An intriguing hyperelliptic Shimura curve quotient of genus 16

TL;DR

This paper constructs a genus 16 hyperelliptic Shimura curve with exceptional properties, including special CM points, a Galois number field related to 2-torsion, and hyperellipticity over both its base field and over .

Contribution

It identifies a new hyperelliptic Shimura curve with unique CM points, Galois properties, and hyperellipticity over , expanding understanding of Shimura curves and their arithmetic features.

Findings

The curve has genus 16 and is hyperelliptic with an exceptional involution.

It has 34 Weierstrass points, half of which are CM points defined over a specific Hilbert class field.

The 2-torsion field of its Jacobian's simple factors has Galois closure equal to the Harbater field N.

Abstract

Let $F$ be the maximal totally real subfield of $\mathbf{Q}(\zeta_{32})$, 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 $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. 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 $\mathbf{Q}(\zeta_{64})$, 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 $N/\mathbf{Q}$ unramified outside $2$ and $\infty$, with Galois group $\mathrm{Gal}(N/\mathbf{Q})\simeq F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. In fact, the Jacobian $\mathrm{Jac}(X_0^D(1))$ 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 $\mathbf{Q}$.

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