An intriguing hyperelliptic Shimura curve quotient of genus 16

Dembélé, Lassina

doi:10.2140/ant.2020.14.2713

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An intriguing hyperelliptic Shimura curve quotient of genus 16

Lassina Dembélé

Vol. 14 (2020), No. 10, 2713–2742

DOI: 10.2140/ant.2020.14.2713

Abstract

Let $F$ be the maximal totally real subfield of $ℚ (ζ_{32})$ , 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 $𝒪$ be a maximal order in $D$ , and $X_{0}^{D} (1)$ the Shimura curve attached to $𝒪$ . Let $C = X_{0}^{D} (1) ∕ ⟨ w_{D} ⟩$ , 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 $ℚ (ζ_{64})$ , 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 $N ∕ ℚ$ unramified outside $2$ and $\infty$ , with Galois group $Gal (N ∕ ℚ) ≃ F_{17} = ℤ ∕ 17 ℤ ⋊ {(ℤ ∕ 17 ℤ)}^{\times}$ . In fact, the Jacobian $Jac (X_{0}^{D} (1))$ 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 $ℚ$ .

Keywords

abelian varieties, Hilbert modular forms, Shimura curves

Mathematical Subject Classification 2010

Primary: 11F41

Secondary: 11F80

Milestones

Received: 24 July 2019

Revised: 27 February 2020

Accepted: 28 March 2020

Published: 19 November 2020

Authors

Lassina Dembélé
Department of Mathematics University of Luxembourg Luxembourg

Vol. 14, No. 10, 2020