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Combinatoire des arbres planaires et arithmétique des courbes hyperelliptiques. (Combinatorics of planar trees and arithmetics of hyperelliptic curves). (French) Zbl 0911.14013

This paper contains several interesting results relating certain trees and (hyper-)elliptic curves with an \(n\)-division point defined over some number field. These results are obtained in the framework of “dessins d’enfants” and Belyi functions.
The basic concept of the paper is what the author calls the abelian stratification of the set \(\widetilde \Sigma\) of equivalence classes of pairs \((P,I)\) consisting of a polynomial \(P\in \mathbb{C}[z]\) and a segment \(I\) joining two different complex numbers \(a,b\); here \((P,I)\) and \((P_1,I_1)\) are equivalent if they are related by linear polynomials \(\gamma\), \(\gamma_1\) through \(P_1= \gamma_1 \circ P\circ \gamma\) and \(I_1= \gamma_1(I)\). For \(g\) and \(n\) in \(\mathbb{N}\), the stratum \(\widetilde \Sigma_{g,n}\) consists of those pairs \(\sigma= (P,I)\) with \(\deg P=n\) such that the graph \(G_\sigma =P^{-1}(I)\) can be written as a union of \(g+1\) linear subgraphs, but not as a union of \(g\) such subgraphs. For \(g=0\), the set \(\widetilde \Sigma_{0,n}\) consists of a single point, but for \(g\geq 1\), \(\widetilde \Sigma_{g,n}\) can be mapped bijectively to the moduli space \(\widetilde H_{g,n}\) of hyperelliptic curves of genus \(g\) with a point \(\rho\) of order \(n\) on \(C\) (i.e. \(n \cdot( [\rho]- [\overline\rho])\) is a principal divisor, where \(\overline\rho\) is the image of \(\rho\) under the hyperelliptic involution). This map is obtained as follows: For \(\sigma= (P,[a,b]) \in \Sigma_{g,n}\) the author shows that \(P\) satisfies Abel’s equation \((P(z)-a) (P(z)-b)=Q^2_\sigma (z)\cdot R_\sigma (z)\), with a polynomial \(R_\sigma\) of degree \(2g+2\) that has only simple roots. Then to \(\sigma\) he associates the curve \(C\) with the affine equation \(y^2= R_\sigma (z)\) and one of the two points of \(C\) lying over \(\infty\). This concept is then applied to the subset \(S\widetilde \Sigma_{g,n}\) of Shabat polynomials, i.e. pairs \((P,I)\) where \(P\) has only two critical values and \(I\) is the segment joining them. It is a consequence of Belyi’s theorem that \((P,I) \mapsto P^{-1}(I)\) induces a bijection between \(S\widetilde \Sigma_{g,n}\) and the set \(\Lambda_{g,n}\) of trees with \(n\) edges and \(2g+2\) vertices of odd valency.
In the case of elliptic curves (i.e. \(g=1)\), there are only two possible types of trees: “\(Y\)-shaped” and “\(X\)-shaped”. In both cases the author calculates the exact order of the associated \(n\)-division point. The known results on torsion points on elliptic curves over number fields then yield estimates for the degree over \(\mathbb{Q}\) of the field of definition of the elliptic curve associated to a given tree. The author also uses old results on modular forms to describe the “\(Y\)-shaped locus” on the modular curve \(X_1(N)\) for any \(N\geq 1\). – Finally he uses his technique to construct hyperelliptic curves defined over \(\mathbb{Q}\) with a \(\mathbb{Q}\)-rational \(n\)-division point for any genus and any \(n\) in a certain range.

MSC:

14H25 Arithmetic ground fields for curves
14H30 Coverings of curves, fundamental group
14H52 Elliptic curves
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
11R32 Galois theory
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