1,1

The Artin transfer homomorphisms of the assigned second 3-class group M with SmallGroups identifier <729,49> [Besche, Eick, O'Brien] determine the capitulation type (0,1,2,2) (TKT without fixed points) of the real quadratic field K in its four unramified cyclic cubic extensions N_i|K (i=1,...,4) and the abelian type invariants of the 3-class groups Cl(3,K)=(3,3) (whence A269322 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,9),(3,3,3),(9,3),(9,3)] (TTT or IPAD). Conversely, the group M=<729,49> is determined uniquely not only by its Artin pattern AP(G)=(TTT,TKT) but even by the TTT component alone [Mayer, 2015], where TKT, TTT, IPAD are abbreviations for transfer kernel type, transfer target type, index-p abelianization data, respectively [Mayer, 2016]. Consequently, the MAGMA program has to determine only the TTT component of the Artin pattern. This is an instance of the "Principalization algorithm via class group structure" [Mayer, 2014] and saves a considerable amount of CPU time, since the determination of the TKT component is very delicate.

An important Theorem by I.R. Shafarevich [Mayer, 2015, Thm.5.1] disables the metabelian group M=<729,49> as a candidate for the 3-class tower group G, since the relation rank of M is too big. In [Mayer, 2015] it is proved that exactly the two non-metabelian groups <2187,284> and <2187,291> [Besche, Eick, O'Brien] are permitted for G, and the decision is possible with the aid of iterated IPADs of second order (which require computing 3-class groups of number fields with absolute degree 18). Since the derived length of both groups is equal to 3, the Hilbert 3-class field tower of all these real quadratic fields has exact length 3.

The MAGMA program requires A269319 as its input data.

H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a library of groups of small order, 2005, an accepted and refereed GAP package, available also in MAGMA.

M. R. Bush, private communication, 11 July 2015.

D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471-505.

D. C. Mayer, Principalization algorithm via class group structure, J. Thèor. Nombres Bordeaux 26 (2014), no. 2, 415-464.

D. C. Mayer, New number fields with known p-class tower, Tatra Mt. Math. Publ. 64 (2015), 21-57.

D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66-104.

The leading term, 534824, and thus the first real quadratic field K with capitulation type c.18, (0,1,2,2), has been identified on 20 August 2009. However, it required six further years to determine the pro-3 Galois group G=<2187,291>, with metabelianization M=G/G''=<729,49>, of the Hilbert 3-class field tower of K in August 2015. The first 28 terms of A269322 up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014]. The 347, resp. 4318, terms up to 10^8, resp. 10^9, have been computed by [Bush].

Concerning the two possibilities for the 3-class tower group, 534824 is the smallest term with associated group G=<2187,291> and 1030117 is the smallest term with associated group G=<2187,284>. (See [Mayer, 2015] for more details.)

(Magma) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do

ZX<X>:=PolynomialRing(Integers()); K:=NumberField(X^2-d); O:=MaximalOrder(K); C, mC:=ClassGroup(O); sS:=Subgroups(C: Quot:=[p]); sI:=[]; for j in [1..p+1] do Append(~sI, 0); end for; n:=Ngens(C); g:=(Order(C.(n-1)) div p)*C.(n-1); h:=(Order(C.n) div p)*C.n; ct:=0;

for x in sS do ct:=ct+1; if g in x`subgroup then sI[1]:=ct; end if; if h in x`subgroup then sI[2]:=ct; end if; for e in [1..p-1] do if g+e*h in x`subgroup then sI[e+2]:=ct; end if; end for; end for;

sA:=[AbelianExtension(Inverse(mQ)*mC) where Q, mQ:=quo<C|x`subgroup>: x in sS]; sN:=[NumberField(x): x in sA]; sR:=[MaximalOrder(x): x in sA];

sF:=[AbsoluteField(x): x in sN]; sM:=[MaximalOrder(x): x in sF];

sM:=[OptimizedRepresentation(x): x in sF];

sA:=[NumberField(DefiningPolynomial(x)): x in sM]; sO:=[Simplify(LLL(MaximalOrder(x))): x in sA];

TTT:=[]; epsilon:=0; polarization1:=3; polarization2:=3; for j in [1..#sO] do CO:=ClassGroup(sO[j]); Append(~TTT, pPrimaryInvariants(CO, p));

if (3 eq #pPrimaryInvariants(CO, p)) then epsilon:=epsilon+1; end if;

val:=Valuation(Order(CO), p); if (2 eq val) then polarization2:=val; elif (4 le val) then if (3 eq polarization1) then polarization1:=val; else polarization2:=val; end if; end if; end for; if (3 eq polarization2) and (4 eq polarization1) and (1 eq epsilon) then printf "%o, ", d; end if; end for;

Subsequence of A269319

Sequence in context: A235076 A344944 A344945 * A227497 A183650 A273812

Adjacent sequences: A269319 A269320 A269321 * A269323 A269324 A269325

nonn,hard

Daniel Constantin Mayer, Mar 09 2016

approved