Abstract
Let K be a field finitely generated over its prime field, and let w ( K ) 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 ) w(K) , such that for any elliptic curve E / K E/K without complex multiplication over K, there is an elliptic curve E ′ / K E\prime /K isogenous to E such that E ′ ( K ) tors E\prime {(K)_{{\text {tors}}}} is of order dividing D. In case K admits a real embedding, we show D = 2 D = 2 , and a nonuniform result is proved in positive characteristic.
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