## 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
*y*^{3} + (μ_{1}x + μ_{4}) y^{2}
+ (μ_{2}x^{2} + μ_{5} x
+μ_{8}) y = x^{4} + μ_{3}x^{3}
+μ_{6}x^{2} + μ_{9}x +
μ_{12}

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
*P*_{ijkl} 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 *P*_{ijk} functions: known_q.txt .

The following file contains some of the bilinear relations satisfied
by the 2- and 3-index *P*_{ij},
*P*_{ijk} 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