ON THE RANK OF LEOPOLDT’S AND GROSS’S REGULATOR MAPS

A BSTRACT . We generalize Waldschmidt’s bound for Leopoldt’s defect and prove a similar bound for Gross’s defect for an arbitrary extension of number ﬁelds. As an application, we prove new cases of Gross’s ﬁniteness conjecture (also known as the Gross-Kuz’min conjecture) beyond the classical abelian case, and we show that Gross’s p -adic regulator has at least half of the conjectured rank. We also describe and compute non-cyclotomic analogues of Gross’s defect.


INTRODUCTION
Let p be a prime number.Given a number field K , we denote by S p (K ) and S ∞ (K ) the sets of p-adic places and archimedean places of K respectively.Fix an algebraic closure Q p of Q p and let A ∧ = Q p ⊗ Z p lim ← − −n A/p n A for any abelian group A. The Leopoldt regulator map is the Q p -linear map (1) Consider the Z p -hyperplane H of P|p Z p given by the equation P|p s P = 0, and the map (2) where O K [ 1 p ] × is the group of p-units of K , N P is the local norm map for the extension K P /Q p , and log p : Q × p → Q p is the usual Iwasawa p-adic logarithm.By the usual product formula, L K is well-defined.We shall refer to L K as the (cyclotomic) Gross regulator map.
Like Leopoldt's conjecture, the Gross-Kuz'min conjecture plays a central role in the formulation of p-adic analogues of Dirichlet's class number formula.Leopoldt's regulator appears in Colmez's formula on the residue at s = 1 of the p-adic Dedekind zeta function of a totally real number field [Col88], whereas Gross's regulator plays the role of an L-invariant in the celebrated Gross-Stark conjecture over a CM number field [Gro81].When K is neither totally real nor CM, a conjectural interpretation of these regulators in terms of p-adic Artin L-functions is still available (see [Mak21]).Besides their potential applications to the equi- variant Tamagawa number conjecture as in [BKS17], the Gross-Kuz'min conjecture and its non-cyclotomic analogue discussed below also yield information on the fine structure of class groups attached to Z p -extensions of K (see e.g.[FG81,Kol91,FMD05,Jau17]).
The Gross-Kuz'min conjecture is true when K /Q is abelian as shown by Greenberg [Gre73].More recently, Kleine [Kle19] proved the conjecture for any K which has at most two p-adic primes.(We note that Kleine's approach does not use p-adic transcendence theory).
Our first result gives an upper bound for the Gross defect δ G K = dim coker L K as well as a slight generalization of Theorem 1.2.Theorem 1.4.Let K /k be an extension of number fields.The following inequalities hold: Theorem 1.5.Let ρ ∈ Art Q p (G Q ) be an irreducible representation and let d = d(ρ), d + = d + (ρ) Otherwise, we have the following inequalities.
By Artin formalism, this yields the upper bound (e.g. for Leopoldt's defect) δ L k (ρ) ≤ d + (ρ)/2 for an arbitrary representation ρ ∈ Art Q p (G k ).We immediately recover Theorem 1.4 by choos- The first bound in Theorem 1.5 is Laurent's main theorem in [Lau89], but we will provide a much shorter proof of this result via a lemma on local Galois representations (Lemma 3.2.7).The second bound, however, does not seem to follow from the classical methods employed by Laurent [Lau89] and Roy [Roy92] to study the p-adic closure of S-units of K , for a given finite set of places S.
Theorem 1.5 together with Artin's formalism easily implies Leopoldt's conjecture for abelian extensions of an imaginary quadratic field.In the same vein, we indicate the two main applications of Theorem 1.5.Corollary 1.6.Let k be a totally real field and let V be a totally odd Artin representation of G k .Then Gross's p-adic regulator matrix R p (V ) defined in [Gro81, (2.10)] has rank at least half of its size.
This corollary strengthens Gross's classical result stating that the matrix R p (V ) has positive rank [Gro81, Prop.2.13].Corollary 1.7.The Gross-Kuz'min conjecture holds for abelian extensions of imaginary quadratic fields.It also holds for abelian extensions of real quadratic fields having at least one real place.
Theorem 4.3.2provides a more extensive list of number fields for which the Gross-Kuz'min conjecture holds unconditionally.The first part of Corollary 1.7 is also proven in [Mak21] when p = 2 using the language of Selmer groups.
We highlight in the last part of this article some interesting connections between noncyclotomic analogues of the Gross-Kuz'min conjecture and algebraic independence of p-adic logarithms of units of number fields.
Given an arbitrary Z p -extension K ∞ of K , we will define a map L K ∞ /K specializing to L K if K ∞ is the cyclotomic extension of K .As noted in [Kis83,JS95], there do exist examples of Z p -extensions K ∞ /K for which δ G K ∞ /K > 0, but a conjectural description of all such K ∞ /K is still missing.
In the next theorem, we fix an embedding Q ⊂ Q p and we let Λ be the Q-linear subspace of Q p generated by 1 and by p-adic logarithms of non-zero algebraic numbers.Theorem 1.8.Let k be an imaginary quadratic field, and K an abelian extension of k in which p splits completely.Then there exist at most finitely many distinct does not vanish on any 7-tuple (a, b, c, d, x, y, z) ∈ Λ 7 which form a Q-linearly independent set.This last condition should be true according to the weak p-adic Schanuel conjecture.In Proposition 3.1.1we illustrate Theorem 1.8 with a classical application to the semi-simplicity of Iwasawa modules attached to K k ∞ /K .Theorem 1.8 can be generalized to arbitrary base fields k having at most r linearly disjoint Z p -extensions with r ≤ 2 (Theorem 5.1.1).The main idea is that, under our assumption on p, one can parameterize Z p -extensions of k by points on a (r − 1)-dimensional linear subspace subvariety C of L given by polynomial equations with coefficients in We then exploit the fact that any linear (resp.algebraic) independence between elements of Λ 0 implies strong conditions on the Q-points (resp.the Q p -points) of C .
Theorem 1.8 was inspired by Betina-Dimitrov's work [BD21] where the authors show the non-vanishing of a certain L-invariant for Katz's p-adic L-function restricted to the anticyclotomic Z p -extension.In fact, their result generalizes to any Z p -extension with nontranscendental slope.We expect that our techniques can give further results on the nonvanishing of L-invariants in more general contexts.
The paper is structured as follows.In Section 2 we recall all the classical results in p-adic transcendence theory which we make use of.In Section 3 we describe Leopoldt's and Gross's defects via class field theory and we show that they are compatible with Artin formalism.Our main results and corollaries are proven in Section 4, except for Theorem 1.8 whose proof is postponed to Section 5.
Acknowledgments.The author would like to thank Dominik Bullach and Lassina Dembélé for reading and providing comments on a preliminary version of this paper.This research is supported by the Luxembourg National Research Fund, Luxembourg, INTER/ ANR/18/12589973 GALF.

p-ADIC TRANSCENDENCE THEORY
Throughout this section we fix an embedding ι p : Q → Q p , allowing us to view algebraic numbers as p-adic numbers.The following very strong conjecture describes the algebraic dependence between logarithms of algebraic numbers (see [CM09, Conjecture 3.10]).Conjecture (Weak p-adic Schanuel conjecture).Let α 1 , . .., α n be nonzero algebraic numbers.If log p (α 1 ), . .., log p (α n ) are linearly independent over Q, then they are algebraically independent over Q.
We recall some classical results of Brumer, Waldschmidt and Roy and deduce some consequences that turn out to be useful in the study of the Gross-Kuz'min conjecture.
Recall that log p is normalized so that we have log p (p) = 0.
Proposition 2.1.2.Let H ⊂ Q be a number field.The Q-linear extension log p : ) of the p-adic logarithm has kernel the line p Q spanned by 1 ⊗ p.
Proof.Let H p be the completion of ι p (H) inside Q p , let O × H p be its unit group and consider the abelian group Moreover, the p-adic logarithm map is injective on O × H p modulo the torsion, so multiplicatively independent numbers α 1 , . .., α n ∈ T have Q-linearly independent p-adic logarithms by the Baker-Brumer theorem.This shows that the restriction of log p to Q⊗T is injective, hence ker(log p ) = p Q .
2.2.Waldschmidt's and Roy's theorem.Recall that Λ is the Q-linear subspace of Q p generated by 1 and by p-adic logarithms of non-zero algebraic numbers.Extensions of Baker's method due to Waldschmidt and Roy give a lower bound for the rank of matrices with coefficients in Λ.To each matrix M with coefficients in Q p , of size m × ℓ, they assign a number θ(M) defined as the minimum of all ratios ℓ ′ m ′ where (m ′ , ℓ ′ ) runs among the pairs of integers satisfying 0 < m ′ ≤ m and 0 ≤ ℓ ′ ≤ ℓ, for which there exist matrices P ∈ GL m (Q) and Q ∈ GL ℓ (Q) such that the product P MQ can be written as m with equality if all the entries of M are Q-linearly independent.The following theorem is Roy's sharpening of Waldschmidt theorem ([Wal81, Théorème 2.1.p],[Roy92, Corollary 1]).Theorem 2.2.1.Let M be a matrix with coefficients in Λ, of size m × ℓ with m, ℓ > 0, and let n be its rank.We have Roy also deduced a useful corollary for 3×2 matrices from Theorem 2.2.1 in [Roy92, Corollary 2].Corollary 2.2.2 (Strong six exponentials theorem).Let M be a (3×2)-matrix with coefficients in Λ.If the rows of M are Q-linearly independent, and if the columns of M are also Q-linearly independent, then M has rank 2.

REGULATOR MAPS AND CLASS GROUPS
3.1.Galois cohomology.For all fields L ⊂ Q and all finite sets S of places of L containing S p (L), we let X (L) (resp.X ′ S (L)) be the Galois group of the maximal abelian pro-p extension of L which is unramified everywhere (resp.unramified everywhere and totally split at all v ∈ S).If S = S p (L), we simply put , the transition maps being the restriction maps.Therefore, X (K ∞ ) and X ′ S (K ∞ ) are modules over the Iwasawa algebra Z p [[Γ]].They are finitely generated torsion as shown by Iwasawa [Iwa73].We let Let S 0 ⊇ S p (K ) S ∞ (K ) be a finite set of places of K .For any extension L of K and any discrete (resp.compact) G L -module M which is unramified outside the places of L above S 0 , we consider for all i ≥ 0 the S 0 -ramified i-th cohomology group (resp.continuous cohomology group) H i S 0 (L, M) = H i (Gal(L S 0 /L), M), where L S 0 /L is the largest extension of L which is unramified outside the places of L above S 0 .Given any subset S ⊂ S 0 , let where L ur v /L v denotes the maximal unramified extension of L v (so R ur = R in particular) and the maps above are the usual localization maps.Note that the definition of X i S (L, M) does not depend on the choice of S 0 .We simply write Proof.The first isomorphism is given by the Poitou-Tate duality theorem [Mil86, Theorem 4.
The isomorphisms provided by Lemma 3.1.2are functorial in L in the sense that, given a finite extension L ′ /L of number fields, the norm map X (L ′ ) → X (L) corresponds to the corestriction map (resp.to the Pontryagin dual of the restriction map) We now make use of the inflation-restriction exact sequence to study the problem of Galois descent.We have a commutative diagram with exact rows where the places v (resp.w) of the second row run through all the p-adic and archimedean places of K (resp. of K ∞ ).Here, we have used the fact that Γ has cohomological dimension one as it is pro-cyclic.
Proof.By Lemma 3.1.2,the kernel of the right vertical map of (3) is equal to the Pontryagin dual of is equal to the rank of the Pontryagin dual of the kernel of the natural map To end the proof, it suffices to notice that for G = Gal(K S /K ), Gal(K P /K P ), Γ or Γ P , the , where r 2 is the number of complex places of K ([Was97, Theorem 13.4]).In particular, Proposition 3.1.3yields an upper bound δ G K ∞ /K ≤ r 2 + δ L K .Therefore, Leopoldt's conjecture for a totally real field K implies the Gross-Kuz'min conjecture for K , as already noticed by Kolster in [Kol91, Corollary 1.3].
Given a prime P ∈ S p (K ), let Γ P be the decomposition subgroup of Γ at P and denote by rec Γ P : K × P → Γ P the corresponding local reciprocity map.Define also the Z p -module By the usual product formula in class field theory the regulator map (4) is well-defined, and it extends to a , where χ cyc is the cyclotomic character.Therefore, L K cyc /K is essentially the same as the map L K of the introduction, and we easily see that δ G Proof.By Kummer theory and local class field theory, Tate's local pairing H 1 (K In particular, dim ker(Loc , so Proposition 3.1.3yields the desired equality.
3.2.Isotypic components.We consider in this paragraph the situation where the Given an algebraically closed field Q of characteristic zero (typically, Q or Q p ), the Q-valued representations of G are semisimple and the regular representation of G splits as where (W, ρ) runs through the set of all the Q-valued irreducible representations of G and (5) is the usual idempotent attached to ρ.
For any finite set of places S of k containing S ∞ (k), let O K [1/S] × be the group of S-units of K .Dirichlet's unit theorem implies that we have a decomposition of where (W, ρ) runs through the set of all non-trivial irreducible representations of G and ).It will be convenient to introduce the following invariants: We record in the next lemma a list of useful properties satisfied by the invariants introduced in (6) and which we make use of in Sections 4 and 5. Recall that a rule ρ → a(ρ) ∈ Z, where ρ runs among all the representations of Galois groups of finite extensions of number fields, is said to be compatible with Artin formalism if, for all finite Galois extensions M/L/E: Lemma 3.2.1.Let L/E be a finite extension of number fields and let a ∈ {d, d + , f }.
(1) The rule ρ → a(ρ) is compatible with Artin formalism. (2 (5) Let M/L be a finite Galois extension, let θ ∈ Art Q p (Gal(M/L)) be irreducible and let χ be a multiplicative character of G L .Then we have Moreover, if θ = 1 L , then we have Assume now that M has at least one real place w and let v (resp.v 0 ) be the place of L (resp. of E) lying below w.Then we clearly have L ρ, then the Frobenius reciprocity implies that there exists a subrepresenta- so this ends the proof of claim (3).We now prove claim (5).The upper bounds on a(χ |G M ) directly follow from claim (4), so we only prove the lower bounds.Since θ is irreducible, the representation (θ ⊗χ) ⊕dim θ (and even Artin formalism then yields the lower bounds of claim (5), as a(Ind L M χ |G M ) = a(χ |G M ) and a takes non-negative values.
We now describe the isotypic components of the map L K ∞ /K .For g ∈ G, P ∈ S p (K ) and η a place of K ∞ above P, the map by a lift g ∈ Gal(K ∞ /k) of g) is a field isomorphism which yields a left G-action x → g(x) (resp.γ → gγ g−1 ) on P|p K × P and on P|p Γ P respectively.This action also restricts to , and G acts trivially on the quotient ( × and the action on H K ∞ /K described above. Fix any (W, ρ) ∈ Art Q p (G) and let Hom G (X , Y ) be the Q p -vector space of all G-equivariant For all p ∈ S p (k), fix a place P 0 of K above p, let G p be the decomposition subgroup of G at P 0 , let W 0 p = W G p and denote by res p the restriction-to-W 0 p map.
Proposition 3.2.2.The map L k ∞ /k (1) can be naturally identified with L k ∞ /k .If 1 ⊂ ρ, then the map L k ∞ /k (ρ) can be naturally identified with the composite map Here, p runs over S p (k) in each sum, and we implicitly used the fact that K ×,∧ Proof.Let j : H K ∞ /K → P|p Γ P be the inclusion map and let j(ρ) be its ρ-isotypic component.Given a prime p ∈ S p (k) and a fixed prime P 0 |p of K as before, we have P|p K and P|p Γ P = Ind G G p Γ P 0 as G-modules and Frobenius reciprocity shows that Hom G (W, P|p K ), the isomorphisms being the natural projection maps.Therefore, j(ρ) • L k ∞ /k (ρ) can be identified with the composite map where the last identification is induced by ⊕res p .Note that res p and rec Γ P 0 commute.Furthermore, letting [n p ] : Γ ∧ P 0 → Γ ∧ p be the multiplication by n p = [K P 0 : k p ] map, the functoriality of Artin's reciprocity law shows that rec p coincides with [n p ] • rec P 0 on k ×,∧ p .Hence, if 1 ⊆ ρ, then the map j(ρ) • L k ∞ /k (ρ) coincides with the map of (7) under the identification [n p ] : Γ ∧ P 0 ≃ Γ ∧ p .Since j(ρ) is an isomorphism in this case, we obtain the desired description of L k ∞ /k (ρ).In the case where ρ = 1, the map L K ∞ /K (1) is nothing but the restriction

by by Frobenius reciprocity and by
).An easy adaptation of the proof of Proposition 3.1.5then shows that L k ∞ /k (ρ) coincides with the localization map H 1 W 0 p ) under these identifications.Corollary 3.2.4.
p , Q p ) , the map being the restriction-to-W 0 p map followed by the post-composition by log p •ι p .Proof.The first claim directly follows from Propositions 3.1.5and 3.2.2.The second claim follows from from Proposition 3.2.2 and from the fact that, if Proof.Part (b) is obvious from the definition of L k ∞ /k (ρ).Part (a) is true if ρ is trivial by Corollary 3.2.4(1).Let K ′ /K /k be Galois extensions, take 1 ⊆ ρ ∈ Art Q p (Gal(K /k)) and de- note by ρ its inflation to Gal(K ′ /k).Then the maps of (7) attached to ρ and ρ coincide As for (c), take any splitting field K of ρ which is Galois over k ′ and put G = Gal(K /k), G ′ = Gal(K /k ′ ).Then Frobenius reciprocity identifies the ρ ′ -isotypic component of L K ∞ /K (seen as a G ′ -equivariant map) with the ρ-isotypic component of L K ∞ /K (seen as a G-equivariant map).Hence, the last property follows from Proposition 3.2.2.
We next define an analogous invariant for Leopoldt's conjecture.For any Galois extension K /k with Galois group G, the localization map Here (and as in the definition of L k ∞ /k (ρ)), P 0 is a fixed place of K above p for every place p of k.The last isomorphism is given by Frobenius reciprocity and is induced by the natural projection map.As in Corollary 3.2.5, it is easy to see that the rule ρ → δ L k (ρ) is compatible with Artin formalism.Remark 3.2.6.In terms of Bloch-Kato Selmer groups for q W, the injectivity of ι k (ρ) is equivalent to that of the localization map H 1 The following lemma on local Galois representations with finite image will help us de-

Here, we let G
as a Galois module, it is enough to show that m is injective.Choose any finite Galois extension L/Q p which contains E and over which ρ is realizable, i.e., there exist a L[G Q p ]-module W L and an isomorphism W L ⊗ L Q p ≃ W. Then we have to show that the map Hom L so this follows from the Q p -admissibility in the sense of Fontaine of Galois representations of finite image (see e.g.[FO08, Proposition 2.7]).Proposition 3.2.8.Let (W, ρ) ∈ Art Q p (G Q ), let K ⊂ Q be a splitting field of ρ and let P 0 be the p-adic place of K defined by a fixed embedding ι p : Q → Q p .We have As in the proof of Theorem 4.1.1,fix a Q-structure W Q of W. Fix also a basis w 1 , . .., w f of the subspace W 0 ) such that the elements Since e(ρ) kills p Q , we deduce from Proposition 2.1.2that the entries of the matrix M ′ = (log p (ι p (Ψ j (w Corollary 4.2.2. (1) Let ρ ∈ Art Q p (G k ), let K be the field cut out by ρ and let f = f (ρ).Then the following inequalities hold true: (2) Let K /k be a finite extension of number fields.Then the following inequalities hold true: Proof.We know that ρ → δ G k (ρ) is compatible with Artin formalism by Corollary 3.2.5.We now explain how to prove (1).Again by Artin formalism, it suffices to prove (1) with ρ replaced by any irreducible subrepresentation For such a θ, Lemma 3.2.1 implies that d + (θ) ≥ 1 if K contains at least one real place, and d + (θ) = d(θ) if K is totally real.Therefore, the inequalities in (1) for θ directly follow from Theorem 4.2.1.Finally, (2) follows from (1) as in the proof of Corollary 4.1.2.Remark 4.2.3.The matrices M and M ′ appearing in the course of the proof of Theorems 4.1.1 and 4.2.1 have full rank under the p-adic Schanuel conjecture.Therefore, Artin formalism shows that Leopoldt's and Gross-Kuz'min's conjectures hold in great generality under the p-adic Schanuel conjecture.4.3.Applications.Theorem 4.3.1.Let k be a totally real number field and let (V , ρ) ∈ Art Q p (G k ) be such that d + (ρ) = 0. Then Gross's p-adic regulator matrix R p (V ) [Gro81, (2.10)] is of size f (ρ) and of rank at least f (ρ)/2.
Proof.Let K be the CM field cut out by ρ and let (Hom Q p (V , Q p ), ρ * ) be the contragredient representation of ρ.Gross's regulator map λ p defined in [Gro81,(1.18)]can be identified with the "minus part" of L K , which is, by definition, the restriction of L K to the subspace where the complex conjugation acts by −1.This means that λ p and L K share the same In the next theorem, we write k + for the maximal totally real subfield of a number field k, and Q ab for the maximal abelian extension of Q. Theorem 4.3.2.Let K /k be an abelian extension of number fields.The Gross-Kuz'min conjecture holds for K in each of the following cases.
(a) Either |S p (K )| ≤ 2, or |S p (K )| ≤ 3 and K has at least one real place, or k is an imaginary quadratic field, or k is a real quadratic field and K has at least one real place.
Proof.Recall that δ G (−) is compatible with Artin formalism by Corollary 3.2.5.We shall often appeal to Lemma 3.2.1 and to the following consequence of Theorem 4.2.1 without further notice.For any irreducible representation it follows from Corollary 4.2.2 that δ G K = 0 for K satisfying one of the two first assumptions in case (a).Consider the two last assumptions in (a) and assume that K /Q is Galois.We claim that δ G Q (θ) = 0 for all irreducible θ ∈ Art Q p (Gal(K /Q)).We may assume that dim θ ≥ 2, so f (θ) ≤ ( f (1 K ) − f (1 Q ))/(dim θ) ≤ (|S p (K )| − 1)/2.The two last assumptions in (a) ensure that we either have f (θ) ≤ 1, or f (θ) ≤ 2 and d + (θ) = d(θ) ≥ 1, so we indeed have δ G Q (θ) = 0. Therefore, δ G K = 0 in case (a).Let G = Gal(K /k) and let Ĝ = Hom(G, Q × p ) be the group of characters of G.We place ourselves in cases (b), (c) and (d), we fix χ ∈ Ĝ and we show that . Suppose now we are in case (c).Then χ descends to a character χ Q of G Q .Moreover, as k/Q is Galois, any irreducible subrepresentation ρ of Ind Moreover, if K is totally real, then any such ρ satisfies d + (ρ) = d(ρ) ≥ 1, so we can conclude We now assume to be in case (d).Then Ind Q k χ has dimension 2, so it is either irreducible or it is the sum of two characters of G Q , say η 1 and η 2 .In the latter case, we already know that δ is irreducible, then the assumptions on K and k imply that f (χ) ≤ 2 and d + (χ) ≥ 1, yielding δ G k (χ) = 0. Therefore, δ G K = 0 in the cases (b), (c) and (d).We now make the assumptions in (e) and we assume without loss of generality that K is Galois over Q with Galois group G.By the Schur-Zassenhaus theorem, G is the semi-direct product of H := Gal(k/Q) acting on G. Let ρ ∈ Art Q p (G) be irreducible and let us prove that δ G Q (ρ) = 0.By [Ser78, Chap II § 8.2], ρ can be written as Ind where k ′ /Q is a subextension of k/Q, θ an irreducible representation of Gal(k/k ′ ) and χ a character of Gal(K /k ′ ).Note that f (ρ) ≤ |S p (k)|/(dim θ) ≤ 2/(dim θ), so we may assume without loss of generality that , and otherwise we have d + (ρ) ≥ 1.In any case, δ G Q (ρ) = 0 so we may infer δ G K = 0 in case (e) as well.

VANISHING LOCUS OF GROSS'S DEFECT
5.1.Preliminaries.This section is devoted to the proof of the following theorem which, in turn, implies Theorem 1.8 stated in the introduction.Theorem 5.1.1.Let k be a number field and let ϕ : G k → Q × p be a finite-order character.Assume also that p completely splits in the number field cut out by ϕ.
(1) Assume that r 2 + δ L k ≤ 1, where r 2 is the number of complex places of k.If there exists at least one (2) Assume that k is an imaginary quadratic field.There is at most one Z p -extension k ∞ /k for which δ G k ∞ /k (ϕ) = 0, and it has a transcendental slope (see Example 5.2.3 for a definition of the slope of k ∞ /k).Moreover, if ϕ cuts out an abelian extension of Q or if the polynomial X Y Z 2 − (A X − BY )(C X − DY ) does not vanish on any 7-tuple (a, b, c, d, x, y, z) ∈ Λ 7 which form a Q-linearly independent set, then δ G k ∞ /k (ϕ) = 0 for any Z p -extension k ∞ of k.
Proof of Theorem 1.8, assuming Theorem 5.1.1.Let K be an abelian extension of an imaginary quadratic field k and let K ab ⊂ K be its maximal absolutely abelian subfield.By Artin formalism (Corollary 3.2.5)and by Theorem 5.1.1,we have δ G K k ∞ /K > 0 for a given Z p -extension k ∞ /k if and only if there exists a character ϕ of Gal(K /k) such that δ G k ∞ /k (ϕ) > 0. Such a character cannot be a character of Gal(K ab /k), and moreover, k ∞ is uniquely determined by ϕ.Therefore, we have In the rest of Section 5, we fix once and for all an abelian extension K /k with Galois group G such that p totally splits in K .We let n be the degree of k and (r 1 , r 2 ) its signature, and we put r = r 1 + r 2 − 1 − δ L k so that the maximal multiple Z p -extension of k has rank n − r.Write p 1 , . .., p n for the p-adic primes of k.Finally, denote by Z (k) the set of all Z p -extensions of k.Instead of working with the map L k ∞ /k (ϕ), it will be more convenient to consider the Proof.This follows from Poitou-Tate duality as in the proof of Proposition 3.1.5where one replaces the G K -module Q p by the module Q p (ϕ) on which G k acts by ϕ.
Since ϕ is a multiplicative character, the ϕ-isotypic component of a 5.2.Matrices in logarithms of algebraic numbers.For any P ∈ S p (K ), we identify H 1 (K P , Q p ) with Hom(K × P , Q p ) ≃ Hom(Q × p , Q p ) via local class field theory.We also see log p and the p-adic valuation map ord p as additive characters K In order to describe elements in the domain of the map (9) we make use of the short exact sequence of where A is induced by the Artin map.
Let 1 ≤ i ≤ n and fix a prime P i of K above p i .We define a basis {η i,ϕ , ηi,ϕ } of the ϕcomponent of Hom( P|p i K × P , Q p ) as follows.First define characters η i and ηi of P|p i K × P by imposing that they are supported on K × P i and that η i|K × P i = − log p and ηi|K × P i = ord p .We then define Let u i be any P i -unit of K which is not a unit (take for example a generator of P h i , where h is the class number of K ).The choice of u i with a given P i -valuation is unique, up to multiplication by a unit of K .Consider It is clear that u i,ϕ is a unit away from the primes above p i , and u 1,ϕ , . .., u n,ϕ form a basis of (Q We also fix a basis {ε 1,ϕ , . .., ε r(ϕ),ϕ } of (Q ⊗ O × K )[ϕ] modulo the kernel of Leopoldt's map ι k (ϕ) of (8), where r(ϕ) = d + (ϕ)−δ L k (ϕ).For all j = 1, . .., n, one can see via ι P j : K → K P j = Q p the elements u i,ϕ and ε i,ϕ inside Q ⊗ Q × p .We then define two matrices L ϕ = (L i, j,ϕ ) and M ϕ = (M i, j,ϕ ) of respective sizes n × n and r(ϕ) × n by letting (11) L i, j,ϕ = log p (ι P j (u i,ϕ )) where we extended log p to Q ⊗ Q × p by linearity.Notice that M ϕ has full rank by construction.Let η ′ be an element in the ϕ-component of n i=1 Hom( P|p i K × P , Q p ), which we write as t i η i,ϕ + ti ηi,ϕ in the basis {η i,ϕ , ηi,ϕ : 1 ≤ i ≤ n}.Denote by T and T the column matrices of respective coordinates (t 1 , . .., t n ) and ( t1 , . .., tn ).Lemma 5.2.1.η ′ belongs to the image of the map A of (10) if and only if T = L ϕ T and M ϕ T = 0.
Proof.By the exactness of (10) such an η ′ is characterized by its vanishing at all the u i 's and the ε i,ϕ 's.The lemma then follows from a straightforward computation, using that η j,ϕ (ι P j (u i )) = − log p (ι P j (u i,ϕ )) and η j,ϕ (ι P j (u i )) = ord p (ι P j (u i )) for all 1 ≤ i, j ≤ n.
In what follows we repeatedly use the following elementary fact.For all compact topological groups G, any non-trivial continuous group homomorphism η : G → Q p factors through a quotient Z η isomorphic to Z p , and two such homomorphisms η and η ′ are proportional if and only if Z η = Z η ′ .Conversely, any topological group Z isomorphic to Z p which arises as a quotient of G defines a continuous homomorphism η : G → Q p , which is unique up to scaling. (2 Proof.Let us prove (1).Notice first that all S ∈ ker M lie in the kernel of N(S).By Proposition 5.2.2 with ϕ = 1 and K = k, C (k) is in bijection with the set of all S ∈ P n−1 (Q p ) such that MS = 0 and rk N(S) < n − 1. Assume first that k is quadratic, so n = 2 and r = 0. Given any S = (s 1 , s 2 ) ∈ P 1 (Q p ), the matrix N(S) = Diag(S)L − Diag(LS) has the form As log p is injective on Z × p , log p (ι p 1 (u 2 )) and log p (ι p 2 (u 1 )) both are non-zero, so at least one of the two non-diagonal entries of N(S) is non-zero.Therefore, this matrix has rank one for any S ∈ P 1 (Q p ), and C (k) = .
We no longer assume that k is imaginary quadratic, but we still assume that n − r ≤ 2.
The case where n − r = 1 is trivial, because it forces Z (k) = {k cyc }.We may then assume that n − r = 2. Since M has full rank r, there exist invertible matrices P,Q such that P MQ = (I r | 0), where I r is the identity matrix of size r.The change of variables S ′ = Q −1 S induces a linear bijection between ker M ⊂ P n−1 (Q p ) and the projective line {0} × P 1 (Q p ) ⊂ P n−1 (Q p ).Now consider the list P 1 (S ′ ), . .., P t (S ′ ) of all (n − 1) × (n − 1)-minors of the matrix All the P k 's are two-variable homogeneous polynomials of degree ≤ n − 1.In particular, if , then at least one of the P k 's is not the zero polynomial and hence, it has at most n − 1 zeros in {0} × P 1 (Q p ), so we can conclude that |C (k)| ≤ n − 1.
The proof of point (2) very similar to the previous one.Indeed, by Proposition 5.2.We end the proof of Theorem 5.1.1 with the case where k is imaginary quadratic.We let τ be the complex conjugation of k.Recall that τ acts on ϕ via ϕ τ (g) = ϕ(τgτ) and that ϕ τ = ϕ if and only if ϕ cuts out an extension of k which is abelian over Q.
Proof.Let k ∞ ∈ Z (k) of coordinates S = (s 1 , s 2 ).Take K to be the Galois closure over Q of the field cut out by ϕ.Note that K = k and that p totally splits in K .By Proposition 5.2.2, k ∞ belongs to C ϕ (k) if and only if the matrix has rank 1.The definition of L ϕ (and also L ϕ τ ) involves the choice, for i = 1, 2, of a prime P i of K above p i and a P i -unit u i of non-zero valuation.Since τ(p 1 ) = p 2 , one may take and let M = Q ker ρ be the field extension cut out by ρ.Then d + (ρ) = d(ρ) if and only if M is totally real.If d + (ρ) = 0, then L is totally real and M is a CM field.If M has at least one real place, then d + (ρ) ≥ dim ρ and any subrepresentation θ
2, C ϕ (k) is in bijection with the set of all S ∈ P n−1 (Q p ) such that MS = 0 and rk N ϕ (S) < n.The same argument with the (n−1)×(n−1) minors of N(S) replaced by the n×n minors of N ϕ (S) shows that, if C ϕ (k) = Z (k), then |C (k)| ≤ n.
and that L K cyc /K (1) can be naturally identified with L k ∞ /k .
Remark 3.2.3.The map L k ∞ /k (ρ) admits a more intrinsic description in terms of Bloch-⊂ H 1 (k p ,W) under Tate's local pairing.Then Kummer theory and the Inflation-Restriction exact sequence provide natural isomorphisms H