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Journal Article

An intriguing hyperelliptic Shimura curve quotient of genus 16


Dembélé,  Lassina
Max Planck Institute for Mathematics, Max Planck Society;

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Dembélé, L. (2020). An intriguing hyperelliptic Shimura curve quotient of genus 16. Algebra & Number Theory, 14(10), 2713-2742. doi:10.2140/ant.2020.14.2713.

Cite as: http://hdl.handle.net/21.11116/0000-0007-A39A-3
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 $\mathbf{Q}$.