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.
@article{1057864664,
author = {Duke, W. and T\'oth, \'A.},
title = {The Splitting of Primes in Division Fields of Elliptic Curves},
journal = {Experiment. Math.},
volume = {11},
number = {3},
year = {2002},
pages = { 555-565},
language = {en},
url = {http://dml.mathdoc.fr/item/1057864664}
}
Duke, W.; Tóth, Á. The Splitting of Primes in Division Fields of Elliptic Curves. Experiment. Math., Tome 11 (2002) no. 3, pp. 555-565. http://gdmltest.u-ga.fr/item/1057864664/