Eigenforms of hyperelliptic curves with many automorphisms
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Eduard Duryev
Abstract
Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve.
The underlying curves are hyperelliptic curves with many automorphisms. For each curve X, the automorphism group Aut(X) acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of the automorphism of maximal order.
Funding statement: During this work, the first author was supported by a FSMP postdoc grant, and the second author was supported by the EPSRC Early Career Fellowship EP/R023379/1.
Acknowledgements
This research has emerged from a question of Curt McMullen on representations of eigenforms as translation surfaces. We also thank the anonymous referee whose comments helped to improve the exposition of the paper.
Communicated by: M. Joswig
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Articles in the same Issue
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
Articles in the same Issue
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
