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These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of <jats:italic>p<\/jats:italic>-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.<\/jats:p>","DOI":"10.1112\/s1461157013000235","type":"journal-article","created":{"date-parts":[[2014,4,16]],"date-time":"2014-04-16T12:06:58Z","timestamp":1397650018000},"page":"1-23","source":"Crossref","is-referenced-by-count":5,"title":["Computing fundamental domains for the Bruhat\u2013Tits tree for , -adic automorphic forms, and the canonical embedding of Shimura curves"],"prefix":"10.1112","volume":"17","author":[{"given":"Cameron","family":"Franc","sequence":"first","affiliation":[]},{"given":"Marc","family":"Masdeu","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2014,4,1]]},"reference":[{"key":"S1461157013000235_r21","volume-title":"Springer Monographs in Mathematics","author":"Serre","year":"2003"},{"key":"S1461157013000235_r6","volume-title":"Rational points on modular elliptic curves","author":"Darmon","year":"2004"},{"key":"S1461157013000235_r1","doi-asserted-by":"publisher","DOI":"10.1090\/ulect\/045"},{"key":"S1461157013000235_r19","doi-asserted-by":"publisher","DOI":"10.24033\/asens.2139"},{"key":"S1461157013000235_r13","doi-asserted-by":"publisher","DOI":"10.1137\/080734467"},{"key":"S1461157013000235_r26","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0091027"},{"key":"S1461157013000235_r20","unstructured":"20. 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