Circular sets of prime numbers and p-extension of the rationals

Original · EN

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group Gₛ(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that Gₛ(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension Qₛ(p)/Q and we relate the cohomology of Gₛ(Q)(p) to the etale cohomology of the scheme Spec(Z)-S. Finally, we calculate the dualizing module.