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Session S24 - Symbolic Computation: Theory, Algorithms and Applications

Tuesday, July 20, 17:00 ~ 17:25 UTC-3

A sage package for n-gonal equisymmetric stratification of $\mathcal{M}_g$

Anita Rojas

Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will present an algorithm running over the software SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus $\mathcal{B}_g$ of the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g\geq 2$, corresponding to group actions with orbit genus $0$.

Our approach is towards studying inclusions and intersections of (closed) strata of $\mathcal{B}_g$. We apply our algorithm to describe part of the geometry of the branch locus $\mathcal{B}_9$, in terms of equisymmetric stratification. We also use it to compute all group actions up to topological equivalence for genus $5$ to $10$, this completes the lists existing in the literature. Finally, we add an optimized version of a known algorithm, which allows us to identify Jacobian varieties of CM-type. As a byproduct, we obtain a Jacobian variety of dimension $11$ which is isogenous to $E_i^{9}\times E_{i\sqrt{3}}^{2}$, where $E_i$ and $E_{i\sqrt{3}}$ are elliptic curves with complex multiplication.

Joint work with Antonio Behn (Pontificia Universidad Católica de Chile) and Miguel Tello-Carrera (Colegio Pedro de Valdivia).

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