An intriguing hyperelliptic Shimura curve quotient of genus 16

Abstract

Let $F$ be the maximal totally real subfield of $\mathbb{Q}\left({\zeta }_{32}\right)$, the cyclotomic field of $32$-nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{𝒪}$ be a maximal order in $D$, and $X_0^D\left(1\right)$ the Shimura curve attached to $\mathcal{𝒪}$. Let $C=X_0^D\left(1\right)∕⟨w_D⟩$, where $w_D$ is the unique Atkin–Lehner involution on $X_0^D\left(1\right)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptional. Second, there are $34$ Weierstrass points on $C$, and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension $E∕F$ of class number $17$ contained in $\mathbb{Q}\left({\zeta }_{64}\right)$, the cyclotomic field of $64$-th roots of unity. Third, the normal closure of the field of $2$-torsion of the Jacobian of $C$ is the Harbater field $N$, the unique Galois number field $N∕\mathbb{Q}$ unramified outside $2$ and $\infty$, with Galois group $Gal\left(N∕\mathbb{Q}\right)\simeq F_{17}=\mathbb{Z}∕17\mathbb{Z}⋊{\left(\mathbb{Z}∕17\mathbb{Z}\right)}^{\times }$. In fact, the Jacobian $Jac\left(X_0^D\left(1\right)\right)$ has the remarkable property that each of its simple factors has a $2$-torsion field whose normal closure is the field $N$. Finally, and perhaps the most striking fact about $C$, is that it is also hyperelliptic over $\mathbb{Q}$.