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)


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 .


Normal Subgroup Short Exact Sequence Abelian Subgroup Open Subgroup Normal Abelian Subgroup 
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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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