Chapter

A Database of Group Actions on Riemann Surfaces

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Abstract

The automorphism group of a Riemann surface is an important object in a number of different mathematical fields. An algorithm of Thomas Breuer determines all such groups for a fixed genus given a complete classification of groups up to a sufficiently large order, but data generated from this algorithm did not include the generators of the corresponding monodromy group, another crucial piece of information for researchers. This paper describes modifications the author made to Breuer’s code to add the generators, as well as other new code to compute additional information about a given Riemann surface. Data from this project has been incorporated into the L-functions and Modular Forms Database (http://www.lmfdb.org) and we also describe the relevant data which may be found there.KeywordsRiemann surfacesAutomorphism groupsGroup actionsSurface kernel epimorphismsFuchsian groupsAlgebraic curvesAutomorphismsAMS classification14H3720H1030F20

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... , [g r ]) in C r G /Σ r . Note that this definition is slightly different from those of [24] and [29]: we do not assume that a refined passport comes from a datum. ...
... There is another useful criterion, already used by Breuer [6] and Paulhus [29]. Indeed, for some elements c, one can ascertain a priori that π −1 (c) = p −1 (c) does not contain any system of generators at all. ...
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The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of Ag{\mathsf {A}}_g generically contained in the Jacobian locus. Counterexamples are known for g7g\le 7. They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: (a) the Galois group is cyclic, (b) it is abelian and the family is 1-dimensional, or c) g9g\le 9. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for g100g\le 100 there are no other families than those already known.
... Note that this definition is slightly different from those of [22] and [28]: we do not assume that a refined passport comes from a datum. ...
... There is another useful criterion, already used by Breuer [5] and Paulhus [28]. Indeed, for some elements c, one can ascertain a priori that π −1 (c) = p −1 (c) does not contain any system of generators at all. ...
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The Coleman-Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of Ag\mathsf{A}_g generically contained in the Jacobian locus. Counterexamples are known for g7g\leq 7. They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: a) the Galois group is cyclic, b) it is abelian and the family is 1-dimensional, and c) g9g\leq 9. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for g100g\leq 100 there are no other families than those already known.
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