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Computing theta functions in quasi-linear time in genus two and above. (English) Zbl 1361.14028
In this paper the authors extend the results in [H. Labrande, “Computing Jacobi’s $$\theta$$ in quasi-linear time”, Preprint (2015), arXiv:1511.04248] and compute the theta function $$\theta(z,\tau)$$ in genus two in quasi-linear time [R. Dupont, Math. Comput. 80, No. 275, 1823–1847 (2011; Zbl 1221.65075)]. They first give the necessary background on genus g theta functions and the algorithms needed to compute them [B. Deconinck et al., Math. Comput. 73, No. 247, 1417–1442 (2004; Zbl 1092.33018)]. They provide a careful analysis in the case of genus two. They also give a table which compares favorably their algorithm to MAGMA’s Theta function [A. Enge and E. Thomé, Exp. Math. 23, No. 2, 129–145 (2014; Zbl 1293.11107)]. Finally, they provide a strategy to generalize to theta functions of any genus.

##### MSC:
 14K25 Theta functions and abelian varieties 14H42 Theta functions and curves; Schottky problem
theta functions
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##### References:
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