claimroutine

Quaternionic Shimura curve quotient has five rational branch values

Gives genus 5 Shimura curve with X/W≅P1X/\mathcal W\cong\mathbf P^1, five branch values
supplies the exceptional set C
Claim·§4.1 The five-point quaternionic quotient·inside thmHeight bounded by radical of contact primes

Statement (paper)

Let B/QB/\Q be the indefinite quaternion algebra of discriminant 210210, O\mathcal O a maximal order, G=B×G=B^\times, X=Sh⁡O^ ×(G,H±)X=\operatorname{Sh}_{\widehat{\mathcal O}^{\,\times}}(G,\mathfrak H^\pm) and W≃(Z/2Z)4\mathcal W\simeq(\Z/2\Z)^4 the Atkin--Lehner group over Q\Q. Then XX has genus 55, and X∗=X/WX^*=X/\mathcal W has genus 00 and is isomorphic over Q\Q to P1\mathbf P^1. The map X→X∗X\to X^* has exactly five geometric branch values, one for each of the labels m=30,42,70,105,210m=30,42,70,105,210, and each is rational. Fixing an isomorphism that sends one branch value to ∞\infty, these five branch values form the set C⊂P1(Q)\mathcal C\subset\mathbf P^1(\Q).

In plain English

For the indefinite quaternion algebra BB of discriminant 210210 with maximal order O\mathcal O, the Shimura curve XX has genus 55, and its quotient X∗=X/WX^*=X/\mathcal W by the Atkin–Lehner group W≃(Z/2Z)4\mathcal W\simeq(\Z/2\Z)^4 is isomorphic over Q\Q to P1\mathbf P^1. The map X→X∗X\to X^* has exactly five geometric branch values, one for each label m=30,42,70,105,210m=30,42,70,105,210, all rational, and after sending one to ∞\infty they form the set C⊂P1(Q)\mathcal C\subset\mathbf P^1(\Q).

Why it's here

This claim supplies the geometric input for the height bound by producing the five-point set C\mathcal C, including ∞\infty, as the branch locus of a rational Shimura curve quotient. The later claims build an abelian surface family over this P1\mathbf P^1 whose bad behaviour is governed by C\mathcal C, which is what links the radical condition in the height bound to these five points.}
Classified routine:
The genus follows from the Eichler/Shimura genus formula for quaternionic Shimura curves. The branch values and their rationality come from Ogg's counts of Atkin–Lehner fixed points and the known classification of genus-0 Atkin–Lehner quotients. This is a direct computation with these known tools, not a new idea.

Uses

Nothing else in the paper

Used by

claimAbelian surface family with quaternionic multiplication

Sources

PaperOpen
Section
§4.1 The five-point quaternionic quotient
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sections/04-height.tex:43
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