## Index of elliptic units

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The elliptic units are a naturally arising subgroup of units of any given abelian extension of an imaginary quadratic field K; their definition is motivated by the desire to find units which play the same role as cyclotomic units in abelian extensions of the rationals. The definition outlined in the thesis uses division values of elliptic functions, and provides a simpler stating point of the theory than Robert's original exposition. The properties of thesis units are established using the theory of good reduction of elliptic curves rather than the classical basis of Robert's proofs. The index of the elliptic units is calculated for various abelian extensions of K, particularly for ray class fields modulo an h of K, and fields of division points on an elliptic curve a defined over K. Here it is assumed that K has class number one and the h is prime to 6- the relaxation of these assumptions introduces inessential technical complications into the result. There is a further restriction on h, which seems essential for the method of proof: h is not divisible by any rational prime which splits in K. Thus these results' include the earlier results of Robert form prime power conductors h . These ray class field results are subsequently used to calculate the p-adic value of the index for a field K(E)g of g-division points on an elliptic curve E (over K) which has good reduction at all primes dividing g. Here p is any rational prime not in the finite set of primes dividing 6 or the degree of ray class field modulo the conductor of E, and E itself has complex multiplication by the ring of integers of K. A similar p-adic result for the elliptic units of an arbitrary finite abelian extension of K is proved. The special case with g a prime power is important for current work on the arithmetic of elliptic curves. The p-adic result above is used to prove a new result relating the rank of the group of points on E over the field K(E)p (p a split prime of K) to the invariants of the Iwasawa module attached to the p-adic L-functions.