The $l$-class group of the Z$_p$-extension over the rational field

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

“…), e ≥ 1, whose p-rank is a multiple of the residue degree ρ N of p in Q p (µ N )/Q p ; thus ρ N → ∞ as N → ∞, which is considered "incredible" for arithmetic invariants, as class groups, for totally real fields. Indeed, interesting examples occur more easily when p totally splits in Q(µ N ) (i.e., p ≡ 1 (mod N )) and this "explains" the result of [38] and [39] claiming that # C Q(ℓ n ) is odd in Q(ℓ ∞ ) for all ℓ < 500, that of [37,51,52] and explicit deep analytic computations in [5,10,11,14,36,37,38,39,48,49,51,52,59] (e.g., Washington's theorem [59] claiming that for ℓ and p fixed, # C K is constant for all n large enough, whence C * K = 1 for all n ≫ 0, then [14, Theorems 2, 3, 4, Corollary 1]); mention also the numerous pioneering Horie's papers proving results of the form: "let ℓ 0 be a small prime; then a prime p, totally inert in some Q(ℓ n 0 0 ), yields C Q(ℓ n 0 ) = 1 for all n". In [5], a conjecture (from "speculative extensions of the Cohen-Lenstra-Martinet heuristics") implies C * Q(ℓ n ) = 1 for finitely many layers (possibly none).…”

“…PROGRAM II. STRUCTURE OF T_K, K=Q(el^n), FOR ANY el, n, p<Bp {el=2;n=3;Bp=2*10^5;if(el==2,P=x;for(j=1,n,P=P^2-2));if(el!=2, P=polsubcyclo(el^(n+1),el^n));Ex=6;K=bnfinit(P,1);forprime(p=2,Bp, KpEx=bnrinit(K,p^Ex);HpEx=KpEx.cyc;L=List;e=matsize(HpEx) [2];R=0; for(k=1,e-1,c=HpEx[e-k+1];w=valuation(c,p); if(w>0,R=R+1;listinsert(L,p^w,1))); if(R>0,print("el=",el," n=",n," p=",p," rk(T)=",R," T=",L)))} el=2 n=1 p=13 rk(T)=1 T= [13] el=2 n=3 p=29 rk(T)=1 T= [29] el=2 n=1 p=31 rk(T)=1 T= [31] el=2 n=3 p=521 rk(T)=1 T=[521] el=2 n=2 p=13 rk(T)=2 T= [169,13] el=3 n=1 p=7 rk(T)=1 T=[7] el=2 n=2 p=31 rk(T)=1 T= [31] el=3 n=1 p=73 rk(T)=1 T=[73] el=2 n=2 p=29 rk(T)=1 T= [29] el=3 n=2 p=7 rk(T)=1 T=[7] el=2 n=2 p=37 rk(T)=1 T= [37] el=3 n=2 p=73 rk(T)=1 T=[73] el=2 n=3 p=3 rk(T)=2 T= [3,3] el=5 n=1 p=11 rk(T)=2 T= [11,11] el=2 n=3 p=31 rk(T)=1 T= [31] el=5 n=2 p=11 rk(T)=2 T= [11,11] el=2 n=3 p=13 rk(T)=2 T= [169,13] el=5 n=2 p=101 rk(T)=1 T=[101] el=2 n=3 p=37 rk(T)=1 T= [37] Remark 2.2. These partial results show that p-ramification aspects are more intricate since, for instance for ℓ = 2, the divisibility by p = 29 only appears for n = 2 and, for p = 13, the 13-rank and the exponent increase from n = 1 to n = 2.…”

“…For instance, the fields K = Q(5) and Q (15) give rise to two annihilators, which is specified by the structure T K ≃ (Z/11Z) 2 and T K ≃ (Z/31Z) 2 , but we compute that C K = 1 in these cases. In [37] it is proved that for ℓ < 131, 109, 101, the p-class group of Q(ℓ ∞ ) is trivial for p = 7, 11, 13, respectively. This confirms that for N = 5, p = 11, T K = R K ≃ (Z/11Z) 2 .…”

“…Field K=Q(5) T= [11,11] Field K=Q(6) T= [7,7] T= [13,13] T= [31] T= [43] T=[73] Field K=Q(12) T= [9,9] T= [7,7] T= [169,169,13,13] T= [29] T= [31] T= [37] T= [43] T=[73] Field K=Q( 14) T= [13] T= [31] T= [113] Field K=Q( 15) T= [7] T= [11,11] T= [31,31] T= [73] Field K=Q(21) T= [49,7] Field K=Q(30) T= [7,7] T= [11,11] T= [13,13] T= [31,31,31] T= [43] T= [73] Field K=Q(42) T= [49,49,7,7] T= [13,13] The composite K = Q(42) has some interest for p = 7 since T K ≃ (Z/7Z) 2 ; so we know that T ...…”

“…Indeed, one may ask if the arithmetic of these fields is as smooth as it is conjectured (for the class group C Q(N ) ) by many authors after many verifications and partial proofs [2,5,10,11,12,13,14,15,34,35,36,37,38,39,40,46,47,48,49,50,51,52,59]. The triviality of C Q(ℓ n ) has, so far, no counterexamples as ℓ, n, p vary, but that of the Tate-Shafarevich group T Q(ℓ n ) (or more generally T Q(N ) ) is, on the contrary, not true as we shall see numerically, and, for composite N , few C Q(N ) = 1 have been discovered.…”