Vol. 14, No. 10, 2020

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An intriguing hyperelliptic Shimura curve quotient of genus 16

Lassina Dembélé

Vol. 14 (2020), No. 10, 2713–2742

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 .

abelian varieties, Hilbert modular forms, Shimura curves
Mathematical Subject Classification 2010
Primary: 11F41
Secondary: 11F80
Received: 24 July 2019
Revised: 27 February 2020
Accepted: 28 March 2020
Published: 19 November 2020
Lassina Dembélé
Department of Mathematics
University of Luxembourg