We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8204-2-2016,
author = {Ruthi Hortsch},
title = {Counting elliptic curves of bounded Faltings height},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {239-253},
zbl = {06602739},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8204-2-2016}
}
Ruthi Hortsch. Counting elliptic curves of bounded Faltings height. Acta Arithmetica, Tome 172 (2016) pp. 239-253. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8204-2-2016/