We give a global description of the Frobenius for the division fields of an elliptic curve E that is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in a subfield of such a division field. These subfields include a large class of nonsolvable quintic extensions and our application provides an arithmetic counterpart to Klein's "solution" of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the reduced curve modulo p.
"The Splitting of Primes in Division Fields of Elliptic Curves." Experiment. Math. 11 (4) 555 - 565, 2002.