Minimal torsion in isogeny classes of elliptic curves

Ross, Raymond

doi:10.1090/S0002-9947-1994-1250824-8

Minimal torsion in isogeny classes of elliptic curves
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Trans. Amer. Math. Soc. 344 (1994), 203-215 Request permission

Abstract:

Let K be a field finitely generated over its prime field, and let $w(K)$ denote the number of roots of unity in K. If K is of characteristic 0, then there is an integer D, divisible only by those primes dividing $w(K)$, such that for any elliptic curve $E/K$ without complex multiplication over K, there is an elliptic curve $E\prime /K$ isogenous to E such that $E\prime {(K)_{{\text {tors}}}}$ is of order dividing D. In case K admits a real embedding, we show $D = 2$, and a nonuniform result is proved in positive characteristic.

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