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Counting elliptic curves of bounded Faltings height

Volume 173 / 2016

Ruthi Hortsch Acta Arithmetica 173 (2016), 239-253 MSC: Primary 11G05. DOI: 10.4064/aa8204-2-2016 Published online: 11 May 2016

Abstract

We give an asymptotic formula for the number of elliptic curves over $\mathbb Q$ 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 $\mathbb R^2$.

Authors

  • Ruthi HortschDepartment of Mathematics, 2-239A
    Massachusetts Institute of Technology
    77 Massachusetts Avenue
    Cambridge, MA 02139, U.S.A.
    e-mail

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