On the l-Adic Iwasawa λ-Invariant in A p-Extension

On the λ-stability of p-class groups along cyclic p-towers of a number field

“…The computations up to N = 3 or 4 (n ∈ [0, N]) are only for verification when a λ-stability exists; otherwise some examples do not λ-stabilize in the interval considered and no conclusion is possible. For the program, one must precise p, ell, the modulus mod, s = ±1 defining real or imaginary quadratic fields k, the length N of the tower and the interval for m: {p=2;ell=257;mod=5;s=-1;N=3;bm=2;Bm=10^3; for(m=bm,Bm,if(core(m)!=m || Mod(m,ell)==0,next); P=x^2-s*m;lambda=0;if(s==-1&kronecker(s*m,ell)==1,lambda=1);\\lambda print();print("p=",p," mod=",mod," ell=",ell," sm=",s*m," lambda=",lambda); for(n=0,N,Pn=polcompositum(polsubcyclo(ell,p^n),P) [1];kn=bnfinit(Pn,1);\\layer kn knmod=bnrinit(kn,mod);v=valuation(knmod.no,p);Cn=knmod.cyc;\\ray class group of kn C=List;d=matsize(Cn) [2];for(j=1,d,c=Cn[d-j+1];w=valuation(c,p); if(w>0,listinsert(C,p^w,1)));\\end of computation of the p-ray class group of kn print("v",n,"=",v," p-ray class group=",C)))} IMAGINARY QUADRATIC FIELDS, p=2, ell=257, mod=1: m=-2,lambda=1 m=-11,lambda=1 m=-14,lambda=0 m=-17,lambda=1 m=-15,lambda=1 v0=0 [] v0=0 [] v0=2 [4] v0=2 [4] v0=1 [2] v1=3 [8] v1=2 [4] v1=4 [4,4] v1=5 [8,2,2] v1=2 [4] v2=7 [16,4,2] v2=7 [8,8,2] v2=8 [4,4,4,4] [4] v0=6 [16,4] v0=4 [16] v0=5 [16,2] v0=6 [16,2,2] v0=5 [8,8] v1=2 [4] v1=6 [16,4] v1=4 [16] v1=5 [16,2] v1=8 [16,…”

“…For m = −55 (successive structures [4], [4,4], [8,4,4,2], [16,8,8,4]), m = −115 [8,8,8,8]), we have two examples of stabilization from n = 2 (rank 4).…”

“…m=-15 m=-403 m=-259 m=-95 m=-195 m=-895 v0=1 [2] v0=1 [2] v0=2 [4] v0=3 [8] v0=2 [2,2] v0=4 [16] v1=2 [4] v1=5 [8,2,2] v1=3 [8] v1=4 [16] v1=4 [4,2,2] v1=5 [32] v2=3 [8] v2=8 [16,4,4] v2=4 [16] v2=5 [32] v2=9 [8,8,8] v2=6 [64] v3=4 [16] v3=11 [32,8,8] v3=5 [32] v3=6 [64] v3=12 [16,16,16] v3=7 [128] If k ⊆ K ⊆ k, for a Z p -extension k of k, and if λ(1) = 0 (e.g., k is any quadratic field and C = 1) then rk p (C n ) = rk p (C ) as soon as this is true for n 1 (this may holds from some layer). See [12,Section 7] for many examples.…”

“…Numerical experiments are out of reach as soon as n > 3 or 4, but a reasonable heuristic is a tendency to stabilization in totally real p-towers, by analogy with the study of Greenberg's conjecture carried out in [11]. In a different framework, let k n be the nth layer of the cyclotomic Z p -extension k ∞ of k and let C n be the whole class group of k n ; then, for ℓ = p, #(C n ⊗ Z ℓ ) stabilizes in k ∞ (a deep result in the abelian case [33], extended in [4] to Iwasawa context in the Z p × Z ℓ -extension of k).…”

ON THE 2-ADIC IWASAWA LAMBDA INVARIANTS OF THE p-CYCLOTOMIC FIELDS AND THEIR QUADRATIC TWISTS

“…(III) Let p = 3. Let L be an imaginary abelian field, and L n the nth layer of the cyclotomic Z 3 -extension over L. In this special setting, Friedman and Sands [3] obtained an explicit version of Friedman's theorem for the minus part of the ℓ-adic lambda invariant of L n . Here, ℓ is a prime number with ℓ ≥ 5.…”