Generalizations of Thomae's Formula for Zn Curves by Hershel M. Farkas

Algebraic Geometry

By Hershel M. Farkas

This ebook offers a accomplished evaluate of the idea of theta capabilities, as utilized to compact Riemann surfaces, in addition to the required heritage for figuring out and proving the Thomae formulae.

The exposition examines the homes of a selected type of compact Riemann surfaces, i.e., the Zn curves, and thereafter makes a speciality of tips on how to turn out 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 features, whereas younger researchers, whose pursuits are with regards to complicated research, algebraic geometry, and quantity conception, will locate new wealthy components to discover. Mathematical physicists and physicists with pursuits additionally in conformal box concept would definitely savour the great thing about this subject.

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In fact we have actually shown a lot more. In place of the differentials we chose, zl dz/wk , we could just as well have chosen the differentials (z − λi )l dz/wk for some 1 ≤ i ≤ nr − 2 and the conclusion would be a basis adapted to the point Pi . When writing the differentials this way we need not assume that λ0 = 0 and the conclusion holds just as well. We also note that for any such i, k and l, the order at P∞ of (z − λi )l dz/wk is (kr − l − 1)n − 1 − k, and dividing it by n and again checking the integral and fractional values of the result gives ⌊ ⌋ { } (kr − l − 1)n − 1 − k (kr − l − 1)n − 1 − k 1+k = kr − 2 − l, = 1− .

6]. 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 .

Rn−1 Proof. The divisor of the function w is ∏rn−2 , and hence the image of i=0 Pi /P∞ this divisor by φP0 is 0. By the previous lemma we also have φP0 (P∞n ) = 0 and thus also φP0 (P∞rn ) = 0, and it is also clear that φP0 (P0 ) = 0. Combining all this we obtain the assertion of the lemma. ⊓ ⊔ We note [( here that )what ]we shall really need is the vanishing of the expresn−2 sion φP0 ∏rn−2 P∞n−2 . This fact can be also proved in another way, using i=1 Pi the differential (z − λ0 )r−2 dz/w, whose (canonical) divisor is simply the divisor n(r−1)−2 ( rn−2 n−2 ) n−2 P0 P∞ (this is not an integral divisor if r = 1, but it makes no ∏i=1 Pi difference).

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