Let p be a prime number. Suppose that K is an algebraic number field with p-class group and p-elementary class group. By class field theory ( Cor. 3.1, p. 838), there exist precisely
distinct (but not necessarily non-isomorphic) unramified cyclic extensions, , of
degree p, if K possesses the p-class rank. For each, let denote the class extension homomorphism induced by the ideal extension monomorphism ( 1, p. 74). We let, resp., be the group of units of K, resp..
Proposition 1.1. (Order of)
The kernel of the class extension homomorphism associated with an unramified cyclic extension of degree is a subgroup of the p-elementary class group and has the -dimension
Proof. The proof of the inclusion was given in ( 1, p. 74) for, and generally in ( Prop. 4.3.(1), p. 484). The relation for the unramified extension is equivalent to the Theorem on the Herbrand quotient ( Thm. 3, p. 92) and was proved in [ Prop. 4.3, pp. 484-485]. According to Hilbert’s Theorem 94 ( p. 279), the kernel cannot be trivial. □
Definition 1.1. For each, the elementary abelian p-group is called the p-capitulation kernel of. We speak about total capitulation   if, and partial capitulation if
If is an odd prime, and is a quadratic field with fundamental discriminant and p-class rank, then there arise the following possibilities for the p-capitulation kernel in any of the un- ramified cyclic relative extensions of degree p, which are absolutely dihedral extensions of degree 2p, according to ( Prop. 4.1, p. 482).
Corollary 1.1. (Partial and total p-capitulation over with)
The p-capitulation over K is total if and only if K is real with, and is of type.
The organization of this article is the following. In §2, basic theoretical prerequisites for the new capitulation algorithm are developed. The implementation in Magma  consists of a sequence of computational techniques whose actual code is given in §3. The final §4 demonstrates the results of an impressive application to the case, presenting statistics of all 3-capitulation types, Artin patterns, and second 3-class groups of the 34631 real quadratic fields with discriminants and 3-class group of type, which beats our own records in  §6 and  §6. Theorems concerning 3-tower groups with derived length perfect the current state of the art.
2. Theoretical Prerequisites
In this article, we consider algebraic number fields K with p-class rank, for a given prime number p. As explained in §1, such a field K has unramified cyclic extensions of relative degree p.
Definition 2.1. By the Artin pattern of K we understand the pair consisting of the family of the p-class groups of all extensions as its first component (called the transfer target type) and the p-capitulation type as its second component (called the transfer kernel type),
Remark 2.1. We usually replace the group objects in the family, resp., by ordered abelian type invariants, resp. ordered numerical identifiers ( Rmk. 2.1).
We know from Proposition 1.1 that each kernel is a subgroup of the p-elementary class group of K. On the other hand, there exists a unique subgroup of index p such that, according to class field theory. Thus we must first get an overview of the connections between subgroups of index p and subgroups of order p of.
Lemma 2.1. Let p be a prime and A be a finite abelian group with positive p-rank and with Sylow p-subgroup. Denote by the complement of such that. Then, any subgroup of index p is of the form with a subgroup of index p.
Proof. Any subgroup S of is of the shape with and. We have. Since is coprime to p, we conclude that and. □
An application to the particular case and shows that with.
Three cases must be distinguished, according to the abelian type of the p-class group. We first consider the general situation of a finite abelian group A with type invariants having p-rank, that is, , , , but for. Then the Sylow p-subgroup of A is of type with integer exponents, and the p-elementary subgroup of A is of type. We select generators of such that and.
Lemma 2.2. Let p be a prime number.
Suppose that G is a group and is an element with finite order divisible by p.
Then the power with exponent is an element of order.
Proof. Generally, the order of a power with exponent is given by
This can be seen as follows. Let, and suppose that and, then . We have, and thus is a divisor of. On the other hand, , and thus divides. Consequently, divides, and thus necessarily divides n, since. This yields, as claimed.
Finally, put, then. □
Now, we apply Lemma 2.2 to the situation where A is a finite abelian group with type invariants having p-rank, that is, , ,.
Proposition 2.1. (p-elementary subgroup)
If A is generated by, then the p-elementary subgroup of A is given by.
Proof. Let generators of A corresponding to the abelian type invariants be, in particular, the trailing two generators have orders and divisible by p. Ac- cording to Lemma 2.2, the powers and have exact order p and thus generate the p-elementary subgroup of A. □
Proposition 2.2. (Subgroups of order p)
If the p-elementary subgroup of A is generated by, then the subgroups of of order p can be given by and for.
Proof. According to the assumptions, is elementary abelian of rank 2, that is, of type, and con- sists of the elements, in particular, is the neutral element. A possible
selection of generators for the cyclic subgroups of order p is to take and
for, since the two cycles of powers of and for meet in the neutral element only. □
Proposition 2.3. (Connection between subgroups of index p, resp. order p)
1) If, which is equivalent to, then
2) If, then there exists a unique bicyclic subgroup of index p which contains. The other p subgroups U of index p are cyclic of order, and they only contain the unique subgroup of generated by the th powers.
3) If, then each subgroup of index p completely contains the p -elementary subgroup.
Proof. If, then. Thus, implies , for each proper subgroup U.
If, then a subgroup U of index p is either of type, i.e., cyclic, or of type.
If, then each subgroup U of index p is either of type or of type .
Theorem 2.1. (Taussky’s conditions A and B, see Formula (5))
Let be an unramified cyclic extension of prime degree p of a base field K with p-class rank. Suppose that and are the subgroups of index p associated with, according to class field theory.
Then, we generally have, and in particular:
1) If, then
L is of type A if either or, and
L is of type B if.
2) If, let denote the unique bicyclic subgroup of index p, then
L is of type A if either or or and, and
L is of type B if and.
3) If, then L is always of type A.
Proof. This is an immediate consequence of Proposition 2.3. □
Theorem 2.2. (Orbits of TKTs expressing the independence of renumeration)
1) If, then if and only if for some permutation and its ex- tension with.
2) If, then if and only if for two permutations and the ex- tensions with, , and with.
3) If, then if and only if for two permutations and the ex- tension with.
Proof. The proof for the case was given in ( p. 79) and ( Rmk. 5.3, pp. 87-88). It is the unique case where subgroups of index p coincide with subgroups of order p, and a renumeration of the former enforces a renumeration of the latter, expressed by a single permutation and its inverse.
If, then the distinguished subgroups of index p, and of order p, should have the fixed subscript. The other p subgroups, resp., can be renumerated completely independently of each other, which can be expressed by two independent permutations. For details, see ( Rmk. 5.6, p. 89).
In the case, finally, the subgroups of index p of and the subgroups of order p of can be renumerated completely independently of each other, which can be expressed by two in- dependent permutations. □
3. Computational Techniques
In this section, we present the implementation of our new algorithm for determining the Artin pattern of a number field K with p-class rank in MAGMA   , which requires version V2.21-8 or higher. Algorithm 3.1 returns the entire class group of the base field K, together with an invertible mapping from classes to representative ideals.
Algorithm 3.1 (Construction of the base field K and its class group C)
Input: The fundamental discriminant d of a quadratic field.
Output: The conditional class group of the quadratic field K, assuming the GRH.
Remark 3.1. By using the statement K: =QuadraticField(d); the quadratic field is constructed directly. However, the construction by means of a polynomial executes faster and can easily be generalized to base fields K of higher degree.
For the next algorithm it is important to know that in the MAGMA computational algebra system , the composition, , of an abelian group A is written additively, and abelian type invariants of a finite abelian group A are arranged in non-decreasing order.
Given the situation in Proposition 2.1, where A is a finite abelian group having p-rank, Algorithm 3.2 defines a natural ordering on the subgroups S of A of index by means of Proposition 2.2, if the Sylow p-subgroup is of type.
Algorithm 3.2. (Natural ordering of subgroups of index p)
Input: A prime number p and a finite abelian group A with p-rank.
Output: Generators of the p-elementary subgroup of A, two indicators, NonCyc for one or more non-cyclic maximal subgroups of, Cyc for one or more cyclic maximal subgroups of, an ordered sequence seqS of the subgroups of A of index p, and, if there are only cyclic maximal subgroups of, an ordered sequence seqI of numerical identifiers for the elements S of seqS.
Proof. This is precisely the implementation of the Propositions 2.1, 2.2 and 2.3 in MAGMA . □
Remark 3.2. The modified statement seqS: =Subgroups(A: Quot:=[p,p]); yields the biggest subgroup of A of order coprime to p, and can be used for constructing the Hilbert p-class field of the base field K in Algorithm 3.3, if the p-class group is of type.
The class group in the output of Algorithm 3.1 is used as input for Algorithm 3.2. The resulting sequence seqS of all subgroups of index p in C, together with the pair, forms the input of Algorithm 3.3, which determines all unramified cyclic extensions of relative degree p using the Artin corre- spondence as described by Fieker .
Algorithm 3.3. (Construction of all unramified cyclic extensions of degree p).
Input: The class group of a base field K and the ordered sequence seqS of all subgroups S of index p in C.
Output: Three ordered sequences, seqRelOrd of the relative maximal orders of, seqAbsOrd of the corresponding absolute maximal orders of, and seqOptAbsOrd of optimized representations for the latter.
Remark 3.3. Algorithm 3.3 is independent of the p-class rank of the base field K. In order to obtain the adequate coercion of ideals, the sequence seqRelOrd must be used for computing the transfer kernel type in Algorithm 3.4. The trailing three lines of Algorithm 3.3 are optional but highly recommended, since the size of all arithmetical invariants, such as polynomial coefficients, is reduced considerably. Either the sequence seqAbsOrd or rather the sequence seqOptAbsOrd should be used for calculating the transfer target type in Algorithm 3.5.
Algorithm 3.4. (Transfer kernel type,).
Input: The prime number p, the ordered sequence seqRelOrd of the relative maximal orders of, the class group mapping of the base field K with p-class rank, the generators of the p- elementary class group of K, and the ordered sequence seqI of numerical identifiers for the subgroups S of index p in the class group C of K.
Output: The transfer kernel type TKT of K.
Remark 3.4. In 2012, Bembom investigated the 5-capitulation over complex quadratic fields K with 5-class group of type ( p. 129). However, his techniques were only able to distinguish between permutation types and nearly constant types, since he did not use the crucial sequence of numerical identifiers. We refined his results in ( §3.5, pp. 445-451) by determining the cycle decomposition and, in particular, the fixed points of the permutation types, which admitted the solution of an old problem by Taussky ( §3.5.2, p. 448).
Algorithm 3.5. (Transfer target type,).
Input: The prime number p and the ordered sequence seqOptAbsOrd of the optimized absolute maximal orders of.
Output: The conditional transfer target type TTT of K, assuming the GRH.
With Algorithms 3.4 and 3.5 we are in the position to determine the Artin pattern of the field K. For pointing out fixed points of the transfer kernel type it is useful to define a corresponding weak TKT which collects the Taussky conditions A, resp. B, of Theorem 2.1, for each extension:
Algorithm 3.6. (Weak transfer kernel type, , containing Taussky’s conditions A, resp. B)
Input: The indicators NonCyc, Cyc, and the TKT.
Output: The weak transfer kernel type TAB of K.
Proof. This is the implementation of Theorem 2.1 in MAGMA . □
4. Interpretation of Numerical Results
By means of the algorithms in §3, we have computed the Artin pattern of all 34,631 real quadratic fields with in the range of fundamental discriminants. The results are presented in the following four tables, arranged by the coclass of the second 3-class group. Each table gives the type designation, distinguishing ground states and excited states, the transfer kernel type, the transfer target type, the absolute frequency AF, the relative frequency RF, that is the percentage with respect to the total number of occurrences of the fixed coclass, and the minimal discriminant MD ( Dfn. 5.1). Additionally to this experimental information, we have identified the group by means of the strategy of pattern recognition via Artin transfers ( §4), and computed the factorized order of its automorphism group and its relation rank . Groups are specified by their names in the SmallGroups Library  . The nilpotency class and coclass were determined by means of ( Thm. 3.1, p. 290, and Thm. 3.2, p. 291), resp. ( Thm. 3.1).
4.1. Groups of Coclass
The 31,088 fields whose second 3-class group is of maximal class, i.e. of coclass, constitute a contribution of 89.77%, which is dominating by far. This confirms the tendency which was recogized for the
restricted range already, where we had in ( Tbl. 2, p. 496) and ( Tbl. 6.1, p.
451). However, there is a slight increase of 0.37% for the relative frequency of in the extended range.
Theorem 4.1. (Coclass 1) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group is of coclass has exact length, that is, the 3-class tower group is isomorphic to, and.
Proof. This is Theorem 5.3 in . □
In Table 1, we denote two crucial mainline vertices of the unique coclass-1 tree by and, and we give the results for.
The large scale separation of the types a.2 and a.3, resp. a.2 and a.3, in Table 1 became possible for the first time by our new algorithm. It refines the results in ( Tbl. 2, p. 496) and ( Tbl. 6.1, p. 451), and consequently also the frequency distribution in ( Fig. 3.2, p. 422).
Inspired by Boston, Bush and Hajir’s theory of the statistical distribution of p-class tower groups of complex quadratic fields , we expect that, in Table 1 and in view of Theorem 4.1, the asymptotic limit of the relative frequency RF of realizations of a particular group is proportional to the reciprocal of the order of its automorphism group. In particular, we state the following conjecture about three do-
Table 1. Statistics of 3-capitulation types of fields K with.
minating types, a.3*, a.3 and a.2.
Conjecture 4.1. For a sufficiently extensive range of fundamental discriminants, both, the absolute and relative frequencies of realizations of the groups, and, resp., and, as 3-class tower groups of real quadratic fields satisfy the proportion.
Proof. (Attempt of an explanation) A heuristic justification of the conjecture is given for the ground states by the relation for reciprocal orders
which is nearly fulfilled by, resp., for the bound
, and disproves our oversimplified conjectures at the end of ( Rmk. 5.2).
For the first excited states, we have the reciprocal orders
but here no arithmetical invariants are known for distinguishing between and, whence we
have, resp., with cumulative factor. □
4.2. Groups of Coclass
The 3328 fields whose second 3-class group is of second maximal class, i.e. of coclass, constitute a moderate contribution of 9.61%. The corresponding relative frequency for the restricted range
is, which can be figured out from ( Tbl. 4-5, pp. 498-499) or, more easily, from
( Tbl. 6.3, Tbl. 6.5, Tbl. 6.7, pp. 452-453). So there is a slight decrease of 0.49% for the relative frequency of in the extended range.
Theorem 4.2. (Section D) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group is isomorphic to either of the two Schur -groups or has exact length, that is, the 3-class tower group is isomorphic to, and .
Proof. This statement has been proved by Scholz and Taussky in ( 3, p. 39). It has been confirmed with different techniques by Brink and Gold in ( Thm. 7, pp. 434-435), and by Heider and Schmithals in ( Lem. 5, p. 20). All three proofs were expressed for complex quadratic base fields K, but since the cover ( Dfn. 5.1, p. 30) of a Schur -group consists of a single element, , the statement is actually valid for any algebraic number field K, in particular also for a real quadratic field K. □
Table 2 shows the computational results for, using the relative identifiers of the ANUPQ package  for groups of order, resp. G of order. The possibilities for the 3-class tower group G are complete for the TKTs c.18, c.21, E.6, E.8, E.9 and E.14, constituting the cover of the corresponding metabelian group. For the TKTs c.18, c.21, the cover is given in ( Cor. 7.1, p. 38, and Cor. 8.1, p. 48), and for E.6, E.8, E.9 and E.14, it has been determined in ( Cor 21.3, p. 187). A selection of densely populated vertices is given for the sporadic TKTs G.19* and H.4*, according to ( Tbl. 4-5). We denote two important branch vertices of depth 1 by for.
Whereas the sufficient criterion for in Theorem 4.4 is known since 1934 already, the following statement of 2015 is brand-new and constitutes one of the few sufficient criteria for, that is, for the long desired three-stage class field towers .
Theorem 4.3. (Section c) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group is one of the six groups, , , , , has exact length, that is,
Proof. This is the union of Thm. 7.1, Cor. 7.1, Cor 7.3, Thm 8.1, Cor 8.1, and Cor 8.3 in . □
Table 2. Statistics of 3-capitulation types of fields K with.
A sufficient criterion for similar to Theorem 4.3 has been given in ( Thm. 6.1, pp. 751-752) for complex quadratic fields with TKTs in section E. Due to the relation rank of the involved groups, only a weaker statement is possible for real quadratic fields with such TKTs.
Theorem 4.4. (Section E) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group is one of the twelve groups, , , has either length, that is, , or length, that is,.
Proof. This is the union of Thm. 4.1 and Thm. 4.2 in . □
Example 4.1. That both cases occur with nearly equal frequency has been shown for the ground states in Thm. 5.5 and Thm. 5.6 of . Due to our extended computations, we are now in the position to prove that the same is true for the first excited states. We have for the two fields with, type E.14, and, type E.6, but only for the three fields with, type E.9, and, both of type E.8,
Recently, we have provided evidence of asymptotic frequency distributions for three-stage class field towers, similar to Conjecture 4.1 for two-stage towers.
Conjecture 4.2. For a sufficiently extensive range of fundamental discriminants, both, the absolute and relative frequencies of realizations of the groups and, resp. and as 3-class tower groups of real quadratic fields satisfy the proportion.
Proof. (Attempt of a heuristic justification of the conjecture)
For the first two groups, which form the cover of, we have the reciprocal order relation
which is nearly fulfilled by the statistical information, resp., given in ( Thm. 7.2, pp. 34-35) for.
For the trailing two groups, which form the cover of, only arithmetical invariants of higher order are known for distinguishing between and. It would have been too time consuming to compute these invariants for ( Thm. 8.2, p. 45). □
Conjecture 4.3. For a sufficiently extensive range of fundamental discriminants, both, the absolute and relative frequencies of realizations of the groups, , and as 3-class tower groups of real quadratic fields satisfy the proportion.
Proof. (Attempt of an explanation) All groups are contained in the cover of. We have the following relations between reciprocal orders
Unfortunately, no arithmetical invariants are known for distinguishing between and. Therefore, we must replace the two values in the middle of the proportion by a cumulative value, resp.. The resulting proportion is fulfilled approximately by the statistical information, resp., given in ( Thm. 5.7) for. However, a total of 24 individuals cannot be viewed as a statistical ensemble yet. □
4.3. Groups of Coclass
There are 190 fields whose second 3-class group is of coclass. They constitute a very small con-
tribution of 0.55%. The corresponding relative frequency for the restricted range is,
which can be figured out from ( Tbl. 5, p. 499) or, more easily, from ( Tbl. 6.2, p. 451). Thus, there is a slight increase of 0.15% for the relative frequency of in the extended range.
For the groups of coclass, the problem of determining the corresponding 3-class tower group G is considerably harder than for, and up to now it is still open.
In Table 3, we denote two important mainline vertices of the coclass-2 tree by and, and we give the statistics for.
4.4. Groups of Coclass
We only have 25 fields whose second 3-class group is of coclass. They constitute a negligible contribution of 0.07%. The corresponding relative frequency for the restricted range is
, which can be seen in ( Tbl. 6.9, p. 454). So there is a slight decrease of 0.03% for the relative
frequency of in the extended range.
In Table 4, we denote some crucial mainline vertices of coclass-4 trees by
and, , ,
a sporadic vertex by, and we give the computational results for.
For the essential difference between the location of the groups as vertices of coclass trees for the types d.25* and d.25, see ( Thm. 3.3-3.4 and Exm. 3.1, pp. 490-492).
The single occurrence of type H.4 belongs to the irregular variant (i), where. This is
Table 3. Statistics of 3-capitulation types of fields K with.
Table 4. Statistics of 3-capitulation types of fields K with.
explained in ( p. 498) and ( pp. 454-455). It is the only case in Table 4 where is determined uniquely.
The author gratefully acknowledges that his research is supported by the Austrian Science Fund (FWF): P 26008-N25.
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