Divisibility conditions on the order of the reductions of algebraic numbers

English

Sgobba, Pietro[University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]

Undated

No

[en] reductions ; distribution of primes ; multiplicative order ; Chebotarev density theorem

[en] Let K be a number field, and let G be a finitely generated subgroup of K*. Without relying on (GRH) we prove an asymptotic formula for the number of primes \p of K such that the order of (G mod \p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes \p for which the order is k-free, and those for which the order has a prescribed \ell-adic valuation for finitely many primes \ell. An additional condition on the Frobenius conjugacy class of \p may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.