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Our implementation shows a large speed-up for precisions as low as a few thousand decimal digits. We also lay out a strategy to generalize this algorithm to genus\u00a0$g$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1112\/s1461157016000309","type":"journal-article","created":{"date-parts":[[2016,8,26]],"date-time":"2016-08-26T11:29:25Z","timestamp":1472210965000},"page":"163-177","source":"Crossref","is-referenced-by-count":7,"title":["Computing theta functions in quasi-linear time in genus\u00a0two and above"],"prefix":"10.1112","volume":"19","author":[{"given":"Hugo","family":"Labrande","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emmanuel","family":"Thom\u00e9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2016,8,26]]},"reference":[{"key":"S1461157016000309_r9","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.2013.878675"},{"key":"S1461157016000309_r2","first-page":"275","article-title":"The arithmetic-geometric mean of Gauss","volume":"30","author":"Cox","year":"1984","journal-title":"Enseign. 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Enge , M. Gastineau , P. Th\u00e9veny and P. Zimmerman , \u2018GNU MPC. INRIA, September 2012. Release\u00a01.0.1\u2019, http:\/\/mpc.multiprecision.org\/."},{"key":"S1461157016000309_r10","first-page":"243","article-title":"Fast genus\u00a02 arithmetic based on theta functions","volume":"1","author":"Gaudry","year":"2007","journal-title":"J.\u00a0Math. Cryptol."},{"key":"S1461157016000309_r15","unstructured":"15. H. Labrande , \u2018Computing Jacobi\u2019s $\\unicode[STIX]{x1D703}$ in quasi-linear time\u2019, Preprint, 2015, arXiv:1511.04248\u00a0[math.NT]."},{"key":"S1461157016000309_r11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01342938"},{"key":"S1461157016000309_r3","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-03-01609-0"},{"key":"S1461157016000309_r4","unstructured":"4. R. Dupont , \u2018Moyenne arithm\u00e9tico-g\u00e9om\u00e9trique, suites de Borchardt et applications\u2019, PhD Thesis, \u00c9cole polytechnique, Palaiseau, 2006, http:\/\/www.lix.polytechnique.fr\/Labo\/Regis.Dupont\/these_soutenance.pdf."},{"key":"S1461157016000309_r14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511619878"},{"key":"S1461157016000309_r13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65315-5"},{"key":"S1461157016000309_r17","article-title":"Computing Igusa class polynomials","volume":"83","author":"Streng","year":"2014","journal-title":"Math. 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