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

Abstract

Let F be the maximal totally real subfield of (ζ32), 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 X0D(1) the Shimura curve attached to 𝒪. Let C=X0D(1)wD, where wD is the unique Atkin–Lehner involution on X0D(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 EF of class number 17 contained in (ζ64), 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 unramified outside 2 and , with Galois group Gal(N)F17=17(17)×. In fact, the Jacobian Jac(X0D(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 .

Citation

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

Information

Received: 24 July 2019; Revised: 27 February 2020; Accepted: 28 March 2020; Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4190416
Digital Object Identifier: 10.2140/ant.2020.14.2713

Subjects:
Primary: 11F41
Secondary: 11F80

Keywords: abelian varieties , Hilbert modular forms , Shimura curves

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.14 • No. 10 • 2020
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