An Elliptic Curve over Q has an Isogenous Quadratic Twist if and Only if it has complex Multiplication
Date
2014-09-23
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Abstract
In this thesis we prove that, for every elliptic curve E over the rational numbers, E is isogenous to a quadratic twist if and only if E admits complex multiplication. To prove this, we use a famous result of Faltings comparing local and global isogenies of elliptic curves over number fields, and a famous theorem proven by Serre on the density of supersingular primes for elliptic curves over the rational numbers. While the result of this thesis is certainly known to experts, a proof seems to not appear in the literature. The thesis includes background on elliptic curves, isogenies, twists and complex multiplication.
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B.Langlois, M. (2014). An Elliptic Curve over Q has an Isogenous Quadratic Twist if and Only if it has complex Multiplication (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/24851