The splitting of primes in division fields of elliptic curves

Details The splitting of primes in division fields of elliptic curves The splitting of primes in division fields of elliptic curves

2001-03-24

arxiv.org/abs/math/0103151v1

Mathematics

Number Theory

14 pages

Scientific paper

In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve E which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in subfields of such a division field. Such fields include a large class of non-solvable 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 elliptic curve reduced modulo p.

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