Calculating the Period Matrix of a Shiga Curve, $y^3 = x^4-1$.

Thanks to Dami Lee for patiently walking through how to compute the period matrix of this 12-fold cyclic cover of a thrice punctured sphere, and thanks Matthias Weber for showing me how to write the 3 fold cover of a 5-punctured sphere as a 12-fold cyclic cover of a thrice punctured sphere. Most of the figures are either hand-drawn or made using Geogebra. Note that I will sometimes use $\tau$ to denote $2 \pi$. All errors are mine and mine alone.

Motivational Sidenote:
This is part of my project in attempting to understand the notion of height in terms of the symmetry of the underlying variety in formal group law theory.

Though it may seem disparate, this post is the computation of the Jacobian of a Shiga curve shown to have height 3 properties by Sebastien Thyssen and Hanno van Woerden. I suspect that the Jacobian of this curve is indeed my abelian variety constructed as a variety with complex multiplication in my last post under the guidance of Kato. Continue reading Calculating the Period Matrix of a Shiga Curve, $y^3 = x^4-1$.