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Note on the Gonality of Abstract Modular Curves

  • Anna Cadoret
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 2)

Abstract

Let S be a curve over an algebraically closed field k of characteristic \(p \geq 0\). To any family of representations \(\rho= ({\rho }_{\mathcal{l}}\, :\ {\pi }_{1}(S) \rightarrow {\mbox{ GL}}_{n}({\mathbb{F}}_{\mathcal{l}}))\) indexed by primes \(\mathcal{l} \gg0\) one can associate abstract modular curves \({S}_{\rho ,1}(\mathcal{l})\) and \({S}_{\rho }(\mathcal{l})\) which, in this setting, are the modular analogues of the classical modular curves \({Y }_{1}(\mathcal{l})\) and Y (). The main result of this paper is that, under some technical assumptions, the gonality of \({S}_{\rho }(\mathcal{l})\) goes to \(+\infty \) with \(\mathcal{l}\). These technical assumptions are satisfied by \({\mathbb{F}}_{\mathcal{l}}\)-linear representations arising from the action of π1(S) on the étale cohomology groups with coefficients in \({\mathbb{F}}_{\mathcal{l}}\) of the geometric generic fiber of a smooth proper scheme over S. From this, we deduce a new and purely algebraic proof of the fact that the gonality of \({Y }_{1}(\mathcal{l})\), for \(p \nmid\mathcal{l}({\mathcal{l}}^{2} - 1)\), goes to \(+\infty \) with .

Keywords

Normal Subgroup Short Exact Sequence Abelian Subgroup Open Subgroup Normal Abelian Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Centre Mathématiques Laurent SchwarzÉcole PolytechniquePalaiseauFrance

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