Some remarks and experiments on Greenberg's p-rationality conjecture
R. Barbulescu and J. Ray

    The p-rational fields were defined in Movahhedi's thesis and appear in several branches of number theory. A recent result of Greenberg allows to construct, for any n, explicit continuous representations rho : Gal(M/Q) -> GL(n,Zp) under the assumption of existence of specific p-rational fields, the previously known cases being n=2 and n=3. Greenberg's p-rationality conjecture states that for any odd prime p and any integer t there exists a p-rational number field K such that Gal(K)=(Z/2)^t. 
    In this talk we start by recalling the link between p-rationality on the one hand and class number and p-adic regulator on the other hand. Since the computation of the class number h is costly we recall an algorithm of M.-N. Gras which allows to test if p divides h without determining h. We also propose a new algorithm to certify that p doesn't divide the p-adic regulator when this is the case. Finally we show that results in the literature imply the case t=1 for any p and that, under a series of arithmetic assumptions, there exist infinitely many cyclic cubic fields which are p-rational.
    We end the presentation by recalling the Cohen-Lenstra-Martinet heuristic and by showing the results of large experiments which corroborate with this heuristic and with Greenberg's conjecture.