Duke Mathematical Journal

Exceptional isogenies between reductions of pairs of elliptic curves

François Charles

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Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and we draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether–Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

Article information

Duke Math. J., Volume 167, Number 11 (2018), 2039-2072.

Received: 26 August 2015
Revised: 8 February 2018
First available in Project Euclid: 26 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

elliptic curves Lang–Trotter Hecke correspondences isogenies Frobenius distribution supersingular primes


Charles, François. Exceptional isogenies between reductions of pairs of elliptic curves. Duke Math. J. 167 (2018), no. 11, 2039--2072. doi:10.1215/00127094-2018-0011. https://projecteuclid.org/euclid.dmj/1530000176

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