Equations of hyperelliptic Shimura curves

Abstract

We describe a method for computing explicit models of hyperelliptic Shimura curves attached to an indefinite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation 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 [22]. As an application, we provide a list of equations of Shimura curves and quotients of them obtained by our method that had been conjectured by Kurihara.

The kernel of Ribet's isogeny for genus three Shimura curves

“…Now, we deal with the curve X (v) . Equations for D = 35, 51, 57, 69 can be found in [12]. Moreover, the procedure used in this paper allows us to determine which Weierstrass points come from fixed points of u.…”

“…Since for D = 35 this procedure gives two equivalence classes satisfying the previous condition, we use another argument in this case. Indeed, from [12] we know that…”

“…In [12], the second author gives a method to compute equations for Atkin-Lehner quotients of Shimura curves attached to an odd discriminant D which are hyperelliptic over Q. For the particular case that X D has genus 3, he provides in Table 2 of [12] equations for X D when D = 39, 55 and for X (v) when D = 35, 51, 57, 69.…”

“…For the particular case that X D has genus 3, he provides in Table 2 of [12] equations for X D when D = 39, 55 and for X (v) when D = 35, 51, 57, 69. In this section we show how one can obtain an equation for X D , even if X D is not hyperelliptic over Q, from certain particular equations for X (u) and X (v) .…”

“…In [12], it is given a method to compute equations for Atkin-Lehner quotients of Shimura curves attached to an odd discriminant D which are hyperelliptic over Q and some equations are computed. This method has a computational limitation since it exploits the MAGMA instruction IndexFormEquation that only applies for general number fields of degree at most four.…”