Modular construction of normal basis (Q1342810)

The following strengthening of a result of Taylor is established. Let \(F\) be an imaginary quadratic field, \(p\) an odd prime number which splits in \(F\), \({\mathfrak p}\) a prime ideal of \(F\) dividing \(p\) and \(m\) a positive integer. Let \(k\) and \(K\) be the ray class fields of \(F\) modulo \({\mathfrak p}^ m\) and \({\mathfrak p}^{[ 5m/2]}\) respectively. Then the ring of \(p\)-integers \(O_ K[ 1/p]\) has a normal basis over \(O_ k[ 1/p]\). In the proof techniques from the theory of modular functions are used.