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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
Keywords:
theta functions
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[1] DOI: 10.1090/S0025-5718-2011-01880-6 · Zbl 1221.65075 · doi:10.1090/S0025-5718-2011-01880-6
[2] DOI: 10.1090/S0025-5718-03-01609-0 · Zbl 1092.33018 · doi:10.1090/S0025-5718-03-01609-0
[3] Cox, Enseign. Math. 30 pp 275– (1984)
[4] DOI: 10.1090/S0002-9947-98-02056-X · Zbl 0901.14016 · doi:10.1090/S0002-9947-98-02056-X
[5] Streng, Math. Comp. 83 (2014)
[6] DOI: 10.1007/978-1-4899-2843-6 · Zbl 0509.14049 · doi:10.1007/978-1-4899-2843-6
[7] DOI: 10.1090/S0025-5718-08-02200-X · Zbl 1208.11136 · doi:10.1090/S0025-5718-08-02200-X
[8] DOI: 10.1007/978-3-642-65315-5 · doi:10.1007/978-3-642-65315-5
[9] DOI: 10.1016/0304-3975(85)90067-2 · Zbl 0601.68034 · doi:10.1016/0304-3975(85)90067-2
[10] DOI: 10.1007/BF01342938 · Zbl 0088.28903 · doi:10.1007/BF01342938
[11] Gaudry, J. Math. Cryptol. 1 pp 243– (2007)
[12] DOI: 10.1080/10586458.2013.878675 · Zbl 1293.11107 · doi:10.1080/10586458.2013.878675
[13] DOI: 10.1017/CBO9780511619878 · doi:10.1017/CBO9780511619878
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