Constructing genus-3 hyperelliptic Jacobians with CM

Abstract

Given a sextic CM field K, we give an explicit method for finding all genus-3 hyperelliptic curves defined over C whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339-372], we give an algorithm which works in complete generality, for any CM sextic field K, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field Fp with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo p.

Super-isolated abelian varieties

“…By Lemma 3.19, a is uniquely determined by η and u. Furthermore, the proof of Proposition 5.12 showed that all possible values of u for a fixed η are determined up to sign by the integral points of the curve C η defined by equation (3). Therefore,…”

“…By Lemma 5.8, h(α(u, η)) ≈ h(u) 2 . Since and each k ≡ 2 mod 3 corresponds to 4 Weil generators given by α(±u k 0 , ±η 0 ), 3 It is not always true that T can be chosen as fundamental units of F we have…”

Modular invariants for genus 3 hyperelliptic curves

“…If one acts on the symplectic basis by a matrix in Γ(2), the value of the form given by Lockhart will change by a nonzero constant (the appearance of the principal congruence subgroup of level 2 is related to the use of half-integral theta characteristics to define the form), but if one acts on the symplectic basis by a general element of Sp(6, Z), the value of the form might become zero. As explained in [2], in general to allow for the period matrix to belong to a different Γ(2)-equivalence class, one must attach to the period matrix an element of a set defined by Poor [25], which we call an η-map. Therefore in general one must either modify Lockhart's definition to vary with a map η admitted by the period matrix or use the form Σ 140 , which is nonzero for any hyperelliptic period matrix.…”

“…We begin by describing the maps η that can be attached to a hyperelliptic period matrix. We refer the reader to [25] or [2] for full details.…”

“…We now recall Thomae's formula, which is proven in [5,23] for Mumford's period matrix, obtained using his so-called traditional choice of symplectic basis for the homology group H 1 (C, Z), and in [2] for any period matrix. Let Ω = ( Ω1 Ω2 ) be a Riemann matrix for the curve obtained by integrating the forms ω i of equation (3.1) against a symplectic basis for the homology group H 1 (C, Z) and Z = Ω −1 2 Ω 1 ∈ H g be the period matrix associated to this symplectic basis.…”