = 0 is an open condition on the set of integral divisors of degree g, we say that such a divisor is special if i(∆ ) ≥ 1, and non-special if i(∆ ) = 0.
9. Now, if i (∏/gi=1 Qi ≥)1, then by the Riemann– Roch Theorem we see that the dimension of L 1 ∏gi=1 Qi is ≥ 2, so that there is a nonconstant function h in this space. If P is not any of the points Qi , 1 ≤ i ≤ g, then choosing a number c ∈ C such that P is a zero of( h − c yields ) an integral divisor ∆ of degree g containing P such that φP0 (∆ ) = φP0 ∏gi=1 Qi (by the Abel Theorem), and hence e − φP0 (P) has the presentation φP0 (∆ /P) + KP0 with ∆ /P integral of degree g − 1. 9 yields (together with the fact that the theta function is even) the identical vanishing of θ (φP0 (P) − e, Π ) = 0.
1 First Properties of Theta Functions In order to define the theta function one does not have to begin with Riemann surfaces but rather with the Siegel upper halfplane. For a natural number g ≥ 1, the 14 1 Riemann Surfaces Siegel upper halfplane of degree g is the set consisting of all the complex symmetric g × g matrices Π such that their imaginary part (viewed as a real symmetric g × g bilinear form) is positive definite. This is a complex manifold of dimension g(g + 1)/2 which is denoted by Hg .