Constructing Genus-3 Hyperelliptic Jacobians With Cm
LMS JOURNAL OF COMPUTATION AND MATHEMATICS(2016)
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 F-p, with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo p.
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Key words
hyperelliptic jacobians,cm
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