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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
genus3; SageMath
Full Text:
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