Prime decomposition in the anti-cyclotomic extension

Source abstract

For an imaginary quadratic number field K K and an odd prime number l l , the anti-cyclotomic Z l \mathbb {Z}_l -extension of K K is defined. For primes p \mathfrak {p} of K K , decomposition laws for p \mathfrak {p} in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of K K is Z l \mathbb {Z}_l -embeddable. For some K K and l l , we find explicit polynomials whose roots generate the first step of the anti-cyclotomic extension and show how the prime decomposition laws give nice results on the splitting of these polyniomials modulo p p . The article contains many numerical examples.