1,1

The Artin transfer homomorphisms of the assigned 3-class tower group G with SmallGroups identifier <81,10> [Besche, Eick, O'Brien] determine the capitulation type (1,0,0,0) (TKT with fixed point 1) 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 A269320 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,3),(3,3),(3,3),(3,3)] (TTT or IPAD). Conversely, the group G=<81,10> is determined uniquely by its Artin pattern AP(G)=(TTT,TKT) [Mayer, 2014, Fig.3.1, Tbl.4.1], where TKT, TTT, IPAD are abbreviations for transfer kernel type, transfer target type, index-p abelianization data, respectively [Mayer, 2016]. The Magma program has to determine both components of the Artin pattern, since there are infinitely many 3-groups with TKT a.2, (1,0,0,0), and there are three groups with IPAD [(3,3);(9,3),(3,3),(3,3),(3,3)]. (This is one of the few cases where the "Principalization algorithm via class group structure" [Mayer, 2014] is unable to distinguish between TKT a.2, (1,0,0,0), and a.3, (2,0,0,0). A zero always denotes a total capitulation.) Since the group G=<81,10> has derived length dl(G)=2, the Hilbert 3-class field tower of these real quadratic fields consists of exactly two stages.

It must be pointed out that the Magma program must be executed on a machine with Linux operating system, since the Magma versions starting with V2.21-8 are not available for Mac OS and MS Windows. Magma version V2.21-7 will fail at discriminant 751657. (Bug corrected 13 November 2015 by the Magma group, Univ. of Sydney, on our request. See the Change Log of V2.21-8.)

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.

F.-P. Heider and B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 1-25.

D. C. Mayer, All known examples for principalization types, Memorial 2009.

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

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

The leading two terms, 72329, 94636, have been identified by [Heider, Schmithals] (up to 10^5). The first 34 terms up to 10^6 have been determined in the time between 2006 and 2009 [Mayer, 2009]. The 535 terms up to 10^7, computed 13 January 2016, are not published officially yet. They constitute a refinement of the numerical results in [Mayer, 2014].

(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; TKT:=[]; for j in [1..#sR] do Collector:=[]; I:=sR[j]!!mC(g); if IsPrincipal(I) then Append(~Collector, sI[1]); end if;

I:=sR[j]!!mC(h); if IsPrincipal(I) then Append(~Collector, sI[2]); end if;

for e in [1..p-1] do I := sR[j]!!mC(g+e*h); if IsPrincipal(I) then Append(~Collector, sI[e+2]); end if; end for;

if (2 le #Collector) then Append(~TKT, 0); else Append(~TKT, Collector[1]); end if; end for; TAB:=[]; image:=[]; fixedpoints:=0; capitulations:=0;

for j in [1..#TKT] do if (j eq TKT[j]) then Append(~TAB, "A"); fixedpoints:=fixedpoints+1;

elif (0 eq TKT[j]) then Append(~TAB, "A"); capitulations:=capitulations+1;

else Append(~TAB, "B"); end if;

if not (TKT[j] in image) then Append(~image, TKT[j]); end if; end for;

if (2 eq polarization2) and (3 eq polarization1) and (0 eq epsilon) and (1 eq fixedpoints) then printf "%o, ", d; end if; end for;

// On 04 April 2016, Magma version V2.21-11 was released for Mac OS, and is able to execute the PROG. - Daniel Constantin Mayer, Apr 16 2016

Subsequence of A269319.

Sequence in context: A254904 A257206 A229322 * A126658 A251340 A183788

Adjacent sequences: A269317 A269318 A269319 * A269321 A269322 A269323

nonn,hard

Daniel Constantin Mayer, Mar 09 2016

approved