On unit signatures and narrow class groups of odd degree abelian number fields

. For an abelian number ﬁeld of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are ﬁxed

1. Introduction 1.1.Motivation.Originating in the study of solutions to the negative Pell equation, the investigation of signatures of units in number rings dates back at least to Lagrange.While a considerable amount of progress has been made for quadratic fields [38,21,7], predictions for the distribution of narrow class groups and possible signs of units under real embeddings for certain families of higher degree number fields have only recently been developed [16,14,4].
In this paper, we study unit signatures and class groups of abelian number fields of odd degree.To illustrate and motivate our results, we begin with two special cases.
Conjecture 1.1.1(Conjecture 6.3.3).As K varies over cyclic cubic number fields, the probability that K has a totally positive system of fundamental units is approximately 3%.
For the conjectures presented in this paper, we sidestep the issue of ordering fields: we expect that any fair counting function in the terminology of Wood [41] should be allowed, for example ordering by conductor or by the norm of the product of ramified primes.We are led to Conjecture 1.1.1 by combining structural results established herein with a randomness Date: July 20, 2021.2020 Mathematics Subject Classification.11R29, 11R27, 11R45, 11Y40.hypothesis (H2) in the vein of the Cohen-Lenstra heuristics.This conjecture agrees well with computational evidence (see section 7.1), and it is compatible with the following theorem.
Theorem 1.1.2(Theorem A.1.2, with Elkies).There exist infinitely many cyclic cubic fields with a totally positive system of fundamental units.
The proof of Theorem 1.1.2involves the study the integral points on a log K3 surface.The (infinite) family of simplest cubic fields of Shanks were each shown to have units of all possible signatures by Washington [40, p. 371], the case complementary to Theorem 1.1.2.
Our second illustrative conjecture is as follows.
Conjecture 1.1.3(Conjecture 6.1.1).As K ranges over cyclic number fields of degree 7 with odd class number, the probability that the narrow class number is also odd is 7/9.
We recall that the narrow class group of a field coincides with the class group if and only if there are units of all possible signatures.This conjecture also matches computational evidence well (see section 7.2).
The predictions above are based on the philosophy underlying the Cohen-Lenstra heuristics, which predicts random behavior for arithmetic objects as soon as one accounts for all of the determined structure.Early examples of the need to account for structural properties, including genus theory and ranks of units, were already present in the original paper of Cohen-Lenstra [9].It remains mysterious and important to understand how one must account for additional structure in generalizations of the Cohen-Lenstra heuristics.For example, what makes a prime good [11], and the interaction between p-parts of class groups and the presence of pth roots of unity [32,33,2], remain unresolved.On the other hand, some reflection principles like those of Scholz and Leopoldt [28], seem to be inherently compatible with the Cohen-Lenstra-Martinet conjectures [17,26,27].
In this paper, we propose a model for the distribution of 2-parts of narrow class groups and signatures of units in families of abelian number fields of fixed odd degree and Galois group.For such families, the Galois action and the presence of the 2nd roots of unity suggest additional, nontrivial structure to account for (as confirmed by computations and available function field analogues).The requirement that the degree is odd when p = 2 isolates the "roots of unity problem" from other obstructions to arithmetic randomness, including genus theory.Since the narrow class group is an extension of the class group by an elementary abelian 2-group that measures signatures of units, our efforts are concentrated on 2-parts.
Our contributions are thus twofold.First, for these families we precisely identify and analyze relevant structure, including the relationship to reflection principles.Second, under the hypothesis that what remains behaves randomly, we make exact predictions for the behavior of units and class groups-with corroborating computational evidence.1.2.Structure: class groups.We now set up the structures we study and model in this paper.We build on work of Dummit-Voight [16], who make predictions for fields of odd degree n whose Galois closure has Galois group S n .Here, we instead consider Galois extensions of odd degree.
Attached to K is a finite-dimensional F 2 -vector space V ∞ (K) V 2 (K) equipped with a nondegenerate symmetric bilinear form and a homomorphism called the 2-Selmer signature map.Dummit-Voight [16] showed that the image S(K) := img(ϕ K ) of ϕ K is a maximal, totally isotropic subspace.If K is Galois over Q with Galois group G K , then we observe that the above objects carry a G K -action and in particular S(K) is a G K -invariant, maximal totally isotropic subspace (Corollary 3.1.3).
In preparation for stating our guiding result, we introduce a bit of notation.Let G be a finite abelian group of odd order and let χ be an F 2 -character of G. Then every irreducible F 2 [G]-module is isomorphic to F 2 (χ), the value field of an F 2 -character χ : G → F × 2 taking values in a (fixed) algebraic closure F 2 of F 2 , where G acts through the character χ.For a finitely generated Z[G]-module M , write rk χ M for the multiplicity of the irreducible module F 2 (χ) in the F 2 [G]-module M/M 2 , and let rk 2 M := dim F 2 M/M 2 .For an F 2 -character χ of G, there is a noncanonical F 2 [G]-module isomorphism Hom F 2 (F 2 (χ), F 2 ) F 2 (χ −1 ) (see Lemma 5.1.2and the discussion preceding it), and we write χ * := χ −1 for the corresponding dual character.We say χ is self-dual if F 2 (χ * ) F 2 (χ) as F 2 [G]-modules.For an F 2character χ of G and V an F 2 [G]-module, we write V χ for the F 2 (χ)-isotypic component of V and V χ ± := V χ + V χ * for the sum.If V is equipped with a symmetric, G-invariant F 2 -bilinear form, then the decomposition of V into the spaces V χ ± is orthogonal (Lemma 3.2.7),giving a canonical decomposition as F 2 [G]-modules.Now let K be a Galois number field with abelian Galois group G K of odd order; then the class group Cl(K) and narrow class group Cl + (K) are Z[G K ]-modules.For an F 2 -character χ, we define the following nonnegative integers: ρ χ (K) := rk χ Cl(K); (1.2.1) We refer to k + χ (K) as the χ-isotropy rank.When K is clear from context, we drop it from the notation.Our main theorem, governing the above structures and quantities, is as follows.
Theorem 1.2.2 (Theorem 5.4.2).Let K be a Galois number field with abelian Galois group G K of odd order.Then for each F 2 -character χ, there are exactly 6 possibilities for Theorem 1.2.2 follows from an investigation of the F 2 [G K ]-module structure of the 2-Selmer signature map together with a classification of invariant, maximal isotropic subspaces in a bilinear space with group action.The six possibilities are given in Table 1.2.3: we write q := #F 2 (χ) = #F 2 (χ * ), and we write # Isom G (V ) for the number of G-equivariant isometries of V (K) := V ∞ (K) V 2 (K).All cases occur (see Example 6.4.1), so the statement is optimal in this sense.
In Tables 1.2.3 and 1.2.4 we observe parallel relations when V ∞ (K) is replaced by V 2 (K) and Cl + (K) is replaced by Cl 4 (K), the ray class group of K of conductor 4, with the quantities ρ 4,χ (K) and k 4,χ (K) defined analogously as in (1.2.1); we restrict attention to narrow class groups in this introduction.

.4: Possibilities for the class group and isotropy rank
The following corollary is then immediate.Corollary 1.2.5.Under the hypotheses of Theorem 1.2.2, we have 2.5 can be seen as a Spiegelungssatz or reflection theorem as in Leopoldt [28] for p = 2, and therefore Theorem 1.2.2 can be seen as a precise refinement of it.A precursor to Corollary 1.2.5 is the theorem of Armitage-Fröhlich [1], generalized by Taylor [39] and Oriat [34,35].Gras then proved a very general T -S-reflection principle [23,Théorème 5.18] (see also the presentation in his book [24, Chapter II, Theorem 5.4.5]);however, certain corollaries for p = 2 [24, Chapter II, Corollary 5.4.6(ii)](details [24, Chapter II, (5.4.9)] added in the second printing) are incorrect: case D of Table 1.2.3 does not appear.
We show that rank inequalities like Corollary 1.2.5 for a Galois number field K of odd degree follow from Kummer duality and the G K -module structure of the 2-Selmer group (and its intersection with coordinate subspaces in the 2-Selmer signature space).In particular, the relevant reflection principles are already encoded.In particular, we recover easily several classical results from the literature.Our results can also instead be seen to fit into a much more general context (Poitou-Tate duality of Selmer groups); however, in view of the subtleties indicated in the previous paragraph, one advantage of our approach is it provides a self-contained, uniform, and transparent proof of these corollaries.At the same time, the concrete description in Theorem 1.2.2 states the precise structure (in particular, the image of the 2-Selmer group under the signature map is a G K -invariant, maximal totally isotropic subspace) which must be respected in a random model and thereby serves as the foundation for our heuristics, which we present in sections 1.4-1.5.1.3.Structure: units.The structural result in Theorem 1.2.2 has the following consequence for units.Let O K be the ring of integers of K.The archimedean signature map sgn ∞ : K × → v|∞ {±1} F n 2 is the surjective group homomorphism recording the signs of elements of K × under each real embedding; its kernel K × >0 := ker(sgn ∞ ) is the group of totally positive elements of K.
where the sum indexes over isomorphism classes of F 2 -characters χ.The structure on unit signature ranks imposed by the Galois module structure is summarized in the following result, keeping the notation (1.2.1).
Theorem 1.3.3(Theorem 5.5.2).Let K be an abelian number field of odd degree with Galois group G K , and let χ be an F 2 -character of G K .Then the following statements hold.
When the degree of K is prime, summing over χ gives the following corollary.
Corollary 1.3.4(Corollary 5.5.4).Let K be a cyclic number field of odd prime degree , and let f be the order of 2 modulo .Then sgnrk(O × K ) ≡ 1 (mod f ), and the following statements hold. (a For example, if 2 is a primitive root modulo and the class number of K is odd, then sgnrk(O × K ) = ; this result for = 3 was observed by Armitage-Fröhlich [1, Theorem V]. 1.4.Heuristics: narrow class groups.We begin by applying the results in the previous section to make predictions for narrow class groups and signatures of units for odd-degree abelian number fields.We keep the notation of (1.2.1).
Let G be a finite abelian group of odd order.A G-number field is a Galois number field K, inside a fixed algebraic closure of Q, equipped with an isomorphism G K G, where G K := Gal(K | Q).Such a field K is totally real, so ±1 are the only roots of unity in K.
Returning to Theorem 1.2.2 and Table 1.2.3, we see that the quantities k + χ , k + χ * are uniquely determined by S χ ± in the cases where χ is self-dual or cases B and B when χ is not self-dual.However, when χ is not self-dual and ρ χ = ρ χ * , there is a question about the distribution of cases C, C , and D. Modeling the image of 2-Selmer signature map as a random totally isotropic G-invariant subspace in the 2-Selmer signature space (see heuristic assumption (H1)), we are led to the following conjecture.
Conjecture 1.4.1 (Conjecture 6.1.1).Let G be an abelian group of odd order, and let χ be an F 2 -character of G that is not self-dual and let q := #F 2 (χ).Then as K varies over G-number fields satisfying ρ χ . (1.4.2) A concrete application of Conjecture 1.4.1 is given in Conjecture 6.1.2,as follows.Suppose 2 has order ( −1)/2 modulo a prime ≡ 7 (mod 8): then there are exactly two non-self-dual characters, and if Cl(K) is self-dual then k χ = k χ * = 0.So as K varies over cyclic number fields of degree such that Cl(K) [2] is self-dual, Conjecture 1.4.1 predicts that . (1.4.3) We further expect that the probability in Conjecture 1.4.1 remains the same in certain natural subfamilies, such as when we fix the value rk χ Cl(K) = rk χ * Cl(K) = r.As a special case, we arrive at Conjecture 1.1.3.
1.5.Heuristics: units.Next, we make predictions for signatures of units.Our model can be applied under many scenarios; in this introduction, we consider two simple, illustrative cases.We first examine the situation when the degree is prime and the class number is odd.Modeling O × K /(O × K ) 2 as a random G K -invariant subspace of the 2-Selmer group of K containing −1, and under an independence hypothesis (H2 ), we are led to the following conjecture.
Conjecture 1.5.1 (Conjecture 6.2.4).Let be an odd prime such that the order f of 2 in (Z/ Z) × is odd.Let q := 2 f , and define m := −1 2f ∈ Z >0 .Then as K varies over cyclic number fields of degree with odd class number, Second, we consider the situation when = 3 or 5 with no additional assumption on the class number.In this case, Corollary 5.5.4(b)implies that sgnrk(O × K ) = 1 or .Although complete heuristics for the 2-part of the class group over abelian fields are not known, Malle [32] provides results in the case that = 3 or 5.We use the following notation: for m ∈ Z ≥0 ∪{∞} and q ∈ R >1 , write (q) 0 := 1 and otherwise (q) m := m i=1 (1−q −i ).Combining these results with a uniform random hypothesis (H2), we make the following prediction.
Conjecture 1.5.2 (Conjecture 6.3.3).Let = 3 or 5 and q = 2 −1 .As K varies over cyclic number fields of degree , then Computing the numerical value of the quantity in (1.5.3), we predict that approximately 3% of cyclic cubic fields have sgnrk(O × K ) = 1 which yields Conjecture 1.1.1.For cyclic quintic fields we predict that this proportion drops to below 0.1%.The predictions in the two conjectures above agree with the computational evidence we compiled: see section 7.
Remark 1.5.4.To extend the above conjectures to all odd primes (or more generally, to all abelian groups G of odd order), we would need to refine the heuristics of Malle [32,33] to predict the distribution of rk χ Cl(K).This distribution will depend on the representation theory of Z/ Z (or more generally, of G); in particular, the constraints in Theorem 1.2.2 must be respected.In contrast, when 2 is a primitive root modulo , there is only one nontrivial (necessarily self-dual) F 2 [Z/ Z]-module, so these representation-theoretic complexities are immaterial; in this case, we expect that the generalization of the above conjectures to such to be more straightforward.
We expect that as → ∞ varies over odd primes, we have Prob sgnrk(O × K ) = 1 → 0%, and we plan to give evidence to support this limiting behavior in the future (see also Remark 7.2.3).
Remark 1.5.5.The statements we prove and conjecture above on unit signature ranks in odd degree extensions are quite different than the situation for real quadratic fields, related to solutions to the negative Pell equation.By genus theory, 100% of real quadratic fields have a totally positive unit [21], and the conjectural asymptotic due to Stevenhagen [38] arises from an apparently different heuristic involving Rédei matrices.
Remark 1.5.6.We are not aware of a function field analogue which would bear on the conjectures presented in this section.These conjectures are based on structural properties of the 2-Selmer signature map, which rely in an essential way on the fact that 2 ∈ O K is neither a unit nor zero.1.6.Outline.In section 2, we set up basic notation and background.In section 3, we study these objects in general as Galois modules over F 2 .We then restrict to the case of odd Galois extensions in section 4 and show how reflection principles follow from the Galois action and Kummer duality-these are for completeness (and to indicate that they are not missing from our model).We then further restrict to abelian extensions and in section 5 prove our main structural result, and we see classical reflection principles as a corollary.In section 6 we introduce our heuristic assumptions and present our conjectures, including details on the low-degree cases.In section 7, we carry out computations that provide some experimental evidence for our conjectures.Finally, in appendix A we prove Theorem 1.1.2.1.7.Acknowledgements.The authors would like to thank Edgar Costa, David Dummit, Noam Elkies, Georges Gras, Brendan Hassett, Hershy Kisilevsky, Evan O'Dorney, Arul Shankar, Jared Weinstein, and Melanie Matchett Wood for comments, and Tommy Hofmann for sharing his list of cyclic septic fields.Special thanks go to two anonymous referees for their excellent and detailed feedback and suggestions.Breen was partially supported by an NSF Grant (DMS-1547399).Varma was partially supported by an NSF MSPRF Grant (DMS-1502834) and an NSF Grant (DMS-1844206).Voight was supported by an NSF CAREER Award (DMS-1151047) and a Simons Collaboration Grant (550029).Elkies was partially supported by an NSF grant (DMS-1502161) and a Simons Collaboration Grant.

Properties of the 2-Selmer group and its signature spaces
We begin by setting up some notation and recalling basic definitions and previous results.We quickly prove a standard lemma (for lack of a reference).Let Z (p) := {a/b ∈ Q : p b} be the localization of Z away from a prime p.
Lemma 2.1.1.Let G be a finite group, let p #G be prime, and let M be a finitely generated, torsion Proof.Recall (by Maschke's theorem) that every finitely generated F p [G]-module is semisimple, since p #G.Let m = p r be the exponent of M (as an abelian group), with r ≥ 0. We argue by induction on r.
Suppose the result holds whenever M has exponent dividing p r ; we prove it for M of exponent p r+1 .Multiplication by p gives an exact sequence We can repeat this with pM , giving the following diagram, with exact rows and columns: Here, . By semisimplicity, the left vertical and bottom horizontal maps split, so Since pM has exponent p r , by induction (pM Let K be a number field of degree n = [K : Q] with r 1 real and r 2 complex places, with algebraic closure K and with ring of integers O K .For a prime p, we denote the localization of O K away from (p) by For a place v of K, we let K v denote the completion of K at v and O K,v its valuation ring, and we let ( , 2.2.The 2-Selmer group and its signature spaces.The main object of study is the 2-Selmer group of a number field K, defined as Following Dummit-Voight [16, Section 3], we recall two signature spaces that keep track of behavior at ∞ and at 2, as follows. Definition 2.2.1.The archimedean signature space V ∞ (K) is defined as where the second product runs over all real places of K.The archimedean signature map is By definition, ker sgn ∞ = K × >0 , the totally positive elements of K, which contains (K × ) 2 , and so the map sgn ∞ induces a well-defined map ϕ K,∞ : Sel For the following statements we refer to Dummit-Voight [16, §4].We have dim F 2 V 2 (K) = n and there is an isomorphism of abelian groups Under this identification, the product of Hilbert symbols defines a map , unique up to multiplication by an element of (O × K,(2) ) 2 ; therefore, the map sgn 2 induces a well-defined map ϕ K,2 : Sel Putting these together, we define the 2-Selmer signature space as the orthogonal direct sum and write b := b ∞ ⊥ b 2 for the bilinear form on V (K).The isometry group of (V (K), b) is the product of the isometry groups (or equivalently, the subgroup of the total isometry group preserving each factor).Equipped with b, the 2-Selmer signature space V (K) is a nondegenerate symmetric bilinear space over F 2 of dimension r 1 + n.Similarly, we define the 2-Selmer signature map ).For a number field K, the image of the 2-Selmer signature map ϕ K is a maximal totally isotropic subspace.
Recall from the introduction that the class group of K is denoted by Cl(K), its narrow class group is denoted by Cl + (K), and its ray class group of conductor 4 by Cl 4 (K).Definition 2.2.5.The archimedean isotropy rank of a number field K is and the 2-adic isotropy rank of K is By Dummit-Voight [16, Theorem 6.1], we have hence the nomenclature given in Definition 2.2.5.Moreover, there is a classical equality (see for example, Theorem 2.2 of Lemmermeyer [29] and also Theorem 4.3.3below).

Connections to Sel 2 (K) via class field theory.
There is a natural, well-defined map Sel where a 2 = (α); this map is surjective and fits into the exact sequence In addition, the 2-Selmer signature map arises naturally in class field theory as follows.Let H ⊇ K be the Hilbert class field of K. Class field theory provides an isomorphism Gal(H | K) Cl(K); let H (2) denote the fixed field of the subgroup Cl(K) 2 .The Kummer pairing Gal(H α is (well-defined and) perfect [16, (3.11)].The Artin reciprocity map provides a canonical isomorphism Gal(H (2) | K) Cl(K)/Cl(K) 2 and so we can rewrite the above map instead as The pairing (2.3.2) is the first of four perfect pairings [16, Lemma 3.10] (see also Lemmermeyer [29, Theorem 6.3]); the other three perfect pairings are where Cl + 4 (K) denotes the ray class group of K of conductor 4 • ∞.

Galois module structures
We next study the Galois module structure on the arithmetic objects introduced in the previous section; we will continue in the next section with more precise results in the odd degree case.Our results overlap substantially with those of Taylor [39].
From now on, suppose that K is Galois over Q, with Galois group G K := Gal(K | Q).We work throughout with left F 2 [G K ]-modules.(We could consider more generally structures implied by the action of a nontrivial automorphism group Aut(K), and many of the results below could be generalized to this setting; we focus here on the extreme case, where Aut(K) is as large as possible.)3.1.Basic invariants.We first prove Galois invariance of the signature spaces in generality.Recall that a F 2 -bilinear form b : Proof.We begin with (a), and suppose that K is totally real.The Galois group G K acts on V ∞ (K) (on the left) via its permutation action on the (index) set of real places of For (b), we follow the proof in Dummit-Voight [16,Proposition 4.4].The map a → 1 + 2a induces an isomorphism which is visibly G K -equivariant.By Lemma 2.1.1,the right hand side is isomorphic to and let v | 2 be a prime of K. Since G K acts transitively (on the left) on the set of places {v : v | (2)} with stabilizers D v := Aut(K v ) the decomposition group, choosing a place v we have well defined.The Hilbert symbol ( , Proof.We show that both sgn ∞ and sgn 2 are G-equivariant which implies that the induced maps ϕ K,∞ and ϕ K,2 are G-equivariant as well.For sgn ∞ , we may suppose that K is totally real, and then To show that sgn 2 is G K -equivariant, we observe that sgn 2 is simply the composition of a natural embedding and projection.3.2.Duals and pairings.We now treat some issues of duality, with an application to the Kummer pairing.Let G be a finite group and let V be a finitely generated (left) The canonical evaluation pairing is nondegenerate and G-invariant, so gives a canonical isomorphism Proof.We work with the first line, the others follow by the same argument.The Kummer isomorphism ), where Q is the maximal subfield whose Galois group has exponent dividing 2 in the Hilbert class field of K.The Artin map defines a canonical 2 .Combining these with the evaluation map then gives a canonical pairing as claimed.This pairing may be explicitly described as where a ⊆ O K is an ideal of odd norm, α ∈ O K is coprime to a, and α a is the Jacobi symbol.
Proof.Restricting b, we obtain an F 2 [G]-module map W → W ∨ by w → b( , w ); by Schur's lemma, this map is either zero or an isomorphism, and the result follows.Lemma 3.2.7,although easy to prove, is fundamental in what follows: it shows that when a decomposition of V into irreducibles is possible, it is already almost an orthogonal decomposition.
To conclude this section, we refine this into a canonical orthogonal decomposition.Suppose G has odd order, so the category of F 2 [G]-modules is semisimple.Let W be an irreducible F 2 [G]-module.We write V W for the W -isotypic component of V in a decomposition of V into irreducibles.Suppose that V is equipped with a symmetric, G-invariant, F 2 -bilinear form.Then by Lemma 3.2.7 we have a canonical decomposition as where the orthogonal direct sum is indexed by irreducibles W up to isomorphism and duals.We call the decomposition given in (3.2.8) the canonical orthogonal decomposition of V .

Galois module structures for odd degree extensions
In this section, we suppose throughout that K has odd degree (but remains Galois).Then K is totally real and the only roots of unity in K are ±1.Moreover, since #G K is odd, the category of left F 2 [G K ]-modules is semisimple.4.1.Basic invariants.We quickly prove two standard lemmas, for completeness.
modules where R has trivial G K action (corresponding to the trace zero hyperplane in the Minkowski embedding).Counting idempotents, we conclude that as Z (2) -modules; tensoring (4.1.2) with Z/2Z and using that {±1} has trivial action gives is semisimple, the short exact sequence (2.3.1)splits as F 2 [G K ]-modules; the result then follows from Lemma 4.1.1.Lemma 4.1.4.For any odd Galois number field K, the G K -invariant subspace of each of the F 2 [G K ]-modules Cl(K) [2], Cl + (K) [2] and Cl 4 (K) [2] is trivial, whereas the G K -invariant subspace of Cl(K) + 4 (K) is isomorphic to F 2 .Proof.Let C(K) denote one of the groups under consideration, and let C(Q) denote the ray class group of the same modulus but over Q.The norm map induces a group homomorphism C(K) [2] → C(Q) [2], and on G K -invariants it is an isomorphism, with inverse extension of ideals, since n is odd.Indeed [2].Since the groups Cl(Q), Cl + (Q), Cl 4 (Q) are trivial and Cl + 4 (Q) Z/2Z, the result follows.As mentioned in the introduction, our reflection theorems (Proposition 4.2.2,Proposition 4.3.6,and Theorem 4.3.3)are special cases of the very general T -S-reflection theorem of Gras [23,Théorème 5.18].Our goal in the next few sections is to give a direct proof of these results: it shows that they can be read off from the 2-Selmer group, i.e., that they are intrinsic to the underlying structure of the image of the 2-Selmer group, as we will see below.
We use the notation Proposition 4.2.2.Let K be a Galois number field of odd degree, and let W be an irreducible Proof.Since the short exact sequence in (2.3.1)splits as a sequence of and negating then gives (4.2.4).
In particular, we see from the proof of Proposition 4.2.2 that the inequality is refined by the equality 4.2.3, with the discrepancy in the inequality being measured by the group S(K).This is the simplest instance of the motivation of our paper: we seek to understand structural properties of the 2-Selmer signature map, from which reflection principles are corollaries.4.3.Isotropy ranks.Similar inequalities govern the narrow class group and its relationship to the class group, encoded in the 2-Selmer group.To measure these contributions, we make the following definitions.Throughout, let W be an irreducible Proof.We have that S(K) ∩ V ∞ (K) ker(ϕ K,2 )/ ker(ϕ K ); since ker(ϕ K,2 ) and ker(ϕ K ) are Kummer dual to Cl + (K) [2] and Cl(K) [2] by Lemma 3.2.3, the first isomorphism follows; taking W -rank and subtracting gives The second isomorphism and equality follow similarly.
A further duality is reflected in the totally positive elements in the 2-Selmer group, as follows.
Theorem 4.3.3.Let K be a Galois number field of odd degree.Then Our proof considers the analogue for Sel + 2 (K) of the exact sequence (2.3.1).Let P K be the group of principal fractional ideals of K, and let P K,>0 be the subgroup of P K consisting of principal fractional ideals generated by a totally positive element.The map The natural G K -equivariant map P K → Cl + (K) defined by (α) → [(α)] has kernel P K,>0 and so we have a canonical injection P K /P K,>0 → Cl + (K).Since P 2 K is a subgroup of P K,>0 the image of the injection is contained in Cl + (K) [2].Therefore, the map ) mapping the class of α ∈ K × to the class of the fractional ideal a such that a 2 = (α) is well-defined; it is also visibly surjective, and so fits into the short exact sequence to both sides of (4.3.5), and using (4.3.4) and Lemma 4.1.1we conclude [2] and cancelling gives the result.
By semisimplicity, we can decompose -modules for some choice S K ⊆ S K , well-defined up to isomorphism.We call S K a coordinate complement to S K in V .With this notation, we immediately turn to our next reflection principle: again, all we use is F 2 [G K ]-module structure and Kummer duality.Proposition 4.3.6.Let W be an irreducible F 2 [G K ]-module, and let S K be a coordinate complement to S K in V .Then , so plugging and rearranging gives Repeating with W replaced by W ∨ gives the inequality K), which gives the equality in (4.3.8) and finishes the proof.Proposition 4.3.11.Let W be an irreducible F 2 [G K ]-module, and let S K be a coordinate complement to S K in V .Then Moreover, S K is self-dual and Proof.We again drop K from the notation.For the equality in (4.3.13), by Theorem 4.3.3,we have both sides gives the result.From (4.3.7)we have ] giving (4.3.12).To restore symmetry, we repeat the same argument with W ∨ and conclude that rk W S = rk W ∨ S , so in fact S is self-dual.
Just as in Proposition 4.2.2, we see from the proof of Proposition (4.3.11) that the real content lies in the equality (4.3.12),i.e., the discrepancy in the upper bound (4.3.13) is measured by the (noncanonically defined) "diagonal subspace" S (K) ⊆ S(K).
We deduce corollaries of these statements in the abelian case in section 5.4.

Galois module structures for odd degree abelian extensions
In this section, we specialize further and suppose that the odd order group G is abelian and prove the main structural results of the paper, first for class groups and then for unit signatures.
5.1.Duality in the abelian case.We begin by revising notation and duality in the abelian setting.Let F 2 be a (fixed) algebraic closure of F 2 .An F 2 -character of G is a group homomorphism χ : G → F × 2 .For an F 2 -character χ, let F 2 (χ) ⊆ F 2 be the subfield generated by the values of χ.Then F 2 (χ) is a finite extension of F 2 : more precisely, if χ has (odd) order d and 2 has order f in (Z/dZ) × , then F 2 (χ) F 2 f as F 2 -vector spaces.The group G acts naturally on F 2 (χ) via multiplication by χ(σ); thus F 2 (χ) is a cyclic, irreducible F 2 [G]-module, generated by 1. Conversely, choosing a cyclic generator, every irreducible F 2 [G]-module is of the form F 2 (χ) for some F 2 -character χ.
By character theory, two such modules F 2 (χ) and F 2 (χ ) are isomorphic if and only if there exists ψ ∈ Gal(F 2 | F 2 ) such that χ = ψ • χ.In particular, since ψ is a power of the Frobenius automorphism, There is also a simple way to understand duality when G is abelian.Let V be a finitely generated , which is a ring automorphism when G is abelian.We define V * (the contragredient representation) to be the F 2 [G]-module with the same underlying F 2 -vector space V but with the action of F 2 [G] under pullback from the involution map.Explicitly, if γ ∈ F 2 [G] and x * ∈ V * denotes the same element x ∈ V then γ(x * ) := (γ * (x)) * ; in particular, for σ ∈ G, then σ(x * ) = σ * (x) * = σ −1 (x) * .We conclude that F 2 (χ) * F 2 (χ −1 ) as F 2 [G]-modules, which explains the notation χ * = χ −1 from the introduction.
Remark 5.1.1.Without the hypothesis that G is abelian, starting with a left Proof.Decomposing into irreducibles up to isomorphism, we may suppose without loss of generality that V = F 2 (χ).Consider the map where Tr : F 2 (χ) → F 2 is the trace map.This map is nonzero and F 2 -linear.We claim it is also G-equivariant: indeed, for all x * ∈ V * , σ ∈ G, and y ∈ V , we have The Galois module structure has concrete implications for ranks.
Example 5.1.8.Let K be a cyclic number field of odd prime degree , and let f denote the order of 2 modulo .Then taking G = G K , applying the decomposition into irreducibles given in Example 5.1.7 and Lemma 4.1.4,we conclude that f divides each of rk 2 Cl(K), rk 2 Cl + (K), and rk 2 Cl 4 (K), and that rk 2 Cl + 4 (K) ≡ 1 (mod f ).5.2.Bilinear forms.For an F 2 -character χ of G, we write V χ for the If V is equipped with a symmetric, G-invariant, F 2 -bilinear form, then V has a canonical orthogonal decomposition (3.2.8) where the orthogonal direct sum is indexed by characters χ taken up to isomorphism and inverses.Consequently, it is enough to understand bilinear forms on the components V χ ± .

Recall the algebra trace Tr
, the element σ acts on F 2 [G] by a permutation matrix with no fixed points when σ = 1.Hence Tr Theorem 5.2.3.Let G be an abelian group of odd order.Then there is a unique G-invariant, symmetric, nondegenerate, F 2 -bilinear form on F 2 [G] up to G-equivariant isometry, given by b :

.2.4)
In the standard basis for F 2 [G], the form b is the standard 'dot product'.
Proof.For x, y ∈ F 2 [G], writing x = a σ σ and y = c σ σ in the standard basis, we have (5.2.5) In other words, this pairing is the dot product in the standard basis, and consequently it is symmetric, nondegenerate, and G-invariant.Note * is the adjoint with respect to b, since Example 5.2.7.As this will be central to our investigation, we write down explicitly the pairing in Theorem 5.2.3 restricted to orthogonal components as in (3.2.8). If F 2 f and the bilinear form is non-alternating when χ is trivial and alternating when χ is non-trivial.When χ is trivial then F 2 (χ) = F 2 and b(1, 1) = 1 so the form is non-alternating.Now suppose that χ has order d > 1 and let ζ ∈ F 2 (χ) be a primitive dth root of unity.Since Tr F 2 (χ)|F 2 (1) = 0 (as f , the order of 2 in (Z/dZ) × is even), we have b(ζ k , ζ k ) = Tr F 2 (χ)|F 2 (1) = 0 for all k, so by linearity we conclude that b is alternating. If and the bilinear form is a sum of hyperbolic planes, pairing dual basis elements nontrivially.Put another way, the canonical pairing (3.2.2) induces a natural pairing on F 2 (χ) ∨ ⊕F 2 (χ), which can be described explicitly as b((f, x), (g, y)) = f (y) + g(x).

5.3.
Maximal totally isotropic subspaces.In this section, we will classify the maximal isotropic subspaces of and study their isometry groups.We continue our hypothesis that G is a finite abelian group of odd order.Let V be a finitely generated F 2 [G]-module equipped with a G-invariant, symmetric, F 2 -bilinear form.We let Isom G (V ) ≤ Aut G (V ) be the group of G-equivariant isometries of V , i.e., the subset of F 2 [G]-module automorphisms of V which preserve the bilinear form.2) for all x, y ∈ F 2 (χ); since b is nondegenerate, this is equivalent to νν * = 1.The map ν → νν * is the norm to the unique subfield of F 2 (χ) of index 2; since the norm is surjective, we conclude that Isom G (F 2 (χ)) is a cyclic group of cardinality (q − 1)/( √ q − 1) = √ q + 1.
Second, suppose χ is not self-dual, and write Then * acts on V by (x, y) * = (y * , x * ).The group Aut G (V ) is given by coordinate-wise multiplication by (µ, ν) The same nondegeneracy argument in the previous paragraph shows this is equivalent to We conclude that Isom G (V ) ≤ Aut G (V ) consists of the elements (ν, (ν −1 ) * ) with ν ∈ F 2 (χ) × , a cyclic group of cardinality q − 1.
Lemma 5.3.4.Suppose the bilinear form on V is nondegenerate.Then all G-invariant maximal totally isotropic subspaces S ⊆ V V such that S ∩(V {0}) = S ∩({0} V ) = {0} are in the same G-equivariant isometry class, and there are exactly # Isom G (V ) of them.
Proof.See Dummit-Voight [16, Lemma A.9] for a proof in the case of F 2 -vector spaces; the method of proof gives the same result for F 2 [G]-modules.For example, the isometry τ in [16, Lemma A.9] is automatically a G-equivariant isomorphism when S is G-invariant: the element g In our main classification in section 5.4, we will need to classify Galois invariant maximal totally subspaces in the setting of the following theorem.
Theorem 5.3.5.Let G be an abelian group of odd order, and let χ be an F 2 -character of G and let q := #F 2 (χ).
with the restriction of the bilinear form (5.2.4) for i = 1, 2.
Then the possible G-invariant, maximal totally isotropic subspaces S ⊆ V are described in Table 1.2.3, each row representing a different G-equivariant isometry class.
Proof.First, suppose χ is self-dual.Then up to G-equivariant isometry, we have V i F 2 (χ) for i = 1, 2 with the (restriction of the) trace bilinear form (5.2.4) (see also Example 5.2.7).A subspace is G-invariant if and only if it is an F 2 (χ)-subspace; so by dimensions, a maximal isotropic subspace S is generated by the F 2 (χ)-span of a single vector.We cannot have S = V 1 or S = V 2 , since each b i is nondegenerate.We finish with Lemma 5.3.1 and Lemma 5.3.4.Alternatively, each subspace is spanned by a unique vector (1, ν) with ν ∈ F 2 (χ) × , and one can verify directly that the F 2 (χ)-span is totally isotropic if and only if νν * = 1, consistent with the calculation in (5.3.2).This covers case A.
So now suppose χ is not self-dual.Now V i F 2 (χ) ⊕ F 2 (χ * ) for i = 1, 2, still with the trace bilinear form b. Let S ⊆ V be a G-invariant, maximal totally isotropic subspace.Since S has half of the F 2 -dimension of the bilinear space, as an F 2 [G]-module the possibilities for χ which is indeed totally isotropic since the restriction b χ of b to V χ is identically zero by Lemma 3.2.7.Similarly for S F 2 (χ * ) 2 ; this handles cases B and B .
We now consider the possibilities for S F 2 (χ) ⊕ F 2 (χ * ).We have S = S χ ⊕ S χ * where for some (x 1 , x 2 ), (y 1 , y 2 ) where both are nonzero.Suppose that the nondegeneracy of b 2 on V 2 then implies that y 2 = 0, and ; by Lemma 5.3.1 and Lemma 5.3.4,the number of subspaces of this form is q − 1 and each subspace is in the same G-equivariant isometry class.Alternatively, a calculation like (5.3.3)shows that S is uniquely determined by the spans of (x 1 , x 2 ) = (1, ν) and (y 1 , y 2 ) = (1, µ) with νµ * = 1, giving indeed q − 1 possibilities.

Main result, and consequences.
It is now a straightforward matter to conclude our main structural result, restated here for convenience.We recall (2.2.3) the 2-Selmer signature map -modules, equipped with the orthogonal direct sum of the bilinear forms (5.2.4).The image of the 2-Selmer group under the signature map is a G K -invariant maximal totally isotropic subspace of V (K) by Corollary 3.1.3.By the canonical orthogonal decomposition (3.2.8), thus we conclude that S(K) χ ± are maximal totally isotropic G K -invariant subspaces of V (K) χ ± .
Theorem 5.4.2.Let K be a Galois number field with abelian Galois group G K of odd order.Then for each F 2 -character χ, there are exactly 6 possibilities for S(K) Proof.In view of the first paragraph, Theorem 5.3.5 applies to classify the possibilities.
Another corollary we obtain is the following result, proven by Oriat [35] (and a special case of the T -S-reflection principle of Gras [23,Théorème 5.18]): see also the survey by by Lemmermeyer [29, Theorem 7.2].
Corollary 5.4.5 (Oriat [35,Corollaire 2c]).Let m ∈ Z ≥1 denote the exponent of the Galois group G K for the abelian number field K of odd degree.If there exists t ∈ Z such that Proof.By Lemma 5.1.5,every F 2 [G]-module is self-dual so the conclusion of Corollary 5.4.3 implies that k + χ (K) = k 4,χ (K) = 0 for all χ in the notation of Definition 4.3.1; the result follows.Alternatively, for the first isomorphism, apply Theorem 4.3.3,given that all modules are self-dual.
Example 5.4.6.If is an odd prime such that 2 is a primitive root modulo , then any cyclic number field K of degree satisfies rk 2 Cl(K) = rk 2 Cl + (K) by Corollary 5.4.5.
Example 5.4.7.More generally, if 2 has even order modulo , then Corollary 5.4.5 applies to cyclic number fields of degree .The first prime for which 2 has even order modulo but 2 is not a primitive root in (Z/ Z) × is = 17.
Example 5.4.8.Corollary 5.4.5 also applies to abelian groups that are not cyclic.For instance, if K is a number field with Galois group G K Z/3Z × Z/3Z, then Corollary 5.4.5 implies that rk 2 Cl(K) = rk 2 Cl + (K).
Remark 5.4.9.Edgar-Mollin-Peterson [18, Theorem 2.5] reprove Corollary 5.4.5, and they additionally make the claim that the corollary holds for all Galois extensions (even though they only give a proof for the abelian case).Lemmermeyer [29, p. 13] observes that this claim is erroneous.We give an explicit counterexample (of smallest degree).Let K be the degree-27 normal closure over Q of the field K 0 of discriminant 3 16 • 37 4 defined by which has LMFDB label 9.9.80676485676081.1.This nonabelian extension K has Galois group isomorphic to the Heisenberg group C 2 3 : C 3 (with label 9T7), which has exponent m = 3.The class group Cl(K) is trivial and Cl + (K) (Z/2Z) 6 .
Corollary 5.5.4.Let K be a cyclic number field of odd prime degree , and let f be the order of 2 modulo .Then sgnrk(O × K ) ≡ 1 (mod f ) and the following statements hold: We conclude by proving statement (a) by considering χ and χ * together.Every nontrivial F 2 [G]-module is non-self-dual by Lemma 5.1.5.We refer to the cases in Table 1.2.3.We claim that for every nontrivial character χ, we have . By symmetry, the same conclusion holds when k + χ * (K) = 1 (cases B/C).In the remaining case D where k . This proves the claim in all cases.Summing (5.5.5) over the ( − 1)/(2f ) pairs of irreducible nontrivial F 2 [G K ]-modules as well as the trivial We record the following special case of Corollary 5.5.4 (observed for = 3 by Armitage-Fröhlich [1, Theorem V]).Corollary 5.5.6.If K is a cyclic number field of prime degree where 2 is a primitive root modulo , then sgnrk(O × K ) = 1 or .If the class number of K is odd, then sgnrk(O × K ) = .The above setup allows us to recover many other related statements.We illustrate with the following.Theorem 5.5.7 (Ichimura [25,Theorem 2]).Let K be an abelian number field of odd degree with Galois group G K .Let χ be an F 2 -character of G K .Then the following statements are equivalent: Proof.This statement is trivially true whenever and then it is equivalent to Proposition 4.3.11.

Conjectures
Even with many aspects determined in a rigid way by the results of the previous section, there still remain scenarios where randomness remains.In this section, we propose a model in the spirit of the Cohen-Lenstra heuristics for this remaining behavior.
6.1.Isotropy ranks.We begin by developing a model for isotropy ranks when K runs over a collection of G-number fields (i.e., Galois number fields K equipped with an isomorphism such that Gal(K | Q) G), where G is a fixed finite abelian group of odd order.In light of Theorem 5.4.2 (and Table 1.2.3) a heuristic is only necessary to distinguish cases C,C from D, i.e., when χ is a non-self-dual F 2 -character of G and the collection is restricted to those K such that ρ χ (K) = ρ χ * (K).For all other cases, the isotropy ranks are determined.
We make the following heuristic assumption: (H1) For the collection of G-number fields K such that ρ χ (K) = ρ χ * (K), the image component S(K) χ ± as defined in (5.4.1) is distributed as a uniformly random G-invariant maximal totally isotropic subspace of F 2 [G] 2 χ ± (see Example 5.2.1).The assumption (H1), combined with the restrictions and masses in Table 1.2.3 lead us to one of our main conjectures.Conjecture 6.1.1.Let G be an odd finite abelian group, and let χ be a non-self-dual F 2character of G with underlying module of cardinality #F 2 (χ) = q.Then as K varies over G-number fields such that ρ χ (K) = ρ χ * (K), we have: The same heuristic implies the same conjecture for the 2-adic isotropy ranks; indeed by Proposition 4.3.11,we have k + χ (K) + k + χ * (K) = k 4,χ (K) + k 4,χ * (K).A particularly simple case of Conjecture 6.1.1 is complementary to Corollary 5.4.10.
• Cases B, B are exactly those where ρ χ = ρ χ * , i.e., Cl(K) [2] is not self-dual, in which case ρ χ − ρ χ * = ±1.In these cases, k • The remaining cases C, C , and D are those where Cl(K) [2] is self-dual.For such fields, we have k In particular, Cl(K) [2] is self-dual if and only if rk 2 Cl(K) is even.Example 6.4.1.We now provide examples of the above three cases for cyclic septic number fields.For each case let K = Q(α) where α is a root of the polynomial f (x).
For unit signature ranks, using the formulas in Conjectures 6.2.2 and 6.2.4 we make the following predictions for class groups of cyclic septic fields with low 2-rank.

Computations
In this section, we present computations that provide evidence to support our conjectures.To avoid redundancy, instead of working with families of G-number fields (which weights each isomorphism class of a field K by # Aut(G K )), we weight each isomorphism class of number fields by 1. (Either weighting evidently gives the same probabilities and moments.) We begin by describing a method for computing a random cyclic number field of odd prime degree of conductor ≤ X. Recall (by the Kronecker-Weber theorem) that f ∈ Z ≥0 arises as a conductor for such a field if and only if f = f or 2 f where f is a squarefree product of primes p ≡ 1 (mod ).Moreover, the number of such fields is equal to ( − 1) ω(f )−2 if ω(f ) ≥ 2 and | f , otherwise the number is ( − 1) ω(f )−1 .Our algorithm generates a random factored integer f ≤ X of this form and a uniform random character with given conductor; then, it constructs the corresponding field by computing an associated Gaussian period.7.1.Cubic fields.We sampled cyclic cubic fields in this manner, performing our computations in Magma [31]; the total computing time was a few CPU days.The class group and narrow class group computations are conjectural on the Generalized Riemann Hypothesis (GRH).Our code generating this data is available online [5].
Let N 3 (X) denote the set of sampled cyclic cubic fields K (having Cond(K) ≤ X), and let N 3 (X, ρ = r) ⊆ N 3 (X) denote the subset of fields K with rk 2 Cl(K) = r.For each of X = 10 5 , 10 6 , and 10 7 , we sampled #N 3 (X) = 10 4 fields.Note that the asymptotic number of cyclic cubic fields with conductor bounded by X is c 3 • X where c 3 ≈ 0.159 [12] (see also the fundamental unit for the real quadratic field Q( √ Aq) of the form = r + s √ 4Aq with r, s ∈ Z >0 and r ≡ 1 (mod 2A), so the solutions (x k , y k ) obtained by multiplying x 0 +y 0 √ 4Aq by the powers k = r k + s k √ 4Aq for k ≥ 1 have r k ≡ 1 (mod 2A) and r k , s k > 0, so x k = r k x 0 + s k y 0 (4Aq) ≡ x 0 (mod 2A) and x k > x 0 > 0. Thus and we obtain infinitely many points (a k , y k ) ∈ C m (Z).
Remark A.2.7.The method for getting infinitely many (a, y) ∈ C m (Z) from an initial solution was already known to Euler [20]; see To find a value of m suitable for applying Proposition A.2.4, we work backwards by first selecting an integral point (a, b, c) ∈ S (by a brute force search or starting with a cyclic cubic field of unit signature rank 1) and then solving for the parameter m of the parabola P m .(Since m occurs linearly in the formula for P m , there is a unique solution; explicitly As it happens the denominator is always positive so we do not even have to worry about dividing by zero at an unfortunate choice of (a, b).) Example A. .It remains to check that 1 − 4m k is not a square for infinitely many k (condition (iii)).This can be done in various ways; for example, once we have checked this for one k 0 , we can find some prime such that 1 − 4m k 0 is not a square mod , and apply Euler's theorem as in A.2.4 to find infinitely many k such that m k ≡ m k 0 (mod ), whence 1 − 4m k is not a square either.For our m = 2/13 we may use m k 0 = 2/21447 and = 5.This gives infinitely many curves C m k each containing infinitely many integral points of S above the shaded region in (A.2.1), thus showing that such points are Zariski-dense in S. (This is the same technique used by Elkies [19] to find a Zariski-dense set of rational points on the Fermat quartic surface A 4 + B 4 + C 4 = D 4 starting from a single elliptic curve on that surface with infinitely many rational points.)A.3.Infinitely many cyclic cubic fields.The construction above produces infinitely many integral points (a, b) that correspond to cyclic cubic fields with totally positive units.We now show that for all but finitely many (a, b), the condition (A.Proof.Let t ∈ Z >0 be cubefree.Then a 2 + 3a + 9 = tz 3 defines a genus 1 curve, so by Siegel's theorem it has finitely many integral points.Therefore there are only finitely many (a, y) ∈ C m (Z) such that a 2 + 3a + 9 = tz 3 for z ∈ Z.But #C m (Z) = ∞ by Proposition A.2.4, so the cubefree part of a 2 + 3a + 9 must take on infinitely many values.
For such primes, the -Newton polygon of M a,b (x) consists of a single segment of slope ord (a 2 +3a+9)/3, and hence the extension defined by M a,b (x) over Q is totally ramified.
We finish with a proof of the theorem in this section.

2. 1 .
Basic notation.If A is a (multiplicatively written) abelian group and m ∈ Z >0 , we write A[m] := {a ∈ A : a m = 1} for the m-torsion subgroup of A. For a prime p, we write rk p (A) := dim Fp (A/A p ) for the p-rank of A; we then have #A[p] = p rkp(A) .

Corollary 3 . 1 . 3 .
For a Galois number field K, the image of the 2-Selmer signature map ϕ K is a G K -invariant maximal totally isotropic subspace.Proof.Combine Theorem 2.2.4 with Proposition 3.1.1and Lemma 3.1.2.

4. 2 .
First reflection principle.In this section, we show that the Galois module structure of the 2-Selmer group and Kummer duality imply rank inequalities on the class group, classically known as a reflection theorem.Let W be an irreducible (left) F 2 [G K ]-module, and for a finitely generated Z[G K ]-module M , let rk W (M ) ∈ Z ≥0 be the multiplicity of W in a decomposition of M/2M into irreducible F 2 [G K ]-modules.We recall Lemma 2.1.1,which gives an isomorphism M/2M M[2] for a torsion, finitely generated Z(2)[G K ]-module M , in particular giving rk W (M ) = rk W (M[2]).

Lemma 5 . 3 . 1 .
Equip F 2 [G] with the trace bilinear form b (5.2.4).Let χ be an F 2 -character of G and let q As in the proof of Theorem 5.2.3, the group Aut G (V ) of F 2 [G]-module automorphisms of F 2 (χ) are given by multiplication by an element ν ∈ F 2 (χ) × .The subgroup Isom G (V ) ≤ Aut G (V ) of isometries are those for which b(x, y) = b(νx, νy) = b(x, νν * y) (5.3. By Example 5.1.7,all nontrivial irreducible F 2 [G K ]-modules have cardinality 2 f , and so together with the trivial component generated by −1 gives the first congruence.The upper bounds in (a) and (b) are immediate since rk 2 V ∞ = .To prove (b), note that all F 2 [G K ]-modules are self-dual by Lemma 5.1.5,hence Corollary 5.4.3 implies that k + χ (K) = 0.By adding up Theorem 5.5.2(b) for all 1 + −1 f irreducible F 2 [G K ]-modules as in Example 5.1.7,we conclude the result.

Proof of Theorem A. 1 . 2 .
Let m ∈ Q satisfy (i)-(iv) of Proposition A.2.4, so that φ m (C m )(Z) contains infinitely many points (a, b, c) ∈ S(Z) with a > 0, and hence b > 0; for example, we may take m = 30/163, 2/13 as in Examples A.2.9 and A.2.10.The intersection of P m with the lines b = a and b = a − 2 removes at most 4 values of a; for the values that remain, f a,b (x) = x 3 − ax 2 + bx − 1 is irreducible over Q.To each of these points we associate the field K a,b = Q(η a,b ) where η a,b is a root of f (x), and consider the set of fieldsK m := {K a,b : (a, b, c) ∈ φ m (C m )(Z) and f a,b (x) is irreducible}.Each K a,b ∈ K m isa cyclic cubic extension because its discriminant is (up to squares) equal to c 2 , and since a, b > 0 its roots are totally positive as in (A.2.1).By Lemma A.3.4, there are infinitely many primes dividing the discriminants of the fields in K and so the set contains fields with arbitrarily large discriminants.By Lemma A.3.1, in the set K there are only finitely many fields where η a,b ∈ K ×2 a,b ; let K + be the infinitely many remaining fields.Since η a,b ∈ K ×2 a,b , then η a,b is a totally positive unit that is not a square.By Corollary 5.5.4,we have sgnrk O × K a,b = 1, 3, so we must have unit signature rank 1, i.e., there is a basis of totally positive units.
Lemma 5.1.5.Let m ∈ Z ≥1 denote the (odd) exponent of the abelian group G. Every irreducible F 2 [G]-module is self-dual if and only if there exists t ∈ Z such that 2 t ≡ −1 (mod m), where m is the exponent of G.Proof.Let F 2 (χ) be an irreducible F 2 [G]-module, and let d be the order of χ.We haveF 2 (χ) F 2 (χ) * = F 2 (χ * ) if and only if χ * = χ 2 k for some k ∈ Z, i.e., 2 k ≡ −1 (mod d).Choosing a character with order d = m then gives the result.Example 5.1.6.The smallest (odd) values of m ∈ Z >0 where −1 ∈ 2 ≤ (Z/mZ) × are m = 7, 15, 21, and 23.Example 5.1.7.Suppose #G = is prime and let f be the order of 2 in (Z/ Z) × .Then there are −1 f distinct, nontrivial F 2 [G]-modules, up to isomorphism.They are all isomorphic as F 2 -vector spaces to F 2 f , a generator of G acts by multiplication by a primitive th root of unity ζ ∈ F 2 f , and two such are isomorphic if and only if ζ = ζ 2 k for some k ∈ Z. Finally, all such modules are self-dual if and only if f is even.
.1.3)By Schur's lemma, the map x → f x * is an isomorphism.Remark 5.1.4.See also Theorem 5.2.3 below, where we revisit the trace pairing on F 2 [G] with its involution * .