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Abelian functions for cyclic trigonal curves of genus 4. (English) Zbl 1211.37082
Summary: We discuss the theory of generalized Weierstrass \(\sigma\) and \(\wp\) functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case [Tokyo J. Math. 27, No. 2, 299–312 (2004; Zbl 1160.11332)]. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve \(y^3=x^5+ \lambda_4x^4+ \lambda_3x^3+ \lambda_2x^2+ \lambda_1x+ \lambda_0\) is discussed in detail, including a list of some of the associated partial differential equations satisfied by the \(\wp\) functions, and the derivation of addition formulae.

MSC:
14H45 Special algebraic curves and curves of low genus
14H55 Riemann surfaces; Weierstrass points; gap sequences
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
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