The Splitting of Primes in Division Fields of Elliptic Curves

Given a Galois extension L/K of number fields with Galois group G, a fundamental problem is to describe the (unramified) primes p of K whose Frobenius automorphisms lie in a given conjugacy class C of G. In particular, all such primes have the same splitting type in a sub-extension of L/K. In general, all that is known is that the primes have density |C|/|G| in the set of all primes, this being the Chebotarev theorem. For L/K an abelian extension Artin reciprocity describes such primes by means of their residues in generalized ideal classes of K. In the special case that L is obtained explicitly by adjoining to K the n-th division points of the unit circle we have that G ⊂ GL1(Z/nZ) = (Z/nZ)∗ and the Frobenius of p is determined by the norm N(p) modulo n. If K = Q (cyclotomic fields) we have that G = GL1(Z/nZ) and any abelian extension of Q occurs as a subfield of such an L for a suitable n (Kronecker-Weber). Here the Chebotarev theorem reduces to the prime number theorem in arithmetic progressions. In a similar manner an elliptic curve E over K gives rise to its n-th division field Ln by adjoining to K all the coordinates of the n-torsion points. Now Ln is a (generally non-abelian) Galois extension ofK with Galois group G, a subgroup of GL2(Z/nZ) (see [13]). 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 primes in fields contained in Ln or, as we shall say, uniformized by E. As observed by Klein (see [8]), such fields include a large class of non-solvable quintic extensions. Our aim in this application is to provide an arithmetic counterpart to Klein’s ”solution” of quintic equations using elliptic functions.