MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Crossing the transcendental divide: from Schottky groups to algebraic curves

Samantha Fairchild and Ángel David Ríos Ortiz


Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus $g\geq 3$ where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an $M$-curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces.


Related publications

2024 Repository Open Access
Samantha Fairchild and Ángel David Ríos Ortiz

Crossing the transcendental divide : from Schottky groups to algebraic curves