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The sigma function for trigonal cyclic curves. (English) Zbl 1441.14145
The Weierstrass semigroup of a complex pointed curve \((X,p)\) is the complement of the Weierstrass gap sequence at \(p\). It determines the Weierstrass normal form of \(X\) with respect to \(p\).
In the present article, the authors use the Weierstrass normal form of a given pointed trigonal cyclic curve \((X,p)\) to obtain a number of detailed results about differentials and the sigma function. For instance, they obtain an explicit description of a basis for the vector space of differential one forms and thus coordinates for the canonical embedding. They also obtain detailed results which apply to differentials of the second and third kinds.
The authors then turn to the study of the Jacobian of \(X\). As one result, they study the sigma function and present a solution to the Jacobi inversion problem. Many of the authors methods are sufficiently general so as to apply to the more general case of \(k\)-gonal cyclic covers of \(\mathbb{P}^1\).

MSC:
14K25 Theta functions and abelian varieties
14H40 Jacobians, Prym varieties
14H55 Riemann surfaces; Weierstrass points; gap sequences
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