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Comp."],"abstract":"<p>We study the essential minimum of the (stable) Faltings\u2019 height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the N\u00e9ron-Tate height of an abelian variety, Faltings\u2019 height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum.<\/p>\n                  <p>One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions.<\/p>\n                  <p>Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.<\/p>","DOI":"10.1090\/mcom\/3286","type":"journal-article","created":{"date-parts":[[2017,9,8]],"date-time":"2017-09-08T11:11:58Z","timestamp":1504869118000},"page":"2425-2459","source":"Crossref","is-referenced-by-count":0,"title":["On the essential minimum of Faltings\u2019 height"],"prefix":"10.1090","volume":"87","author":[{"given":"Jos\u00e9","family":"Burgos Gil","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ricardo","family":"Menares","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan","family":"Rivera-Letelier","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,1,18]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"55","DOI":"10.4171\/RSMUP\/128-4","article-title":"Semi-abelian schemes and heights of cycles in moduli spaces of abelian varieties","volume":"128","author":"Bost, Jean-Beno\u00eet","year":"2012","journal-title":"Rend. Semin. Mat. Univ. Padova","ISSN":"https:\/\/id.crossref.org\/issn\/0041-8994","issn-type":"print"},{"key":"2","unstructured":"[BMRan] Jos\u00e9 Ignacio Burgos Gil, Ricardo Menares, Juan Rivera-Letelier. arXiv:1609.00071, ancillary files."},{"key":"3","unstructured":"[BGPRLS15] Jos\u00e9 Ignacio Burgos Gil, Patrice Philippon, Juan Rivera-Letelier, and Mart\u00edn Sombra. The distribution of Galois orbits of points of small height in toric varieties. arXiv:1509.01011v1, 2015."},{"issue":"5","key":"4","doi-asserted-by":"publisher","first-page":"2145","DOI":"10.5802\/aif.2985","article-title":"Successive minima of toric height functions","volume":"65","author":"Burgos Gil, Jos\u00e9 Ignacio","year":"2015","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"issue":"121-122","key":"5","first-page":"25","article-title":"Preuve des conjectures de Tate et de Shafarevitch (d\u2019apr\u00e8s G. 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Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1007\/BF01187931","article-title":"On algebraic equations with integral coefficients whose roots belong to a given point set","volume":"63","author":"Fekete, M.","year":"1955","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"9","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1515\/crll.2001.043","article-title":"Generalized arithmetic intersection numbers","volume":"534","author":"K\u00fchn, Ulf","year":"2001","journal-title":"J. Reine Angew. 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