Compositio Mathematica

A Formula for the Number of Elliptic Curves with Exceptional Primes

David Grant a1
a1 Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309–0395, U.S.A. E-mail:

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We prove a conjecture of Duke on the number of elliptic curves over the rationals of bounded height which have exceptional primes.

Key Words: elliptic curves; Galois representations.