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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
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Torsion subgroups in families of twists of elliptic curves
The maximum number of quadratic twists with $n$-torsion (n odd) that an elliptic curve over a number field $K$ can have is 2, and here is an easy proof.
You can see this by asking how many twists wi …