Drmota, M.; Giménez, O.; Noy, M.; Panagiotou, K.; Steger, A. Proceedings of the London Mathematical Society Vol. 109, p. 892-920 DOI: 10.1112/plms/pdu024 Data de publicació: 2014-10-01 Article en revista
McDiarmid and Reed ['On the maximum degree of a random planar graph', Combin. Probab. Comput. 17 (2008) 591-601] showed that the maximum degree of a random labeled planar graph with vertices satisfies with high probability (w.h.p.)for suitable constants . In this paper, we determine the precise asymptotics, showing in particular that w.h.p. for a constant that we determine explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.
We describe a method for computing equations of hyperelliptic Shimura curves attached to indefinite quaternion algebras over Q and Atkin–Lehner quotients of them. It exploits Cerednik–Drinfeld 's non-archimedean uniformization of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points, as introduced in Molina [‘Ribet bimodules and specialization of Heegner points’, Israel Journal of Mathematics]. We provide a list of equations of Shimura curves and quotients of them obtained by our method that had been conjectured by Kurihara