On the Gross order of vanishing conjecture for large vanishing orders

We prove the Gross order of vanishing conjecture in special cases where the vanishing order of the character in question can be arbitrarily large. In almost all previously known cases the vanishing order is zero or one. One major ingredient of our proofs is the equivalence of this conjecture to the Gross-Kuz'min conjecture. We present here a direct proof of this equivalence, using only the known validity of the Iwasawa Main Conjecture over totally real fields.


Introduction
One of the major breakthroughs in the theory of L-functions of number fields in recent years was the proof of one part of the Gross-Stark conjecture for onedimensional characters by Dasgupta, Kakde and Ventullo ( [DKV18]). This conjecture compares, in an appropriate setting, the leading terms of the Deligne-Ribet p-adic L-function and the Artin L-function at s = 0 for each character. Although the main result of [DKV18] is unconditional, only the Gross order of vanishing conjecture (abbreviated by GVC) asserts that the vanishing orders of the two Lfunctions are equal.
Currently, GVC is basically only known unconditionally in cases where the vanishing order of the Artin L-function at s = 0 is zero or one, e.g if the base extension K/Q is abelian, by work of Gross [Gro81] or Emsalem [Ems83].
Seemingly completely independent of the above, there is a classical conjecture in Iwasawa theory, called Gross-Kuz'min conjecture (abbreviated by GKC), which asserts that for the cyclotomic Z p -extension of a number field K (p a fixed prime) with Galois group Γ, the quotient of Γ-coinvariants of a certain Iwasawa module is finite. This conjecture (in this generality) has been formulated first by Kuz'min in [Kuz72] and independently by Jaulent in [Jau86]. It is known for abelian extensions K/Q by work of Greenberg ([Gre73]) and it has been studied intensely by Jaulent, Kolster, Kuz'min, Nguyen Quang Do and many others.
In the first part of this manuscript we show that GVC is (under a standard assumption) equivalent to an appropriate characterwise variant of GKC. In particular, we allow the characters to have arbitrary dimension and the base extensions to be non-abelian over Q.
In the second part we construct examples with large vanishing orders for which we prove that GKC and therefore, via the equivalence, also GVC holds.
In the remaining part of the introduction we state our results more explicitly and sketch our general proof strategy.
The Gross order of vanishing conjecture We keep the notation from the last paragraph and additionally introduce a set S which consists of the places above p and above ∞. Then for an irreducible character of G over C, we denote by L S (s, χ) the S-truncated Artin L-function. Moreover, for a totally even character ψ of G over C p we denote by L p,S (s, ψ) the S-truncated Deligne-Ribet p-adic L-function (see e.g. [DR80] and [Gre83]). Then we can recall the following conjecture.
Conjecture GVC(K/R, χ). Let K be a Galois CM-extension of a totally real field R with Galois group G. If χ is a totally odd irreducible character of G over C p , then ord s=0 L p,S (s,χω R ) = ord s=0 L S (s, χ); here we identify χ with a totally odd irreducible character of G over C via a fixed isomorphism j : C ∼ = C p . Moreover,χ is the contragredient character and ω R is the Teichmüller character.
In [Bur20] the inequality '≥' of this conjecture is shown unconditionally. We will recover this inequality (under our hypothesis K ∩ R ∞ = R) in course of the proof that the two conjectures are equivalent in an appropriate setting.
Equivalence of the conjectures Fix K/R and G = Gal(K/R) as above, and recall that R ∞ denotes the cyclotomic Z p -extension of R.
In Section 3, we will prove the following Theorem A. Let K be a finite Galois CM-extension of a totally real number field R with Galois group G, let p be an odd prime and let χ be a totally odd irreducible character of G over C resp. C p . Assume that K ∩ R ∞ = R. Then GKC(K/R, χ) holds if and only if GVC(K/R, χ) holds.
Actually an equivalence of appropriate formulations of the Gross-Kuz'min and the Gross order of vanishing conjectures can be obtained more generally, namely without the assumption K ∩ R ∞ = R, by combining recent and deep results of Burns [Bur20] with work of Kolster [Kol91] (cf. Remark 4.14). We give here a more direct proof, the main ingredients of which are the Iwasawa main conjecture over totally real fields proved by Wiles ([Wil90]), Brauer Induction and a careful analysis of the module Examples with large vanishing order To the knowledge of the authors in almost all proven cases of GVC(K/R, χ) the vanishing order of L S,χ (s, χ) at s = 0 is 0 or 1, e.g. if R = Q.
In view of Theorem A, it should not be a surprise that also (almost, cf. Remark 2.7) all known instances of the Gross-Kuz'min conjecture are cases where K contains only one or two primes above p or where K/Q is abelian. So evidence for these important conjectures for 'higher order of vanishing' is scarce. This is, in part, based on the fact that most of the proofs depend on the famous 'transcendence result' of Baker/Brumer, the necessary generalization of which is only conjectural (for work on the p-adic Schanuel conjecture in connection to the above conjectures see [Kuz18] and [Ems83]).
We also use the Theorem of Brumer/Baker in our results. First, we derive the following result: Theorem B. For infinitely many primes and infinitely many non-abelian extensions K/R of degree 8, GVC(K/R, χ) holds for all characters χ ∈ Ir − (Gal(K/R)).
This includes examples with arbitrarily large vanishing order.
The main input here is the known validity of Leopoldt's conjecture and the equivalence to the Gross-Kuz'min conjecture in this specific setting.
In [Gre73], Greenberg used the Theorem of Brumer/Baker in order to prove the minus part of GKC for a CM-field K, provided that K/Q is abelian and that the prime p is totally split in K (moreover, he showed how to circumvent the latter condition in the abelian case). Using the approach of [Gre73] and building on results from [Kle19] about the asymptotic growth of Galois coinvariants in a Z pextension, we can more generally use Greenberg's approach in order to derive from the Theorem of Brumer/Baker the following Theorem C. Let K be a finite abelian CM-extension of a totally real number field R, and suppose that there exists some prime p of R above p which is totally split in K, and such that R p = Q p . Then where r denotes the number of primes of the maximal totally real subfield K + of K which split in K, and [K : R] = 2s.
We then proceed by explaining how one can derive from Theorem C examples with large vanishing orders (see Corollary 4.13 and Example 4.16) where GVC(K/R, χ) holds; the main idea is to enlarge the totally real field R ⊆ K.
In a subsequent paper the authors plan to describe how one can use this work in order to verify both conjectures numerically in many interesting new cases.
We want to thank Dominik Bullach for suggesting to think about this subject in the first place and many helpful conversations at the initial phase of this project. We are very grateful to Pascal Stucky for many illuminating discussions, in particular with the first author. Moreover, we want to thank Cornelius Greither for several insightful comments and suggestions.

Notation
Unless stated otherwise, p will always denote a rational prime.
We denote by K a fixed algebraic closure of a field K with characteristic 0 and by G K the absolute Galois group.
For a number field K we denote by S ∞ (K) the set of infinite places in K and by S p (K) the set of places above p in K. Let S be a finite set of places of K containing S ∞ (K). Then we denote by E S K the group of S-units of K and by Cl S (K) the S-class group of K. Moreover, we write Cl(K) = Cl S∞(K) (K) and E ′ K = E Sp(K) K (i.e., the group of p-units of K) for brevity. Let K/R be a finite Galois extension with Galois group G. We denote by Ir(G) resp. Ir p (G) the set of irreducible characters of G over C resp. over C p and for each such character χ we fix a representation V χ over C resp. over C p . For each such χ we writeχ for its contragredient. For each χ ∈ Ir p (G), we denote by ε χ = χ(1) |G| · σ∈G χ(σ)σ −1 the corresponding idempotent. Now suppose that K is a CM-field, and that R is totally real. Then we write K + for the maximal real subfield of K and τ for the (unique) non-trivial element of Gal(K/K + ). We say that χ ∈ Ir(G) is even if χ(τ ) = χ(1), and odd if χ(τ ) = −χ(1).
More generally, suppose that χ denotes a character of the absolute Galois group G R of R with finite order. Letting ker(χ) = {g ∈ G R | χ(g) = χ(1)}, we let R χ ⊆ R be the subfield fixed by ker(χ). Then χ is called totally even if R χ is totally real, and it is called totally odd if R χ is CM.
In the special setting of characters of G = Gal(K/R), K a CM-Galois extension of R, the notions odd and totally odd (respectively, even and totally even) coincide, because R χ ⊆ K for each character χ of G.
We write Ir ± (G) and Ir ± p (G) for the subsets of Ir(G) and Ir p (G) that consist of the characters which are totally even or totally odd, respectively. For any G-module M we define M ± := {m ∈ M : τ (m) : ±m}; these are two G-submodules of M .
We denote by K ∞ /K the cyclotomic Z p -extension of a number field K and by K n the unique intermediate field of degree p n over K, n ∈ N 0 .

Two conjectures named after Gross
In this chapter we formulate the Gross-Kuz'min conjecture, the Gross order of vanishing conjecture and characterwise variants of both. Moreover, we recall the cases where the conjectures are unconditionally known.

The Gross-Kuz'min conjecture
Fix a prime p and a number field K. For each n ∈ N, let A ′ n = A n /B n be the quotient of A n = Cl(K n ) p , the p-Sylow subgroup of Cl(K n ), by the subgroup B n generated by the primes of K n dividing p, and let A := lim ← − A n and A ′ := lim ← − A ′ n , where the projective limits are taken with respect to the norm maps. In what follows, we write GKC(K) for "the Gross-Kuz'min conjecture for the cyclotomic Z p -extension K ∞ /K".
Remark 2.2. Let L/K be a finite extension of number fields, let K ∞ /K be the cyclotomic Z p -extension, and let L ∞ = L·K ∞ . If GKC(L) holds, then also GKC(K) is true.
The minus Gross-Kuz'min conjecture If K is a CM-field and p = 2, then we consider minus parts (A ′ ) − = lim ← −n (A ′ n ) − . The following conjecture is then a part of Conjecture 2.1: Conjecture 2.3 (Minus Gross-Kuz'min conjecture). With the above notation, We write GKC − (K) for "the minus Gross-Kuz'min conjecture for the cyclotomic Z p -extension K ∞ /K".
Note: Conjecture 2.3 is true if and only if One can further reformulate this conjecture: Character-wise minus Gross-Kuz'min conjecture Let K be a CM-field, p = 2, and let R ⊆ K be a totally real subfield such that K/R is normal. We write G = Gal(K/R). Recall the notation R ∞ from the Introduction. In the following, we assume that In the following, we will write GKC(K/R, χ) for "the characteristic polynomial f A ′ ,χ (T ) is not divisible by T ".
A ′ , Conjecture 2.3 is equivalent to the validity of Conjecture 2.4 for all χ ∈ Ir − (G).
The next comment will explain why Gross's name appears in the conjecture.
(1) There exists a different formulation of Conjecture 2.1 which dates back to Jaulent's thesis [Jau86]. It asserts that for a number field K a certain invariant, called "Gross-defect" δ K , is zero. Indeed, [Kol91, Theorem 1.14] shows that δ K = 0 if and only if (A ′ n ) Γ is bounded independently of n, which in turn holds if and only if (A ′ ) Γ is finite.
(2) B. Gross's name appears in the conjecture because Conjecture 1.15 in [Gro81] is equivalent to the Gross defect of [Kol91] being zero if one considers CM-fields K and minus-parts of the objects involved.
We now give a list of all (unconditional) results on the Gross-Kuz'min conjecture holds (as far as the authors are aware).
(1) Let K be any number field. If K contains exactly one prime above p, then GKC(K) follows from Chevalley's Theorem (see [Lan90,Lemma 4.1 in Chapter 13]); if K contains two such primes then GKC(K) is known due to [Kle19, Theorem 3.4].
(2) Let K be a number field such that K is normal over Q, Gal(K/Q) ∼ = S 4 , K contains an imaginary quadratic subfield, and p is such that the decomposition group of some prime of K dividing p is isomorphic to a 3-Sylow subgroup of Gal(K/Q). Then [Kuz18,Corollary 4] shows that GKC(K) holds. (3) Let K/Q be a normal extension, G := Gal(K/Q), and let p be a prime number such that the decomposition field F p = K Dp of p is normal over Q and totally real, and such that in the Artin-Wedderburn decomposition of the group ring Q p [G/D p ] only matrix rings M n (R) with n ≤ 2 occur. Then GKC(K) holds for this prime p ([Jau02]). (4) Greenberg proves GKC(K) for any abelian extension K of Q in [Gre73]. (5) Let K be a CM-field. Then GKC − (K) holds if K contains a CM-subfield k such that the prime p does not split in K/k and GKC − (k) holds (see [Gre73, Section 3] and [Jau17, Scolie 10]).

The order of vanishing conjecture of Gross
Now we describe the order of vanishing conjecture originally formulated as Conjecture 2.12 a) in [Gro81]; we use the notation of [Bur20]. Let K be a finite CM Galois extension of a totally real number field R. We fix a finite set S of places in R which contains S ∞ (R) and S p (R). Let χ ∈ Ir(G). Then we denote by L S (s, χ) the S-truncated Artin L-function of χ. By [Tat84, Proposition 3.4] we have where Y K,S denotes the free abelian group on S.
Remark 2.8. Equation (1) implies that r S,χ = r S,χ α for all automorphisms α of C. If ψ ∈ Ir p (G), then we can therefore set r S,ψ := r S,χ , where χ is any character in Ir(G) with χ j = ψ for some field isomorphism j : In what follows, we will usually identify a p-adic character ψ with some fixed χ ∈ Ir(G).
For each character χ in Ir + p (G) we write L p,S (s, χ) for the S-truncated Deligne-Ribet L-function (cf. [DR80]) as discussed by Greenberg in [Gre83]. We write ω R for the Teichmüller character ω R : G R −→ Z × p . Recall that for χ ∈ Ir − p (G) the character χω R is totally even.
Conjecture 2.9. (Gross order of vanishing conjecture) For each χ ∈ Ir − p (G) we have where we again identify χ with a character in Ir − (G) by using some fixed field isomorphism j : Remark 2.10.
(1) In the classical formulation of the conjecture, S = S ∞ (R)∪S p (R). We will write GVC(K/R, χ) for "the Gross order of vanishing conjecture for the extension K/R and the character χ ∈ Ir − p (Gal(K/R))" in this classical setting.
(2) Let R be a totally real field, and let χ : G R −→ Q × be a totally odd finite (one-dimensional!) character. We let K = R ker(χ) (which is a CM-field) and G := Gal(K/R). Then [CD14] and [Spi14] independently show that for each (3) [Bur20, Theorem 3.1 i], which is proved via a different approach, implies that more generally for arbitrary finite CM-extensions K of R and each χ ∈ Ir − p (G) we have ord s=0 L p,S (s,χω R ) ≥ ord s=0 L S (s, χ).
(4) By using Brauer Induction one obtains: Conjecture 2.9 is valid if and only if it is valid for all L-functions of the form L p,SM (s,φω M ), where M is any totally real intermediate field of K/R and φ is a one-dimensional character in Ir − p (Gal(K/M )). This is [Bur20, Remark 2.2]. Now we give a list of all (unconditional) results on Conjecture 2.9 (to the knowledge of the authors).

The equivalence of the conjectures
In this chapter we prove Theorem A from the Introduction, which establishes the equivalence between our characterwise versions of the Gross-Kuz'min conjecture and the order of vanishing conjecture of Gross under a standard hypothesis. Using the notation from Section 2.2, recall that K is a finite Galois CM-extension of a totally real number field R, p is an odd prime and G = Gal(K/R). For any fixed character χ ∈ Ir − p (G), we let (by abuse of notation) χ ∈ Ir − (G) be a character which corresponds to χ under some fixed isomorphism j : C ∼ = C p , and we define R χ is a CM-field, since χ is a totally odd character. We let H := R χ (µ p ) and H ∞ := H · R ∞ , where µ p denotes the group of p-th roots of unity and R ∞ denotes the cyclotomic Z p -extension of R. Note that H is also a CM-field. Finally, we define Γ := Gal(H ∞ /H) and we identify Λ := Z p Γ with the ring of power series Z p T .
Theorem A. Let K be a finite Galois CM-extension of a totally real field R with Galois group G, let p be an odd prime and χ ∈ Ir − p (G). Assume that Then GKC(K/R, χ) holds if and only if GVC(K/R, χ) holds.
In view of Remark 2.5, we derive the following Corollary 3.1. Let K be a CM-field, and let p = 2. Then GKC − (K) holds if and only if for any totally real subfield R of K satisfying K ∩ R ∞ = R, GVC(K/R, χ) holds for each character χ ∈ Ir − p (Gal(K/R)). Proof of Theorem A. The proof splits into two parts, the first of which is purely algebraic. The second part involves as main ingredient the Iwasawa main conjecture.
Fix some χ, and let R χ , H and H ∞ be defined as above. Let L ∞ be the maximal unramified abelian pro-p-extension of H ∞ . Then X = Gal(L ∞ /H ∞ ) is a Gal(H ∞ /R)-module under the usual conjugation action. Note that, via class field theory, we have a Gal(H ∞ /R)-isomorphism where the limit is taken using the norm maps the primes above p are totally split, then class field theory induces isomorphisms Analogously, Iwasawa modules can be defined for the cyclotomic Z p -extension M ∞ of any number field M . In order to distinguish the corresponding Iwasawa modules, we will write A (M∞) , (A ′ ) (M∞) , etc.
As a first step, we will show that GVC(K/R, χ) holds if and only if GVC(H/R, χ) holds (our motivation for proving this reduction is that the Iwasawa main conjecture, which will be used in the second part of the proof, is formulated for the Z p -extension H ∞ /H rather than for K ∞ /K).
In particular, then the following conjectures are equivalent: Proof. Let M ∞ and (R χ ) ∞ denote the cyclotomic Z p -extensions of M and R χ , and write G = Gal(M/R) for this proof. Using the canonical surjection Gal(R/R) ։ G, we can view χ as a character on G. GKC(M/R, χ) concerns the module SinceG := Gal(M/R χ ) = ker(χ| M ), χ factors through G/G = Gal(R χ /R). If I := {σ 1 , . . . , σ l } denotes a set of representatives for the cosets of G/G, then χ(σ i ρ) = χ(σ i ) for each i ∈ {1, . . . , l} and every ρ ∈G. Therefore the above module can be written as The inner sum in (3) yields a group ring element corresponding to the norm map N M/Rχ , and i.e. the characteristic power series of these three Λ-modules differ by at most a power of p. This shows that GKC(R χ /R, χ) holds if and only if GKC(M/R, χ) holds.
The following proposition is due to Sinnott.
Recall that Gal(H ∞ /R) acts on X = Gal(H ∞ /L ∞ ). We have R χ ∩ R ∞ = R by assumption (2). Therefore also R ∞ ∩ H = R (because p does not divide the degree Let ∆ := Gal(H/R). We write ∆ w ⊆ ∆ for the decomposition group of any w ∈ S p (H) and G w ⊆ G = Gal(K/R) for the decomposition group of a w ∈ S p (K). Fix a prime v of R dividing p, and consider the module by Proposition 3.3,(a). In the following lemma, we fix a prime w 1 of H above v and a prime w 2 of K dividing v such that w 1 ∩ R χ = w 2 ∩ R χ , and we denote by p n , for n ∈ N, the number of primes of H ∞ dividing w 1 . Finally, let ω n (T ) = (T + 1) p n − 1, n ∈ N.
(1) Since R χ is contained in both H and K, we can consider χ as a character on ∆ and on G. if w 1 ∩ R χ = w 2 ∩ R χ : if D 1 denotes the decomposition field of w 1 in H/R and D 2 denotes the decomposition field of w 2 in K/R, then D 1 ∩ R χ = D 2 ∩ R χ , because the prime w i ∩ D i ∩ R χ does not split at all in R χ , for i ∈ {1, 2}. Now Gal(K/R χ ) and Gal(H/R χ ) are contained in the kernel of χ. Therefore  By definition of B v,H , ψ = Ind ∆ ∆w1 1 ∆w 1 , and therefore Frobenius reciprocity implies that χ, ψ ∆ = χ| ∆w 1 , 1 ∆w 1 ∆w 1 = dim C (V ∆w 1 χ ) (see also [Tat84,p. 24]). This means that we have an isomorphism of Q p -vector spaces Now we go up the Iwasawa towers. For every prime w 0 | v of H, we note that w|w0 in H∞ wQ p is a Λ-module, and a Q p -vector space of dimension p n = deg(ω n ), by the definition of n. Since ω n annihilates each prime of H ∞ lying above w 0 , it follows that we have an isomorphism In particular, this implies that As we already pointed out in Remark 3.5(4), we may replace the decomposition group ∆ w1 by the decomposition group G w2 ⊆ G in the above formula if w 1 ∩ R χ = w 2 ∩ R χ ; this proves the lemma. Now we start with the analytic part of the proof. Recall that r S,χ = ord s=0 L S (s, χ), as in Section 2.2.
Lemma 3.6. For each χ ∈ Ir − (G), we have where V χ denotes the representation space of χ, and where w ∈ S(K) denotes any prime dividing v, respectively.
Note: the infinite places do not contribute to the above sum, since K is a CMfield.
Recall that X = Gal(L ∞ /H ∞ ). Let V X := X ⊗ Zp Q p , and let V (χ) X = ε χ · V X be the eigenspace corresponding to the action of Gal(H/R) via χ. We define f X,χ (T ) to be the characteristic polynomial of γ − 1, where γ is a fixed generator of Γ (cf. also Section 2.1).
Since the multiplicity of each zero of f X,χ (T ) (in the algebraic closure Q p ) is a multiple of χ(1), we can define a polynomial g X,χ (T ) ∈ Z p [χ][[T ]] as the monic polynomial that satisfies f X,χ (T ) = g X,χ (T ) χ(1) .
Next we look at the connection to p-adic L-functions. Let c : Γ −→ Z × p be the cyclotomic character and set u := c(γ). Recall that S = S ∞ (R) ∪ S p (R). Then for each non-trivial totally even one-dimensional character ψ there exists (e.g. by [DR80]) a unique power series G ψ (T ) ∈ Z p [ψ] T such that Assume now that ψ is a character of arbitrary dimension with R ψ totally real. With the help of Brauer Induction one can use the one-dimensional case to define a p-adic L-function satisfying the desired interpolation property. Moreover, one finds power series G ψ (T ) and H ψ (T ) satisfying an equation analogous to (4).
In this general situation, G ψ (T ) a priori is contained only in the quotient field of Z p [ψ] T , but using [Gre83, Prop. 5], Wiles showed (cf. [Wil90, Thm. 1.1]), by proving the one-dimensional main conjecture, that if p is odd and ψ is such that R ψ is totally real, then Using the Iwasawa main conjecture over totally real fields for one-dimensional characters (i.e. [Wil90, Thm. 1.2]), Brauer Induction and the well-known functoriality properties of the power series involved, one can deduce the Theorem 3.7. (Iwasawa main conjecture) Let p be an odd prime and let χ be a totally odd character of arbitrary dimension satisfying R χ ∩ R ∞ = R. Then we have Here * means the following: if λ ∈ Λ, then by the Weierstraß Preparation Theorem λ is associated to a power of p times a distinguished polynomial, and we denote by λ * this polynomial.
Now we can put the components of the proof together. By the definitions and in view of Proposition 3.3, we have an exact sequence Lemma 3.4 implies that Here w 1 (v) and w 2 (v) are fixed primes of H, respectively, K dividing v, and n(v) denotes the number of primes of H ∞ above w 1 (v), as in the proof of Lemma 3.4. By the multiplicativity of characteristic power series, we obtain that the exact power of T dividing f X,χ = f A,χ equals since ord T =0 (ω m (T )) = 1 for each m ∈ N.
The first summand in (7) is zero if and only if GKC(H/R, χ) holds. In view of Lemma 3.2, this is equivalent to the validity of GKC(K/R, χ). On the other hand, by the main conjecture, GVC(K/R, χ) is true if and only if ord T =0 f X,χ (T ) = χ(1) · r S,χ .
This concludes the proof.
(2) As already mentioned in the Introduction, one can obtain Corollary 3.1 more generally by using [Bur20] and [Kol91]. Indeed, in [Gro81] it is proved that the non-vanishing of the 'Gross regulator' is equivalent to Conjecture 1.15 in [Gro81]. A 'minus-version' of [Kol91, Theorem 1.14] then shows that GKC − (K) is equivalent to Conjecture 1.15 in [Gro81]. Now Burns proved in [Bur20] that the 'Gross regulator' is non-zero if and only if GVC(K/R, χ) holds. Combining these results, one obtains our Corollary 3.1 without the assumption K ∩ R ∞ = R. (3) Conjecture 2.3 can also be formulated for p = 2. Most of the proof of Theorem A can be adapted in order to cover also this case. The only step of the proof which does not carry over is the present lack of a main conjecture in the p = 2 setting.
Using Theorem A and Corollary 3.1, one can try to derive from the known results about GKC new cases of GVC; unfortunately, it turns out that the classically known 'trivial' cases from Remark 2.7,(1) yield only situations where r S,χ ≤ 1, and for which therefore also the GVC is known.
Corollary 3.9. Let K be a normal CM-extension of a totally real number field R and let p = 2. Suppose that R contains exactly one prime above p and that K ∩ R ∞ = R. Then GKC − (K) holds.
Here again we note that this result can be obtained more generally, i.e. without the condition K ∩ R ∞ = R, via the work of Burns in [Bur20], as explained in Remark 3.8(2).
Having in mind the known results mentioned in Section 2.2, it is of particular interest to produce examples where r S,χ is large. Here Corollary 3.1 can be very helpful in order to derive from a known instance of GVC(K/R, χ) for some χ with r S,χ > 0 examples where GVC is known for some character with larger vanishing order: Remark 3.10. Let K denote a Galois CM-extension of a totally real number field R. Suppose that GKC − (K) is known and that we can enlarge R and choose some totally real number fieldR ⊆ K which strictly contains R. In view of Tate's formula 3.6, the value of r S,χ , for suitable χ, can grow if we consider characters of the group Gal(K/R). If, for example, K/R is abelian, then Tate's formula just says that r S,χ equals the number of primes of R above p which split completely in R χ . Now suppose that the primes of R dividing p are totally split inR. Then each characterχ ∈ Ir − (Gal(K/R)) satisfies In the next section, we will prove GKC in new cases, which then also yield new instances of the Gross order of vanishing conjecture.

Proving the conjectures in new cases
In this chapter we first give an equivalent formulation of the minus Gross-Kuz'min conjecture.
Then we describe two approaches for the construction of families of abelian extensions K/R such that GKC(K/R, χ) and GVC(K/R, χ) hold for some χ ∈ Ir − (Gal(K/R)) with large vanishing order r S,χ of the corresponding Artin L-function. The first approach makes use of a relation to Leopoldt's conjecture, and the second one is based on an application of the Brumer/Baker Theorem.
Let p = 2 be a prime, let K be a CM-field with maximal totally real subfield K + , let K + ∞ be the cyclotomic Z p -extension of K + . We write K + ∞ = K + n , and we let K ∞ := K + ∞ · K = K n , where K n = K + n · K for each n ∈ N. We assume that each prime of K + dividing p is totally ramified in K + ∞ . Let r denote the number of primes above p in K + which split in K/K + . We identify Γ n := Gal(K n /K) with Gal(K + n /K + ) for every n ∈ N. Finally, we recall that E ′ K denotes the group of p-units of K. i.e.
, for all n ∈ N.
Proof. Fix a level n. We use Chevalley's Theorem (see [Lan90,Lemma 4.1 in Chapter 13]), or more precisely the variant for A ′ and p-units due to Gras (see [Gra94]), once for the cyclic extension K n /K and once for the extension K + n /K + . This yields: Here e(K n /K) denotes the product of the ramification indices in K n /K of all the primes of K, and e(K + n /K + ) is defined analogously. Now the quotient on the left hand side yields the order of ((A ′ n ) − ) Γn . Recall that GKC − (K) holds if and only if the orders of these groups remain bounded as n → ∞.
Since all the primes dividing p are totally ramified in K ∞ /K and in K + ∞ /K + , the quotient of the ramification numbers equals p rn , where we recall that r denotes the number of primes of K + dividing p which split in K/K + . The statement of the lemma follows: in order to have stabilisation of the orders on the left hand side, it is (see [Kle19, Theorem 2.1]) necessary and sufficient that there exists an index Corollary 4.2. In particular, GKC − (K) holds if no prime of K + dividing p splits in K.
In [Gre73], Greenberg implicitly proves the following reduction theorem; this result has been reproved and used by Jaulent in [Jau17]. In what follows, we will derive two approaches for constructing finite families of extensions K/R such that GKC(K/R, χ) holds for some χ ∈ Ir − (Gal(K/R)) with large vanishing order r S,χ . The first approach is related to Leopoldt's conjecture.
Proposition 4.4. Let K be a CM-field, and suppose that Leopoldt's conjecture holds for K. If the prime p is totally split in K/Q, then GKC − (K) holds.
Proof. Let A = A (K∞) be defined as in Section 2.1. First, it follows that Leopoldt's conjecture also holds for K + , i.e. the quotient A + /(T · A + ) is finite. Moreover, there exist exactly 2r 2 (K) primes in K and r 2 (K) primes in K + . Since Leopoldt's conjecture holds for K, we know that

Now we consider the exact sequence
which follows from the exact sequence (6) in the proof of Theorem A (equivalence of the two conjectures) by specialising to the case R = K + . Here r = r S,χ for the only non-trivial character of Gal(K/K + ), and the integers n i are defined as in Lemma 3.4. Since p is totally split in K/Q, it follows that r equals the number of primes of K + above p, i.e. r = r 2 (K). This proves that ord T =0 f − A ′ (T ) = 0. Now we can use the existing literature on Leopoldt's conjecture for deriving cases of the GKC.
Theorem 4.5. [Kli90, Thm. 2.1 c)] Let K/Q be an imaginary Galois extension with group G, and let τ ∈ G denote the complex conjugation, well defined up to conjugation in G. Then Leopoldt's conjecture is true for all p, provided that for all irreducible characters χ of G the condition holds.
In the case which is of interest to us, namely K being a CM-field, this condition is quite restrictive. In fact, Klingen shows in [Kli90, Lemma 2.3] that under the condition that K is a CM-field hypothesis (10) is satisfied if and only if G/ τ is abelian. So these cases can also be covered by combining the classical facts that Leopoldt's conjecture is proven for abelian extensions over Q and that Leopoldt's conjecture holds for a CM-field K if and only if it holds for its maximal totally real subfield K + (see e.g. [NSW08,Cor. 10 Our interest for these results stems from the following corollary to Theorem 4.5.  Theorem 4.8. Let K be a finite normal CM-extension of a totally real number field R, and suppose that (a) there exists some prime p of R above p which is totally split in K, and such that R p = Q p , (b) G := Gal(K/R) is abelian. Let A ′ = (A ′ ) (K∞) be defined as in Section 2.1. Then where r denotes the number of primes of K + which split in K, and [K : R] = 2s.
This is Theorem C from the Introduction and we note that hypothesis (a) may also be stated in the following way: there exists a prime P ∈ S p (K) such that K P = Q p .
Proof. We will use arguments from the proof of [Gre73,Proposition 3]. In all what follows, we denote by H 0 the group of p-units α of K which are divisible only by powers of ideals of K which lie above p, and which satisfy τ (α) = α −1 , where τ denotes the complex conjugation. Let p 1 , . . . , p s , p 1 , . . . , p s be the primes of K above p.
Let K pi be the completion of K at p i , and let ϕ i : K −→ K pi be the canonical embedding. Note that K pi = Q p for every i, by our assumptions. We define a map Claim 4.9. ψ is injective.
By Hasse's Norm Theorem, an element α ∈ K is a norm from the n-th layer K n of the cyclotomic Z p -extension of K if and only if it is a local norm at each prime dividing p. The element α is a local norm in K pi if and only if ψ i (α) ∈ E p n Kp i . In particular, ψ(N n (K n )) ⊆ U p n . Therefore We let H := ψ(H 0 ), and we denote by H the p-adic closure of H in U. Recall that H 0 is a Z-module of rank s = |G| 2 . Claim 4.10. rank Zp (H) = s.
Proof. Consider the matrix A = (ψ i (α)) α∈H0,i∈{1,...,s} . If rank Zp (H) < s, then the dimension of the row space of A is at most s − 1. This means that the dimension of the kernel of the corresponding Z p -linear map f : Z s p −→ U is at least one. In other words, there exist a 1 , . . . , a s ∈ Z p , not all zero, such that Taking the logarithm yields s i=1 a i log p (ϕ i (α)) = 0 for each α ∈ H 0 , since log p (p) = 0. Recall that we have fixed a prime p of R which is totally split in K; in particular, fixing a prime p 1 of K above p, the 2s = |G| primes of K dividing p can be written as {p 1 , . . . , p s , p 1 , . . . , p s } = {p 1 = σ 1 (p 1 ), . . . , σ s (p 1 ), σ s+1 (p 1 ), . . . , σ 2s (p 1 )}.
for each n ∈ N and some fixed constant C. Since H 0 contains only p-units α such that τ (α) = α −1 , we may conclude that for every n ∈ N.
The Theorem of Chevalley (in the version (9) used in the proof of Lemma 4.1) implies that for all sufficiently large m ∈ N (recall that r equals the number of primes of K + dividing p which split in K). We may assume that m is large enough such that all the primes of K m dividing p are totally ramified in K ∞ . Therefore we may deduce from [Kle19, Theorem 2.2] that for every n ≥ m. This implies that rank Zp ( This also finishes the proof of Theorem 4.8. Corollary 4.13. Let K be a finite abelian CM-extension of a totally real number field R. Let p 1 , . . . , p t be the primes of R dividing p, let G = Gal(K/R), and write |G| = n = 2 k · m with k ≥ 1 and m odd. Suppose that (a) p 1 is totally split in K/R and satisfies R p1 = Q p , and that (b) the primes of K + dividing p 2 , . . . , p t are unsplit in K.
Remark 4.14. Condition (b) from the above corollary is satisfied, for example, if (i) the numbers of primes of K above p 2 , . . . , p t are not divisible by 2, or if (ii) the numbers of primes of K above p 2 , . . . , p t are not divisible by 2 k = n m , and the 2-Sylow subgroup of G = Gal(K/R) is cyclic (indeed, in this case there exists exactly one element of G of order 2, and therefore our assumption implies that the decomposition fields of p 2 , . . . , p t in K are contained in the unique intermediate field of degree n 2 over R, which is K + ). Remark 4.15.
(1) In the situation of Corollary 4.13, each character χ ∈ Ir − (Gal(K/R)) satisfies r S,χ = 1 so these cases can already be derived from the literature.
(2) On the other hand, by using the approach described in Remark 3.10, we can use the above corollary for producing examples of GVC(K/R, χ) with large vanishing orders. Indeed, ifR is a field as in Corollary 4.13, then the p-adic L-functions attached to characters in Ir − (Gal(K/R)) will have vanishing orders at least [R : R], because p 1 is totally split in K/R.
Example 4.16. In order to show that the conditions of Corollary 4.13 are easily realisable we computed some examples. The totally real subfield R is always non-normal over Q. We define the modulus m ∞ to be the product of all the infinite places of R and m fin to be a suitable product of finite primes. Then we set K = R(m ∞ m fin ) and we search for moduli m fin such that the hypothesis from Corollary 4.13 are satisfied for K. Below we give a list of some of the examples we found with absolute degree [K : Q] larger than 10. Here we identify a monic polynomial f (X) = X n + n−1 i=0 a i X i ∈ Z[X] with the vector [a 0 , a 1 , . . . , a n−1 ], and we write p q for a prime ideal of O R above q. In order to keep the list in a concise form we do not specify which of the prime ideals above q we used.
In the last column of the table, we write down the degree of the maximal totally real subfield K + of K over R, which marks a lower bound for r S,χ for the non-trivial character χ of Gal(K/K + ), in the sense of Remark 4.15,(2). Finally we produce an example where K/R is non-abelian. Although the corresponding character χ satisfies r S,χ = 1 and therefore we do not obtain new, previously unknown cases, this theorem nicely illustrates possible further applications of the rank inequality from Theorem 4.8.
Theorem 4.17. Let K be a normal extension of a totally real number field R. Let p 1 , . . . , p t be the primes of R dividing p. We assume that (a) p 1 is totally split in K and R p1 = Q p , (b) the primes of K + dividing p 2 , . . . , p t do not split in K, (c) G := Gal(K/R) is isomorphic to the dihedral group D n with 2n elements, where n ≡ 2 (mod 4), and Then GKC(K/R, χ) holds for some χ ∈ Ir − (G).
Then the complex conjugation τ corresponds to the only non-trivial element a n/2 in the center of G. We letR be the subfield of K which is fixed by the subgroup a generated by a. ThenR is a degree 2 extension of R, and it is totally real becausẽ R ⊆ K + = K τ . It follows from Theorem 4.8, applied to the extension K/R, that where r denotes the number of primes of K + above p which split in K, i.e. = n 2 . Now the representation theory of the dihedral group D n implies that Ir − (G) contains n−2 4 characters of dimension 2 and two characters of dimension 1.