Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

Abstract

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and ∞, is well known including numerical experiments with PARI/GP programs. The case of "incomplete p-ramification" (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S ⊆ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H K,S of any field K given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the "story" of abelian S-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion T K,S of Gal(H K,S /K) (even if S = P) is a fundamental obstruction in many problems. All relationships involving S-ramification, as Iwasawa's theory, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text. Practice of the Incomplete p-Ramification Over a Number Field -History of Abelian p-Ramification -252/280The analogous theory for a local base field has also a long history that we shall not consider in this article.

Tate–Shafarevich groups in the cyclotomic Zˆ-extension and Weber's class number problem

“…The analogous "fixed points formula" for the ℓ-torsion group T Q(ℓ n ) , in Q(ℓ n )/Q gives also T Q(ℓ n ) = 1 ([17, Theorem IV. 3.3], [20,Proposition 6], [28,Appendix A.4.2]); which justifies once again the fact that the notation T always refers to a p-torsion group.…”

“…Proof. Since K m /K is a p-ramified p-extension, the claim comes from the fixed points formula giving T Gal(Km/K) Km ≃ T K ( [17,Theorem IV.3.3], [20,Proposition 6], [28,Appendix A.4.2]).…”

“…When T K = 1 under Leopoldt's conjecture (freeness of G K ), one speaks of p-rational field K; in this case, the Tate-Shafarevich groups are trivial or obvious, which has deep consequences as shown for instance in [3] in relation with our conjectures in [21] on the p-adic properties of the units. For more information on the story of abelian p-ramification and for that of p-rationality, see [28,Appendix A] and its bibliography about the pioneering contributions: K-theory approach [20], p-infinitesimal approach [41], cohomological/pro-p-group approach [53,54]. All basic material about p-rationality is overviewed in [17, III.2, IV.3, IV.4.8].…”

On Galois representations with large image

“…Regarding this conjecture many computations allow us to have some evidence, but very little is known in general. See [8, Chapter IV, §3 and §4] and [9] for a good exposition. Nevertheless, the p-group T K,p is a deep arithmetical object associated to K, as we can see from the following result, for example.…”

Test of Vandiver's conjecture with Gauss sums -- Heuristics

“…Remark 3.3. (i) One says that K is p-rational if T = 1 (same definition for any number field fulfilling the Leopoldt conjecture at p; see [17,21] for more details and programs testing the p-rationality of any number field). For the pth cyclotomic field K this is equivalent to its "p-regularity" in the more general context of "regular kernel" given in [12, Théorème 4.1] (T − = 1 may be interpreted as the conjectural "relative p-rationality" of K).…”