The sigma function for trigonal cyclic curves

Abstract

A recent generalization of the "Kleinian sigma function" involves the choice of a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$. This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at $P$. We exhibit the Riemann constant for a Weierstrass semigroup at $P$ with minimal set of generators $\{3, 2r+s,2s+r\}$, $r<s$, equivalently, non-symmetric, we construct a basis of $H^1(X, \mathbb{C})$ and a fundamental 2-differential on $X\times X$, we give the order of vanishing for sigma on Wirtinger strata of the Jacobian of $X$, and a solution to the Jacobi inversion problem.