GALOIS THEORY OF ITERATED MORPHISMS ON REDUCIBLE ELLIPTIC CURVES AND ABELIAN SURFACES WITH REAL MULTIPLICATION

Abstract

GALOIS THEORY OF ITERATED MORPHISMS ON REDUCIBLE ELLIPTIC CURVES AND ABELIAN SURFACES WITH REAL MULTIPLICATION MAY 2014 DOMENICO AIELLO, B.A., WILLIAMS COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Siman Wong Let F be a number field and let A be an abelian algebraic group defined over F . For a prime ` and a point α ∈ A(F ), we obtain the tower of extensions F ([`n]−1(α)) by adjoining to F the coordinates of all the preimages of α under multiplication by [`]. This tower contains the coordinates of all of the `-power torsion points of A along with a Kummer-type extension. The Galois groups of these extensions encode information about the density of primes P in the ring of integers of F for which the order of α (mod P) is not divisible by `. In this thesis, we determine these Galois groups and explicitly compute the associated density for the cases where A is (1) a reducible elliptic curve; (2) a product of elliptic curves with complex multiplication; (3) an abelian surface with real multiplication.