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Constructing genus-3 hyperelliptic Jacobians with CM. (English) Zbl 1404.11085
Summary: Given a sextic CM field \(K\), we give an explicit method for finding all genus-\(3\) hyperelliptic curves defined over \(\mathbb{C}\) 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 A. Weng [J. Ramanujan Math. Soc. 16, No. 4, 339–372 (2001; Zbl 1066.11028)], 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 \(\mathbb{F}_{p}\) with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo \(p\).

MSC:
11G15 Complex multiplication and moduli of abelian varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H42 Theta functions and curves; Schottky problem
14Q05 Computational aspects of algebraic curves
Software:
genus3; SageMath
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