A note on free pro-$p$-extensions of algebraic number fields

Yamagishi, Masakazu

doi:10.5802/jtnb.86

Cited by 10 publications

(7 citation statements)

References 17 publications

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“…Of course, dim F p (H 2 (G K,S , Z/pZ)), giving the minimal number of relations, is easily obtained only when P ⊆ S (equal to rk p (T K,S ) under Leopoldt's conjecture), which shall explain the forthcoming studies about this: [5], [6,7], [8], [11], [12], [14], [36], [44], Haberland [45], [46], El Habibi-Ziane [47] . .…”

Section: A21šafarevič Formulamentioning

confidence: 97%

“…In general, r K,S is non-obvious and varies from 0 to r 2 + 1 (see Wingberg [6,7], Yamagishi [8], Maire [9,10,11], Labute [12], [13], Vogel [14] for some results and cases where G K,S may be free with less than r 2 + 1 generators and our forthcoming numerical results showing that many Z p -ranks can occur).…”

Section: Notion Of Galois S-ramificationmentioning

confidence: 99%

“…{p=3;print("p=",p);n=8+floor(30/p);g=znprimroot(p);forprime(N=1,10ˆ3, if(Mod(N,p)!=1,next);P=xˆp-N;print();print("P=",P);T=Mod(0,p); for(k=1,(N-1)/2,if(Mod(k,p)==0,next);T=T+k * znlog(k,g));K=bnfinit(P,1); F=idealfactor(K,p);d=matsize(F) [1];F1=component(F,1); for(j=1,d,print(F1[j]));for(z=2ˆd,2 * 2ˆd-1,bin=binary(z);mod=List; for(j=1,d,listput(mod,bin[j+1],j));M=1;for(j=1,d,ch=mod[j]; if(ch==1,F1j=F1[j];ej=F1j [3];F1j=idealpow(K,F1j,ej); M=idealmul(K,M,F1j)));Idn=idealpow(K,M,n);Kpn=bnrinit(K,Idn); Hpn=Kpn.cyc;L=List;e=matsize(Hpn) [2];R=0; for(k=1,e,c=Hpn[e-k+1];w=valuation(c,p);if(w>0,R=R+1; listinsert(L,pˆw,1)));print("S=",mod," rk(A_S)=",R," A_S=",L)))} p=3 [11, [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5]˜, 1, 1, [10,9,8,7,6, 5, 4, 6, 3, 2, 1]˜] [11, [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5]˜, 10, 1, [10,9,8,7,6,5,4 [13, [-4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4]˜, 12, 1, [12,11,10,9,8,7,6, 5, 5, 4, 3, 2, 1]˜] [13, [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4]˜, 1, 1, [12,11,10,9,8,7,6, 5, 2, 4, 3, 2, 1]˜] T=Mod(0,13) S=[0,0] rk(A_S)=1 A_S= [13] T=Mod(0,13) S=[0,1] rk(A_S)=1 A_S= [13] T=Mod(0,13) S= [1,0]…”

Section: Example With a Field Discovered By Jaulent-sauzetmentioning

confidence: 99%

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

Gras¹

2019

Communications in Advanced Mathematical Sciences

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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.

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Section: A21šafarevič Formulamentioning

confidence: 97%

Section: Notion Of Galois S-ramificationmentioning

confidence: 99%

Section: Example With a Field Discovered By Jaulent-sauzetmentioning

confidence: 99%

See 1 more Smart Citation

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

Gras¹

2019

Communications in Advanced Mathematical Sciences

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“…Remark 8.1.2 Leopoldt's conjecture for F implies that the rank k of Γ max verifies 1 ≤ k ≤ r 2 + 1 where r 2 is the number of complex places in F . In the situation of proposition 8.1.1, Yamagishi [Ya1] has determined k explicitely in terms of local datas; in particular k ≤ r 2 + 1 (see also subsection 8.3).…”

Section: Thementioning

confidence: 99%

Computation of a universal deformation ring in the Borel case

Mézard

1999

Math. Proc. Camb. Phil. Soc.

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“…The existence of free pro-p-extensions (Galois extensions with Galois group a free pro-p-group) has been the subject of much study. See for example the list of known results in [15]. Let K = Q(ζ p n ) for some n > 0, and let Ω K denote the maximal pro-p-extension of K which is unramified at all primes not dividing p. Let G K denote the Galois group.…”

mentioning

confidence: 99%