Prime form and sigma function

2012/04/17 by Gibbons, John, Matsutani, Shigeki, Onishi, Yoshihiro
#14K25 (Primary) 14H55 #20C30 #37K15 (Secondary) #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences

paper · doi:10.48550/arxiv.1204.3747

Abstract

In this article, we study some cyclic (r,s) curves X given by y^r =x^s + λ₁ x^s-1 +...+ λ_s-1 x + λ_s. We give an expression for the prime form \cE(P,Q), where (P, Q ∈ X), in terms of the sigma function for some such curves, specifically any hyperelliptic curve (r,s) = (2, 2g+1) as well as the cyclic trigonal curve (r,s) = (3,4), \cE(P,Q) =\fracσ_\natural_r(u - v)√(du₁)√(d v₁), where \natural_r is a certain index of differentials. Here u₁ and v₁ are respectively the first components of u = w(P) and v = w(Q) which are given by the Abel map w: X → \CC^g, where g is the genus of X.

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