Title:

Modular functions, elliptic functions and Galois module structure

We study the Galois module structure of some abelian extensions of an imaginary quadratic field. Our investigation is motivated by a wellknown case in cyclotomic theory and work of M.J. Taylor. We consider extensions N/L where N (resp. L ) is the ray class field of conductor lp^{{m+r}} (resp. {it lp^{r} ) over some imaginary quadratic field ({it l} denotes an odd prime and {it p} belongs to a certain class of prime ideals). We also consider briefly the case where {it p} is composite. The object of study is O_N as a module over its associated order. We describe this order completely by describing its completions at all prime ideals. The primes of tame ramification are handled by a classical result of Noether. For the primes of wild ramification we use a local analogue of the aforementioned result of cyclotomic fields and a new result describing the Galois module structure of division fields attached to a relative LubinTate formal group. There is a precursor of this last mentioned result in the work of Taylor. We prove that in some cases the integrals elements in such a division field form a free module over their associated order. We define a submodule of O_N, R_N, and show that it is free by constructing a generator as a singular value of a certain elliptic modular function. This elliptic modular function is analogous to the function introduced by Fueter and used by Taylor in his work. Lastly, we describe some cases where we are able to lift our result for R_N to O_N.
