Heriot-Watt University

Department of Mathematics

Weierstrass functions for the Trigonal (3,4) curve

This page is intended to contain a repository of results referring to Weierstrass theory for the general trigonal curve of genus 3

y3 + (μ1x + μ4) y2 + (μ2x2 + μ5 x +μ8) y = x4 + μ3x36x2 + μ9x + μ12

See the following papers for details of the theory:

Abelian Functions for Trigonal Curves of Genus Three
J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Ônishi, and E. Previato, International Mathematics Research Notices, 2007, Art.ID: rnm140 (38 pages), 2007, Abstract, PDF Copy.
Abelian functions associated with genus three algebraic curves
J. C. Eilbeck, M. England and Y. Ônishi, Journal of Computation and Mathematics 14, 291-326 (2011). Full journal article and related depository.

The following file contains the first seven terms in the expansion of equation (4.6) in the above paper zw_equations_gen34.txt. These are written in Maple format in an obvious notation, but note we use (z,w) in this file instead of (x,y).

The following file contains all the relations satisfied by the 4-index Pijkl functions: known_4i.txt . These are written in Maple format in an obvious notation.

The following file contains some of the quadratic relations satisfied by the 3-index Pijk functions: known_q.txt .

The following file contains some of the bilinear relations satisfied by the 2- and 3-index Pij, Pijk functions: lin.txt .

The following file contains the terms of the expansion of the σ function up to terms of Sato weight 18: Ci.txt .

The following file contains the Coble hypersurface associated with the curve: Coble4.txt .

Results for the corresponding genus 4 trigonal curve, the (3,5) curve, will be found at
Abelian Functions for Purely Trigonal Curves of Genus four
S. Baldwin, J. C. Eilbeck, J. Gibbons, and Y. Ônishi, math.AG/0612654, (2006).

Maintained by Chris Eilbeck/Heriot-Watt University, Edinburgh/ J.C.Eilbeck@hw.ac.uk