This is the submitted version of the article: Anglès, Bruno; Bandini, Andrea; Bars Cortina, Francesc; [et al.]. «Iwasawa main conjecture for the Carlitz cyclotomic extension and applications». Mathematische Annalen, Vol. 376, Issue 1-2 (February 2020), p. 475-523. DOI 10.1007/s00208- 019-01875-8 This version is available at https://ddd.uab.cat/record/240658 under the terms of the license IWASAWA MAIN CONJECTURE FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS BRUNO ANGLÈS, ANDREA BANDINI, FRANCESC BARS, AND IGNAZIO LONGHI Abstract. We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq (θ) (p is a prime of Fq [θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a p-adic L-function to prove a close analog of the Ferrero-Washington Theorem for F and to provide information on the p-adic valuations of the Bernoulli-Goss numbers β(j) (i.e., on the values of the Carlitz-Goss ζ-function at negative integers). 1. Introduction One of the main topics of modern number theory is the investigation of arithmetic properties of motives over a global field (in any characteristic) and their relation with (or interpretation as) special values of ζ-functions or L-functions. Iwasawa theory offers an effective way of dealing with various issues arising in this context, such as the variation of arithmetic structures in p-adic towers, and is one of the main tools currently available for the knowledge of ζ-values associated to an arithmetic object (see, e.g., [27]). One of the major outcomes of this theory is the construction of p-adic L-functions, which provide a good understanding of both the special values and the properties of the arithmetic object. In particular, the various forms of Iwasawa Main Conjecture (IMC) provide a link between the analytic side and the arithmetic side. The prototype is given by the study of class groups in the extensions generated by p- power roots of unity. Let k be a totally real number field, fix a prime p > 2 and consider the extensions k0 := k(ζp ) and k∞ := ∪n k(ζpn ) (where ζm denotes a primitive m-th root of unity). We briefly recall the statement of IMC in this basic setting. Let ∆ := Gal(k0 /k) (note that ∆ is isomorphic to a subgroup of F∗p ), then kcyc := k∞ ∆ is the cyclotomic Z -extension of k. p We put • G := Gal(kcyc /k) ≃ Zp , the Galois group; • Λ(kcyc ) := Zp [[G]], the associated Iwasawa algebra; n Gp , the n-th layer of k • kn := kcyc cyc with Gal(kn /k) ≃ Z/p Z; n • Cℓn , the p-part of the class group of kn and Cℓcyc := lim Cℓn (limit with respect to the ←− norm maps) the “pro-p class group” of kcyc . The group Cℓcyc admits an action by ∆ so we can consider its χ-part Cℓcyc (χ) for any character χ of ∆. Moreover Cℓcyc is a finitely generated torsion Λ(kcyc )-module. Since Λ(kcyc ) is a noetherian Krull domain (it is noncanonically isomorphic to Zp [[T ]]), a structure theorem allows to define a principal ideal ChΛ(kcyc ) (Cℓcyc (χ)) = (pµχ f χ ) ⊆ Λ(kcyc ) (with f χ a “polynomial” and p - f χ ), called the characteristic ideal of Cℓcyc (χ). By a celebrated theorem of Iwasawa, ChΛ(kcyc ) (Cℓcyc (χ)) gives information on the order of class groups: one 2010 Mathematics Subject Classification. 11R60; 11R23; 11M38; 11S40; 11R58; 14G10; 11G45; 14F30. Key words and phrases. Stickelberger series; Carlitz-Goss ζ-function; L-functions; Bernoulli-Goss numbers; class groups; Iwasawa Main Conjecture; function fields; cyclotomic extensions. F. Bars supported by MTM2016-75980-P. 1 IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 2 has n +deg χ )n+ν (1) |Cℓn (χ)| = pµχ p T (f χ for n ≫ 0 and some νχ ∈ Z. The class number formula provides another way of obtaining the information in (1), namely via special values of Dirichlet L-functions. This fact suggests a deeper relation between the algebraic and the analytic theory, expressed by the Iwasawa Main Conjecture. Let ω be the Teichmüller character, take an odd character χ ̸= ω of ∆ and let κ be the cyclotomic character (so that gζ = ζ κ(g) for any p-power root of unity ζ and any g ∈ G). Then there exists a p-adic L-function Lp (ωχ−1 , s) which interpolates p-adically the special values of the (twisted) Dirichlet L-function, and an element ℓχ ∈ Λ(kcyc ), such that (2) Lp (ωχ−1 , s) = ℓχ (κ(g0 )s − 1). The Iwasawa Main Conjecture states that (3) (ℓχ ) = (pµχ f χ ) and was proved in this form by B. Mazur and A. Wiles in [29] for k = Q (K. Rubin provided a different and more general proof using Kolyvagin’s method, see [42, Chapter 15] for an overview). Another major result was obtained by B. Ferrero and L. Washington in [18], where they showed that the invariant µχ in equation (3) satisfies (4) µχ = 0 for any χ and any abelian number field k. In the last decades Iwasawa theory grew enormously and found fruitful applications in dif- ferent areas of number theory. In particular various instances of the IMC have been formulated and proved for elliptic curves (in relation with the Birch and Swinnerton-Dyer Conjecture, see, e.g., [26], [8] and [35]), for motives in general (connected with the Tamagawa Number Conjecture of S. Bloch and K. Kato [9]), for non-abelian extensions (where the characteristic ideals are substituted by elements in K-theory groups, see [14]) and so on. Nevertheless, most of these developments deal with global fields of characteristic 0. Moving to the function field setting the situation is very different and much less understood (an excellent survey for an updated overview of the various aspects of Iwasawa theory over function fields is [12]). In characteristic p, an IMC like (3) has been proved for Zdp -extensions (see [16] and/or [10] and [11]). However, geometric Zdp -extension of function fields are some- what artificial, if d is finite, when compared with the cyclotomic extension of Q; and even the arithmetic extension is not very satisfactory. Hence, in our opinion, a true function field version of the basic Mazur-Wiles Theorem above was still missing. Providing it is the main goal of this paper: on the way we shall also obtain analogs of (2) (in many versions) and, as an application, of (4). 1.1. The function field setting. As usual in function field arithmetic, the field F := Fq (θ) and the ring A := Fq [θ] play the role of Q and Z. Here and in the following, we assume that q is a power of p. For a lighter notation, we will usually write F for Fq . The symbol ∞ will denote the place of F with uniformizer 1θ . Let F be a fixed algebraic closure of F . The Carlitz module Φ (see Section 2 for a quick review of the relevant theory) is a morphism of F-algebras Φ : A → EndF (F ) given by : Φθ = θ + τ, where τ : F → F , x 7→ xq . If a ∈ A − {0}, we set Φ[a] = {x ∈ F : Φa (x) = 0}. We fix a monic irreducible polynomial πp in A: p = πp A will correspond to (p) ⊂ Z. The function field counterpart of Q(ζpn ) is obtained adding to F the pn -roots of the Carlitz module Φ. The field Fn := F (Φ[πpn+1 ]) is a Galois extension of F with Galois group isomorphic to IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 3 (A/pn+1 )∗ . We recall that Fn /F is unramified outside p and ∞, and totally ramified at p. The “cyclotomic” extension of F we are going to consider is F := ∪n Fn . 1.1.1. The algebraic side. The group Gal(F/F ) is isomorphic to the units of Ap (the com- pletion of A with respect to p). In characteristic p the 1-units in a local field form a free Zp -module of infinite rank: hence we get Gal(F/F ) ≃ ∆ × Z∞ p , where ∆ ≃ Gal(F/F0 ) is a cyclic group of order q − 1, d = degθ πp . We will assume q > 2, in order to guarantee the d d existence of non-trivial characters of the group ∆. To include the values of these characters, we shall use the Witt ring W of the residue field A/p and define our Iwasawa algebra as Λ := W [[Gal(F/F0 )]]. Observe that our Λ is isomorphic to a ring of power series over W in infinitely many variables: therefore it is not noetherian. In this situation, one cannot apply the usual structure theorem for modules over Iwasawa algebras; however, it is still possible to define the Fitting ideal FittΛ (M ) for a finitely generated Λ-module M . 1.1.2. The analytic side. An interesting feature of function field arithmetic is the presence of more L-functions than in the number field setting. First of all, we have the usual, complex-valued L-functions as studied by Artin and Weil: to any continuous character ψ : Gal(F/F ) → C∗ one can attach L(s, ψ). And we have p-adic L-functions arising from p-adic interpolation of L(s, ψ). The genuinely new phenomenon is the appearance of characteristic p L-functions. The first example was discovered by Carlitz already in 1935 ([13]); some decades later Goss developed a full theory around it (see [20] or [21, Chapter 8]). The Carlitz-Goss zeta function ζA (s) will be defined and discussed in detail in Section 3.2. We mention that the special values of this zeta function are given by: ∑ ∑ 1 ζA (n) = ∈ F((θ−1 )) (with n ∈ Z). an m>0 a∈A, a monic degθ a=m In particular, for any positive integer n, ζA (−n) ∈ A, and ζA (−n) = 0 ⇔ n ≡ 0 (mod q − 1). Furthermore, for any n > 1, ζA (n) is transcendental over F ([43]). Goss also defined v-adic L-functions Lv (s, ψ) where v can be any place of F , and ψ a “Dirichlet” character. As one would expect, when v is a finite place, Lv (s, ψ) can be seen as a v-adic interpolation of ζA (s). 1.1.3. Special values. Evaluating ζA (s) at negative integers, one obtains the Bernoulli-Goss numbers β(j) ∈ A. If j ∈ N, j > 1, j ̸≡ 0 (mod q − 1), then β(j) = ζA (−j) ∈ A − {0}. For j ∈ N, j > 1, j ≡ 0 (mod q − 1), ζA (−j) = 0 and the precise formula for β(j) will be given in Definition 3.13. Similarly to the classical ζ, special values of the the Carlitz-Goss zeta have relevant arithmetical interpretations and have been the object of many investigations in recent years (see, for example, L. Taelman’s results in [36] and [37]). 1.2. Our results. In this paper we prove analogs of formulae (2), (3) and (4). We also provide some arithmetic information on the Bernoulli-Goss numbers. 1.2.1. The analytic side. If v is a place of F, we denote by Cv a v-adic completion of an algebraic closure of the v-adic completion of F. The extension F/F is ramified only at S = {p, ∞}.1 Let FS be the maximal abelian extension of F unramified outside S and put GS := Gal(FS /F ). We consider the Stickelberger series ∏ ΘFS /F,S (X) := (1 − Fr−1 q X deg(q) −1 ) ∈ Z[GS ][[X]], 1We shall not distinguish between a prime ideal of A, like p, and the place of F corresponding to it. IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 4 where the product is taken over all places q outside S and Frq ∈ GS denotes the Frobenius at q. For any subextension of FS a similar Stickelberger series is obtained by taking the projection of ΘFS /F,S (X) to the appropriate algebra. This series turns out to be a kind of universal object for L-functions attached to abelian characters unramified outside S. Evaluation at X = 1 (convergence is ensured by Proposition 3.2) yields an element in the Iwasawa algebra Zp [[GS ]] and hence one in Λ: these elements can be seen as p-adic L-functions. If L is any of the fields C, C∞ or Cp and ψ is a continuous character from GS to L∗ , then there are interpolating relations between ψ(ΘFS /F,S ) and • the complex L-function LS (s, ψ) (for L = C) ; • the Carlitz-Goss ζ-function ζA (s) (for L = C∞ ) ; • the p-adic L-function Lp (s, ψ) (for L = Cp ). Details and more precise formulations will be given after introducing the proper notations, in equation (11), Theorem 3.8 and Theorem 3.16 respectively. 1.2.2. The algebraic side. On the algebraic side we study Cℓ 0 (Fn ){p}, that is, the p-part of the group of classes of degree zero divisors of Fn . Let χ be any character defined on ∆ : we recall that such characters are called even if q − 1 divides | ker(χ)| and odd otherwise. As usual, χ0 will denote the trivial character. For any module M we shall denote by M (χ) the χ-component of M . Let Θ∞ (X, χ) denote the χ-component of the projection of ΘFS /F,S (X) to Z[Gal(F/F )][[X]] (see Definition 3.5) and put   Θ∞ (X, χ) if χ is odd Θ#∞ (X, χ) := .  Θ∞ (X,χ) 1−X if χ ̸= χ0 is even Our first main result is the following: an Iwasawa Main Conjecture for the p-cyclotomic extension F/F . Theorem 1.1 (IMC). For any χ ̸= χ0 , C(F)(χ) := lim ←− Cℓ 0 (Fn ){p}(χ) is a finitely generated n torsion Λ-module, and ( ) FittΛ (C(F)(χ)) = Θ# ∞ (1, χ) . Remark 1.2. In this paper we shall provide some results on the χ0 -component as well but we leave the precise statements to the following sections (see, in particular, Corollary 4.10 and Remark 4.14), since they require a few more notations and all the main arithmetical applications will involve only the χ-components for χ ̸= χ0 . The proof of Theorem 1.1 will be given in Section 5 (see Theorems 5.1 and 5.2). Here is a brief summary of the main ideas and steps. The strategy is based on some results of C. Greither and C.D. Popescu (in [23] and [24]). Let XFn be the curve associated with the field Fn and denote by Tp (Fn ) := Tp (Jac(XFn )(F)) the p-adic Tate module of its Jacobian (with F a fixed algebraic closure of F). Taking the limit over n, we get Tp (F). Put GF := Gal(F/F). Using Greither-Popescu results we can compute the Fitting ideal of Tp (Fn )(χ) and then, with some work to check the necessary compatibility conditions, we can compute the Fitting ideal of Tp (F)(χ) over Λ[[GF ]] (Theorem 4.16). The group Cℓ 0 (Fn ){p} can be recovered as GF -coinvariants of Tp (Fn ) (Lemma 4.6). By specializing the arithmetic Frobenius (i.e., the generator of GF ) to 1 in FittΛ[[GF ]] (Tp (F)(χ)), we finally obtain Theorem 1.1. 1.2.3. Special values. Our second main result is the following. Theorem 1.3. For any χ ̸= χ0 , one has Θ# ∞ (1, χ) ̸≡ 0 (mod p). IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 5 Thanks to Theorem 1.1, this can be seen as a close analog of the Ferrero-Washington Theorem. The proof (given in Theorem 6.3) is based on the following ideas. A map defined using a result by Sinnott provides a formula relating Θ∞ (X, χ) with a p-adic L-function Lp (X, y, χ) (Theorem 3.22). It turns out that the Bernoulli-Goss numbers β(j) appear as special values of this Lp (X, y, χ) (see (51) and (52) for the precise statements). Theorem 1.3 then follows observing that the β(j) are nonzero (Lemma 3.14). ep . Theorem 1.3 implies that the index In Section 2.4 we shall define a Teichüller character ω Np (i) := Inf{n > 0 : Θ# epi ) ̸≡ 0 (mod p) } n (1, ω is well defined for any 1 6 i 6 q d − 2. In Corollary 6.9 we will show that Np (i) provides a lower bound for vp (β(j)) (the p-adic valuations of the Bernoulli-Goss numbers) for j > 1, j ≡ −i (mod q − 1) : a relation that, to our knowledge, seems to have no counterpart in the number field setting. In this paper we are focused on the arithmetic of special values of the Carlitz-Goss zeta function at negative integers. If L/F is a finite extension and OL denotes the integral closure of A in L, L. Taelman has introduced a finite A-module H(Φ/OL ) associated to the Carlitz module Φ and the extension L/F ([36]). This A-module is an analogue of the ideal class group of a number field. For H(Φ/OF0 ), Taelman proved an analogue of the Herbrand-Ribet Theorem linking the isotypic components of the p-part of H(Φ/OF0 ) to the arithmetic of the special values of the Carlitz-Goss zeta function at positive integers ([37]). It would be very interesting to study the projective limit (for the trace maps) lim H(Φ/OFn ) ⊗A Ap in the spirit ←− of Iwasawa Theory. Acknowledgments. All the authors thank the MTM 2009-10359, which funded a workshop on Iwasawa theory for function fields in 2010 and supported the authors during their stay in Barcelona in the summer of 2013, and the CRM (Centre de Recerca Matemàtica, Bellaterra, Barcelona) for providing a nice environment to work on this project. The fourth author thanks NCTS/TPE for support to travel to Barcelona in summer 2013. 2. Basic facts on the p-cyclotomic extension We recall here some basic facts (and notations) about what we call the p-cyclotomic ex- tension of the rational function field, including the corresponding Iwasawa algebra and the Iwasawa modules which will be relevant for our work. 2.1. The p-cyclotomic extension. Let F, F , A and the place ∞ be as in the introduction, §1.1. Let Φ be the Carlitz module associated with A: it is an F-linear ring homomorphism Φ : A → F {τ } , θ 7→ Φθ = θτ 0 + τ , where F {τ } is the skew polynomial ring with τ f = f q τ for any f ∈ F . We fix once and for all an algebraic closure of F , which shall be denoted by F . For any ideal a of A write Φ[a] := {x ∈ F : Φa (x) = 0 ∀ a ∈ a} for the a-torsion of Φ. It is an A-module isomorphic to A/a; in particular, if a|b as ideals of A, we have Φ[a] ⊆ Φ[b]. Fix a prime ideal p ⊂ A of degree d > 0 and, for any n ∈ N, let Fn := F (Φ[pn+1 ]) be the field generated by the pn+1 -torsion of the Carlitz module. It is well known (see [32, Chapter 12] or [21, §7.5]) that Fn /F is an abelian extension with Galois group Gn := Gal(Fn /F ) = ∆ × Γn ≃ (A/pn+1 )∗ ≃ (A/p)∗ × (1 + p)/(1 + pn+1 ) , IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 6 where ∆ ≃ Gal(F0 /F ) ≃ (A/p)∗ is a cyclic group of order q d − 1, and Γn = Gal(Fn /F0 ) is the p-Sylow subgroup of Gn . (By a slight abuse of notation, we identify the prime-to-p part of Gn for all n, and denote it always as ∆.) The extension Fn /F is totally ramified at p and tamely ramified at the place ∞, whose inertia group is cyclic of order q − 1; in particular, Fn /F0 is only ramified at p. In the isomorphism Gal(F0 /F ) ≃ (A/p)∗ , the inertia at ∞ corresponds to F∗ . Definition 2.1. We define the p-cyclotomic extension of F as the field ∪ F := F (Φ[p∞ ]) = F (Φ[pn ]) n with abelian Galois group G∞ := Gal(F/F ) = lim ←− Gal(Fn /F ) = ∆ × lim ←− Γn =: ∆ × Γ . n n For any place v of F we denote by Iv,n (resp. Iv ) its inertia group in Gn (resp. in G∞ ). The set of ramified places in F/F is S := {p, ∞} and, for any n, one has Ip,n = Gn , Ip = G∞ and I∞,n = I∞ ,→ ∆ . Denote by Ap the completion of A at p, Fp the completion of F at p and Fp the residue field of Ap . Readers who prefer a more “hands-on” approach might appreciate the equality Ap = Fp [[πp ]], where πp is the monic irreducible generator of p in A. The group of units of the local ring Ap has a natural filtration; we put Un := 1 + pn Ap . Let Cp be the completion of an algebraic closure of Fp ; we also fix once and for all an embedding F ,→ Cp . We have isomorphisms G∞ ≃ A∗p and Γ ≃ U1 , which are induced by the p-cyclotomic character κ. To define κ : G∞ → A∗p , we extend Φ to a formal Drinfeld module which we denote by the same symbol Φ : Ap → Ap {{τ }} (see [31]). Then for any σ ∈ G∞ and any ε ∈ Φ[p∞ ] we have (5) σ(ε) = Φκ(σ) (ε) . This action provides the isomorphisms mentioned above. In particular, Γn corresponds to U1 /Un+1 ≃ (1 + pAp )/(1 + pn+1 Ap ). It is well known that U1 ≃ Z∞ p (a product of countably many copies of Zp ). As mentioned earlier, we define Fp to be the residue field Ap /pAp ; it is the same as the residue field A/p. Since we are in positive characteristic, Fp can be canonically identified with a subring of Ap (by lifting x ̸= 0 to x̃, the unique root of 1 whose reduction mod p is x) and in the rest of the paper we shall generally think of it as such. 2.2. The Iwasawa algebra. Let W be the Witt ring of Fp , which is isomorphic to Zp [µqd −1 ] (where µqd −1 denotes the (q d − 1)-th roots of unity). By definition of Witt ring, we have an identification W/pW = Fp . Moreover, the projection W  Fp has a partial inverse F∗p → µqd −1 , the Teichmüller character (again by lifting x to x̂, the unique root of 1 whose reduction mod p is x). The Iwasawa algebra we shall be working with is the completed group ring Λ := W [[Γ]] = lim W [Γn ] . ←− For any n > 0, put Γ(n) := Gal(F/Fn ). The exact sequence Γ(n) ,→ Γ  Γn induces In ,→ Λ  W [Γn ] . We also put Mn := pn Λ + In−1 for any n > 1. We recall some other basic facts on this non-noetherian algebra: • {Mn }n>0 is a basis of neighbourhoods of zero in Λ ; • Λ/pΛ = Fp [[Γ]] ; IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 7 • Λ is a compact W -algebra and a complete local ring with maximal ideal M1 , so that Λ/M1 ≃ W/pW ≃ Fp . 2.3. Consequences of a theorem of Sinnott. Let C 0 (Zp , Ap ) be the space of continuous functions from Zp to Ap , endowed with the topology of uniform convergence. More generally, we can consider C 0 (Zp , M ) where M is any finitely generated Ap -module: it turns out that C 0 (Zp , Ap ) is the projective limit of C 0 (Zp , A/pn ) as n varies. Following [34], we define the Ap -module of Dirichlet series Dir(Zp , Ap ) as the closure in C 0 (Zp , Ap ) of the module generated by the functions ϑu : Zp → Ap , y 7→ uy , for all u ∈ U1 . (If Fv is the completion of F at a place v, the element u ∈ Fv satisfies |1 − u| < 1 and y ∈ Zp , we put ∑ (y ) y u := (u − 1)n ∈ Fv∗ , n n>0 (y ) where n is the reduction modulo p of the value of the usual binomial.) The next theorem follows from Sinnott’s results and ideas in [34] applied to our setting. Theorem 2.2. There is an injective morphism s : Λ/pΛ ,→ Dir(Zp , Ap ) such that for any γ ∈ Γ, one has s(γ) = ϑκ(γ) . Proof. For any ring R, the algebra of R-valued distributions on Γ can be identified with R[[Γ]]. ∫ an isomorphism Ap [[U1 ]] → Dir(Zp , Ap ) by attaching In [34, Theorem 1] Sinnott constructs to a measure µ the function y 7→ U1 uy dµ(u). In particular, ϑu corresponds to the Dirac delta at u. To complete our proof, one just has to recall that Λ/pΛ = Fp [[Γ]] is a subring of Ap [[Γ]] and compose Sinnott’s isomorphism with the one Ap [[Γ]] ≃ Ap [[U1 ]] induced by the cyclotomic character κ.  It is clear from the proof that the image of s is exactly the closure of the Fp -module generated by the functions ϑu . Proposition 2.3. The morphism s induces a ring homomorphism sn : Fp [Γn ] → C 0 (Zp , A/pn+1 ) . Proof. It suffices to remark that κ(Γ(n) ) = Un+1 . Hence for γ ∈ Γ(n) and y ∈ Zp we have s(γ)(y) = ϑκ(γ) (y) = κ(γ)y ∈ 1 + pn+1 Ap which implies that the ideal (γ − 1 : γ ∈ Γ(n) ) in Fp [[Γ]] is sent by s into C 0 (Zp , pn+1 Ap ). The kernel of the natural projection Fp [[Γ]] → Fp [Γn ] is precisely the closure of this ideal (that is, the image in Fp [[Γ]] of In ⊂ Λ).  We get a commutative diagram s Fp [[Γ]] −−−−→ Dir(Zp , Ap )     y y s Fp [Γn ] −−−n−→ C 0 (Zp , A/pn+1 ) where the vertical maps are the natural ones. By construction one has s = lim ←− sn . n Proposition 2.4. If q d > 2, the map sn is not injective for n > 0. IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 8 Proof. Fix n > 0 and choose a1 , a2 and a3 in Ap so that ai and aj are different modulo p if i ̸= j (this is where we use q d > 2). Consider the elements γi ∈ Γ defined by κ(γi ) = 1 + ai πpn for i = 1, 2, 3 . Our hypothesis on the ai ’s implies that the γi ’s have Fp -linearly independent images in Fp [Γn ]. We need to find xi ∈ Fp so to have x1 κ(γ1 )y + x2 κ(γ2 )y + x3 κ(γ3 )y ∈ pn+1 Ap for all y ∈ Zp . This is equivalent to x1 + x2 + x3 + y(x1 a1 + x2 a2 + x3 a3 )πpn ≡ 0 (mod pn+1 ) , i.e., { x1 + x2 + x3 = 0 . x1 a1 + x2 a2 + x3 a3 ≡ 0 (mod p) For any nontrivial solution of this linear system, the image in Fp [Γn ] of x1 γ1 + x2 γ2 + x3 γ3 is a nontrivial element of the kernel of sn .  2.4. Characters of ∆. Let ωp : A → Ap be the morphism of F-algebras obtained composing A  A/p = Fp with the lift Fp ,→ Ap (i.e., the Teichmüller character in positive characteris- tic 2). Then any a ∈ A − p is uniquely decomposed as (6) a = ωp (a)⟨a⟩p where ⟨a⟩p ∈ U1 = 1 + pAp . The domain of ωp can be extended to all of Ap (note that then the restriction of ωp to Fp is just the identity) and equality (6) holds for any a ∈ Ap − pAp . The restriction of ωp ◦ κ : G∞ → F∗p to ∆ yields an isomorphism ∆ → F∗p , which in the rest of the paper will be simply denoted ωp , by an abuse of notation meant to emphasize the “Teichmüller-like” quality of this characteristic p character. If, for any a ∈ A − p, we let σa ∈ ∆ be the element such that σa (ε) = Φa (ε) ∀ ε ∈ Φ[p] (recall that ∆ ≃ Gal(F0 /F )), then we have ωp (σa ) = ωp (a) . Composition of κ|∆ with the Teichmüller lift F∗p → µqd −1 yields a character ωep : ∆ → W ∗ (the Teichmüller character in characteristic 0). It satisfies ep (σa ) ≡ ωp (a) (mod pW ) . ω A (p-adic) character χ on ∆ is called odd if χ(I∞ ) ̸= 1 and even if χ(I∞ ) = 1. Since all ep , this definition amounts to saying that ω such characters are powers of ω epi is even if and only if q − 1 divides i. b we put 2.4.1. Decomposition by characters. For any p-adic character χ ∈ Hom(∆, W ∗ ) =: ∆ 1 ∑ (7) eχ := χ(δ −1 )δ ∈ W [∆] |∆| δ∈∆ for the idempotent associated with χ. We recall a few basic relations: • for any δ ∈ ∆, (8) eχ δ = χ(δ)eχ ; b • for any ψ ∈ ∆, { 1 if ψ = χ ψ(eχ ) = ; 0 if ψ ̸= χ 2The map ω can also be defined as the morphism of F-algebras such that v (θ − ω (θ)) > 1: it satisfies p p p ωp (a) ≡ a (mod p) and corresponds to the choice of a root of πp in F (because πp = πp (θ) ∈ A = F[θ] and we have πp (θ) ≡ πp (ωp (θ)) (mod p), therefore πp (ωp (θ)) ≡ 0). IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 9 ∑ • eχ = 1. b χ∈∆ As usual, for any W [∆]-module M , we denote by M (χ) the χ-part of M (i.e., the submodule eχ M ) and we have a decomposition ⊕ (9) M≃ M (χ) . b χ∈∆ 3. p-adic interpolation of the Carlitz-Goss L-function In this section we present the analytic side of our work, i.e., the Carlitz-Goss ζ-function ζA and the p-adic L-function we shall use to interpolate ζA at integers. Moreover we introduce the Stickelberger series which will appear also in the computation of Fitting ideals of Tate modules and class groups in Section 4. Actually, the Stickelberger series is the main hero of this section: as we shall see, it plays a universal role in interpolating L-functions attached to abelian characters with no ramification outside a prescribed locus. In the case of C- valued characters and the complex L-functions attached to them, this will be clear from (11). In Theorems 3.8 and 3.16, we shall see how, taking characteristic p-valued characters, the Stickelberger series interpolates the L-functions defined by Goss. We also remark that in [28] the Stickelberger series is used as a p-adic L-function. For the convenience of the reader we will recall different constructions and properties: some of them are known but we lack an explicit reference including all of them. 3.1. The Stickelberger series. Recall that PF is the set of places of F . Places different from ∞ will be often identified with the correponding prime ideals of A. The subset of PF where the extension F/F ramifies is S = {p, ∞}. Define GS as the Galois group of the maximal abelian extension FS of F which is unramified outside S. For any q ∈ PF −S, let Frq ∈ GS denote the corresponding (arithmetic) Frobenius automorphism. Definition 3.1. We define the Stickelberger series by ∏ (10) ΘFS /F,S (X) := (1 − Fr−1 q X deg(q) −1 ) ∈ Z[GS ][[X]] . q∈PF −S More generally, for any closed subgroup U < GS , we define GS ΘF U /F,S (X) := πGS /U (ΘFS /F,S )(X) S ∏ = GS (1 − πG S /U (Fr−1 q )X deg(q) −1 ) ∈ Z[Gal(FSU /F )][[X]] , q∈PF −S GS where πG S /U : Z[GS ] → Z[Gal(FSU /F )] is the map induced by the projection GS  GS /U . The series in (10) is well-defined, since for any k there are only finitely many places of degree k. 3.1.1. Convergence. Let R be a topological ring, complete with respect to a non-archimedean absolute value. The algebra R[[GS ]] is the inverse limit of R[Gal(E/F )] as E varies among finite subextensions of FS /F ; as such, it is endowed with a topological structure. (Topolo- gically each R[Gal(E/F )] is the product of [E : F ] copies of R and R[[GS ]] has the coarsest topology such that all projections R[[GS ]] → R[Gal(E/F )] are continuous.) For any topological ring R the Tate algebra R⟨X⟩ consists of those power series in R[[X]] whose coefficients tend to 0. In particular, R[[GS ]]⟨X⟩ contains all those power series whose image in R[GS /U ][[X]] is a polynomial for any open subgroup U < GS . IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 10 For any unitary R, the natural map Z → R (by 1 7→ 1) allows to think of ΘFS /F,S (X) as an element in R[[GS ]][[X]]. Moreover, for any group homomorphism ( α : )GS → R∗ , the extension by linearity to a map α : Z[GS ] → R yields a power series α ΘFS /F,S (X) ∈ R[[X]]. Proposition 3.2. Let R be a unitary topological Zp -algebra, complete with respect to a non- archimedean absolute value. Then the series ΘFS /F,S (X) defines an element in the Tate algebra R[[GS ]]⟨X⟩. Proof. The proof is essentially the same as in [28, Proposition 4.1] (see also [6, §5.3]), so here we just sketch the basic ideas. Let ψ : GS → C∗ be a continuous character (GS has the profinite topology, so ψ factors through a subgroup of finite index). Then ( )−1 ( ) −s ∏ ψ(Fr−1 q ) (11) ψ ΘFS /F,S (q ) = 1− =: LS (s, ψ) (N q)s q∈PF −S is (possibly up to the Euler factors from places in S) the classical complex L-function attached to ψ (here N q := q deg(q) is the order of the finite field Fq and we assume Re(s) > 1 to ensure convergence of the infinite product). More precisely, one has ∏ (12) L(s, ψ) := LS (s, ψ) · (1 − ψ(v)q −s deg(v) )−1 , v∈S where ψ(v) denotes the value of ψ on the inverse of the Frobenius of v (an element in ker(ψ) ker(ψ) GS / ker(ψ) if v is not ramified in FS /F ), with the usual convention ψ(v) = 0 if FS /F is ramified at v. A theorem of Weil (see, e.g., [38, V, Théorème 2.5]) implies that L(s, ψ) is a polynomial in −s q , unless ψ = ψ0 is trivial, in which case one has 1 (13) L(s, ψ0 ) = −s . (1 − q )(1 − q 1−s ) Thus LS (s, ψ) is a rational function of q −s , with denominator bounded independently of ψ. Choose an auxiliary place q0 ∈ / S and put ΘFS /F,S,{q0 } (X) := (1 − q deg(q0 ) Fr−1 q0 X deg(q0 ) )ΘFS /F,S (X) . ( ) −s By Weil’s theorem, for all ψ as above, ψ ΘFS /F,S,{q0 } (q ) belongs to C[q −s ] (more precisely, to Z[ψ(GS )][q −s ] ). As a consequence, one gets GS πGS /U (ΘFS /F,S,{q0 } )(X) ∈ Z[Gal(FSU /F )][X] for all open subgroups U < GS and hence ΘFS /F,S,{q0 } (X) ∈ Z[[GS ]]⟨X⟩. It follows that also ΘFS /F,S (X) is in Zp [[GS ]]⟨X⟩, because the ratio between ΘFS /F,S,{q0 } (X) and ΘFS /F,S (X) is a unit in the Tate algebra. Finally, for R as in the hypothesis, the natural map Zp → R is extended to a continuous homomorphism Zp [[GS ]][[X]] → R[[GS ]][[X]]. Our proposition follows from the restriction Zp [[GS ]]⟨X⟩ → R[[GS ]]⟨X⟩.  Remark 3.3. We remind readers of two important properties of Stickelberges series (for details see [32, Chapter 15]). Let E/F be a finite subextension of FS /F with G = Gal(E/F ). Then one has (1) ΘE/F,S (X) ∈ (1−qX)|G| 1 Z[G][X] ∩ Z[G][[X]] (where Z is the integral closure of Z in the algebraic closure of Q); (2) the Brumer-Stark element wE/F = (q − 1)ΘE/F,S (1) ∈ Z[G] annihilates Cℓ(E). A different approach to the proof of (2) was proposed in [1]. IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 11 ∗ ) 3.2 and α : GS → R a continuous group homo- Theorem 3.4. Let R be as in( Proposition morphism. The power series α ΘFS /F,S (X) converges on the unit disk {x ∈ R : |x| 6 1}. Proof. The ring homomorphism R[[GS ]] → R induced by α is continuous; hence it ex- tends (to a homomorphism ) of Tate algebras R[[GS ]]⟨X⟩ → R⟨X⟩. Thus, by Proposition 3.2, α ΘFS /F,S (X) is in R⟨X⟩, which, by definition, consists exactly of those power series convergent on the unit disk.  3.1.2. Stickelberger series in the p-cyclotomic tower. In the following, we shall be particularly interested in the image of the Stickelberger series along the p-cyclotomic tower. Define (14) Θ∞ (X) := ΘF /F,S (X) ∈ Z[G∞ ][[X]] and, for all n ∈ N, Θn (X) := ΘFn /F,S (X) ∈ Z[Gn ][[X]] . We shall think of Θ∞ and Θn as power series with coefficients respectively in W [[G∞ ]] and W [Gn ]. Any element in G∞ can be uniquely written as δγ, with δ ∈ ∆ and γ ∈ Γ. Consequently, given χ ∈ Hom(∆, W ∗ ) we can define a group homomorphism G∞ → Λ∗ by δγ 7→ χ(δ)γ. By linearity and continuity, this can be extended to a ring homomorphism (which, by abuse of notation, we still denote by the same symbol) χ : W [[G∞ ]] → Λ. The decomposition (9) applied to W [[G∞ ]][[X]] then yields the following definition. Definition 3.5. For any χ ∈ Hom(∆, W ∗ ), the χ-Stickelberger series for the p-cyclotomic tower is Θ∞ (X, χ) := χ(Θ∞ )(X) ∈ Λ[[X]] . Similarly, we put (15) Θn (X, χ) := χ(Θn )(X) ∈ W [Γn ][[X]] . The series Θn (X, χ) form a projective system: let πnn+1 : W [Γn+1 ][[X]] → W [Γn ][[X]] be the projection induced by the natural map Γn+1  Γn , then we have πnn+1 (Θn+1 (X, χ)) = Θn (X, χ) b Moreover, (8) yields and Θ∞ (X, χ) = lim Θn (X, χ) for all χ ∈ ∆. ←− (16) eχ Θ∞ (X) = Θ∞ (X, χ)eχ and ∑ Θ∞ (X) = Θ∞ (X, χ)eχ b χ∈∆ (of course these relations descend to level n for all n ∈ N). Finally, the proof of Proposition 3.2 shows that Θn (X, χ) ∈ W [Γn ][X] if χ is not the trivial character χ0 and that Θn (X, χ0 ) ∈ 1 1−qX W [Γn ][X]. 3.2. Carlitz-Goss ζ-function and Bernoulli-Goss numbers. We recall the construction of the Goss L-function and the main properties needed in our work (a general reference is [21, Chapter 8]). As usual, F∞ denotes the completion of F at ∞ and C∞ is the completion of an algebraic closure of F∞ . The valuation on F∞ extends to v∞ : C∞ → Q∪{∞}. We also fix an embedding ∗ . of F in C∞ . Finally, let U1 (∞) denote the group of 1-units in F∞ Since we are taking F = F(θ), a somewhat natural choice of uniformizer at ∞ is θ−1 . Fixing ∗ → F∗ , which sends x ∈ F ∗ into the residue a uniformizer establishes a sign function sgn : F∞ ∞ of xθ v∞ (x) , and a projection ∗ xθv∞ (x) F∞ −→ U1 (∞) , x 7→ ⟨x⟩∞ := . sgn(x) IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 12 Note that one has ker(sgn) = θZ × U1 (∞). Remark 3.6. These maps can be made more “concrete” by the following observation. Let a ∈ A − {0} and write it as a = a0 + . . . + an θn , with n = deg(a) and ai ∈ F. Then we have a a (17) sgn(a) = an ∈ F∗ and ⟨a⟩∞ = deg(a) = ∈ 1 + θ−1 F[[θ−1 ]] . θ sgn(a) an θ n 3.2.1. The group S∞ . Let IF denote the group of ideles of F . Then we have ∏ (18) IF /F ∗ ≃ ker(sgn) × A∗q =: D , q∈PF −{∞} where Aq denotes the completion of A with respect to q and the isomorphism is given by the embedding of the right-hand side as a subgroup of IF . The group of C∞ -valued principal quasi-characters on IF /F ∗ is S∞ := C∗∞ × Zp . For s = (x, y) ∈ S∞ , we define a continuous homomorphism ker(sgn) −→ C∗∞ by (19) a 7→ as := x−v∞ (a) ⟨a⟩y∞ . This map is extended to all of IF by the projection to ker(sgn) induced by the isomorphism (18). The group structure on S∞ is given by (x1 , y1 ) + (x2 , y2 ) := (x1 x2 , y1 + y2 ). We have an injection Z ,→ S∞ , by j 7→ cj := (θj , j) . By (17) we get acj = aj for all j ∈ Z and monic a ∈ A. In analogy with the complex half-plane C+ := {z ∈ C | ℜ(z) > 1}, we define a “half-plane” S+ ∞ := {(x, y) ∈ S∞ : |x| > 1} . 3.2.2. From ΘFS /F,S to ζA . Let A+ be the set of monic polynomials in A. Thinking of A as a subset of F∞ , we have A+ = A ∩ ker(sgn). Definition 3.7. The Carlitz-Goss ζ-function is defined as ∑ (20) ζA (s) := a−s , s ∈ S∞ . a∈A+ For s = (x, y), we have as = xdeg(a) ⟨a⟩y∞ , hence |a−s | = |x|− deg(a) . It follows that the series (20) converges on S+ ∞ . (Note the analogy with convergence of the series defining L(s, ψ) for Re(s) > 1.) Class field theory identifies the group U1 (∞) with a factor of GS . Consequently, the con- struction in Section 3.2.1 can be used to define C∞ -valued characters on GS . More pre- cisely, for y ∈ Zp let ψy : GS → C∗∞ be the homomorphism obtained by composing the class field theoretic projection ρ : GS  U1 (∞) with (1, y) ∈ S∞ . Then Theorem 3.4 shows that ψy (ΘFS /F,S )(x) converges for all x ∈ C∞ such that |x| 6 1. Theorem 3.8. For all s = (x, y) ∈ S+ ∞ , we have (21) ψ−y (ΘFS /F,S )(x−1 ) = (1 − πp−s )ζA (s) . (Recall that πp is the monic irreducible generator of the ideal p in A.) Proof. For every place q in PF − S, let πq ∈ A+ denote the monic generator of the corre- sponding prime ideal in A. Then (20) can be rewritten as an Euler product ∏ ∏ (22) ζA (s) = (1 − πq−s )−1 = (1 − ⟨πq ⟩−y ∞x − deg(q) −1 ) . q∈PF −{∞} q∈PF −{∞} IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 13 By class field theory, we have a reciprocity map rec : IF → GS with dense image isomorphic to ker(sgn) × A∗p . Using (18), the composition ρ ◦ rec is just the projection ∏ D = θZ × U1 (∞) × A∗q −→ U1 (∞) . q∈PF −∞ For q in PF −S, let iq ∈ IF denote the idele having πq as its q-component and 1 as component at all other places: then Frq = rec(iq ). By the diagonal embedding F ∗ ,→ IF , we also get rec(iq ) = rec(iq a) for all a ∈ F ∗ . Since πq belongs to F ∗ and iq πq−1 is in the fundamental domain D, we finally obtain ρ(Frq ) = ⟨πq−1 ⟩∞ . −y Thus ψ−y (Fr−1 q ) = ⟨πq ⟩∞ and ∏ ψ−y (ΘFS /F,S )(x−1 ) = (1 − ⟨πq ⟩−y ∞x − deg(q) −1 ) . q∈PF −S Comparison with (22) completes the proof.  Theorem 3.8 can be used to obtain analytical continuation of ζA on the “boundary” of S+ ∞ (that is, {s = (x, y) : |x| = 1}), since, by Theorem 3.4, the left-hand side of (21) converges if |x| = 1. Remark 3.9. Let R be a topological ring: then the ring of R-valued distributions 3 on ker(sgn) is isomorphic to R[[U1 (∞)]][[X]]. This suggests that equation (21) can be interpreted as providing an integral formula for the Carlitz-Goss zeta function (namely, integration of the quasi-character s against the distribution induced by the Stickelberger series; a variant of this will be made explicit in the proof of Theorem 3.22). Integral formulas for ζA and its generalizations were already known (starting with Goss’s foundational paper [20]; see [39, §5.7] for a quick introduction to the topic), but (to the best of our knowledge) were all based on measures on some additive group; our approach instead stresses the role of the multiplicative group ker(sgn) and thus might provide some useful new insight. 3.2.3. Bernoulli-Goss numbers. Our final goal in this chapter is to interpolate the Carlitz- Goss zeta function at negative integers. Lacking a functional equation, we have to use more brute force techniques in order to extend the domain of ζA to all of S∞ . For any n > 1 let A+,n := {a ∈ A+ : deg(a) = n} (note that |A+,n | = q n ). For any j ∈ Z and n ∈ N put ∑ Sn (j) := aj . a∈A+,n Note that we have S0 (j) = 1 for all j ∈ Z. Lemma 3.10. If 1 6 j < q n − 1, then Sn (j) = 0. Proof. This is due to Carlitz. See e.g. [21, Remark 8.12.1.1] for a proof.  Remark 3.11. The statement in Lemma 3.10 is far from being the best possible. Necessary and sufficient criteria for the vanishing of Sn (j) can be found in [41, Theorem 1], which also provides some information on the history of the subject.4 Reorganizing the terms in (20) we can also write the Carlitz-Goss ζ-function as   ∑ ∑ ζA (x, y) :=  ⟨a⟩−y ∞  x−n , (x, y) ∈ S∞ . n>0 a∈A+,n This second formula guarantees the convergence for all s ∈ S∞ because of the following 3By R-valued distributions on a locally profinite group G we mean the linear functionals on the space of compactly supported locally constant functions G → R. 4Readers are warned that the notation in [41] is different from ours: our S (j) becomes S (−j) in [41]. n n IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 14 Lemma 3.12. For any y ∈ Zp and any n > 1, one has   ∑ v∞  ⟨a⟩y∞  > pn−1 . a∈A+,n ′ Proof. The case n = 1 is obvious. Now consider n > 2. If y = pn−1 y ′ , we can write ⟨a⟩y∞ = 1+ã where v∞ (ã) > 1. The claim in this case then follows from ∑ ∑ ′ n−1 ∑ n−1 ∑ n−1 ∑ n−1 ⟨a⟩y∞ = ⟨a⟩y∞p = (1 + ã)p = (1 + ãp ) = ãp a∈A+,n a∈A+,n a∈A+,n a∈A+,n a∈A+,n (since we are in characteristic p and |An,+ | = q n ). If y ̸≡ 0 (mod pn−1 ), then take an integer yn−1 ≡ y (mod pn−1 ) with 1 6 yn−1 6 pn−1 − 1. Since q > p, we get yn−1 < q n − 1 and Lemma 3.10 implies Sn (yn−1 ) = 0. Therefore ∑ ∑ ( a )yn−1 1 ⟨a⟩y∞n−1 = n = nyn−1 Sn (yn−1 ) = 0 . θ θ a∈A+,n a∈A+,n Moreover ( n−1 ′ ) ( ′ )pn−1 n−1 ⟨a⟩y∞ − ⟨a⟩y∞n−1 = ⟨a⟩y∞n−1 ⟨a⟩p∞ y − 1 = ⟨a⟩y∞n−1 ⟨a⟩y∞ − 1 = ⟨a⟩y∞n−1 cp (where v∞ (c) > 1), so that v∞ (⟨a⟩y∞ − ⟨a⟩y∞n−1 ) > pn−1 . Hence     ∑ ∑ ( ) v∞  ⟨a⟩y∞  = v∞  ⟨a⟩y∞ − ⟨a⟩∞ yn−1  > pn−1 .  a∈A+,n a∈A+,n For any j ∈ N and x ∈ C∗∞ , we have the equality (x ) ∑ ( a )j ∑ (23) ζA j , −j = x− deg(a) θj deg(a) deg(a) = Sn (j)x−n , θ θ a∈A+ n>0 which leads to the following Definition 3.13. (1) For any j ∈ N we put ∑ (24) Z(X, j) := Sn (j)X n ∈ A[X] n>0 (it is a polynomial because of Lemma 3.10). (2) For any j ∈ N, the Bernoulli-Goss numbers β(j) are defined as   Z(1, j) if j = 0 or j ̸≡ 0 (mod q − 1) β(j) := .  − dX d Z(X, j)|X=1 if j > 1 and j ≡ 0 (mod q − 1) By definition, for any j ∈ N, we have ζA (−j) = Z(1, j) (by an abuse of notation, we write ζ(−j) for ζ(c−j )). It is known that, for j > 1 with j ≡ 0 (mod q − 1), we have Z(1, j) = 0, which corresponds to a trivial zero in this setting (see [21, Example 8.13.6]). Moreover it is clear that β(j) ∈ A and β(j) = 1 for 0 6 j 6 q − 2. We also have β(q − 1) = 1, as can be deduced from the following lemma. Lemma 3.14. For any j ∈ N, we have β(j) ≡ 1 (mod θq − θ). In particular β(j) ̸= 0. IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 15 Proof. Recall that β(0) = 1 and S0 (j) = 1 for any j > 0. For any α ∈ F we can write a polynomial a ∈ A+,n in terms of powers of θ − α, i.e., a = a0 + a1 (θ − α) + · · · + (θ − α)n . Therefore, for any j > 1, ∑ ∑ j Sn (j) = aj ≡ q n−1 a0 (mod θ − α) . a∈A+,n a0 ∈F Thus Sn (j) ≡ 0 (mod θ − α) for any n > 2 or for n = 1 and j ̸≡ 0 (mod q − 1). Moreover for n = 1 and j ≡ 0 (mod q − 1) one has ∑ S1 (j) ≡ 1 ≡ −1 (mod θ − α) . a0 ∈F Hence   S0 (j) − X ≡ 1 − X (mod θ − α) if j > 1 and j ≡ 0 (mod q − 1) Z(X, j) ≡ .  S0 (j) ≡ 1 (mod θ − α) otherwise The lemma follows by the definition of β(j) (recalling that the terms θ −α are relatively prime and their product is θq − θ).  3.3. p-adic L-function and interpolation. The previous section dealt with the prime at infinity, now we focus on the other place in S. We give here the details of the construction of Goss’s p-adic L-function (see [21] for more). 3.3.1. The group Sp . Similarly to S∞ , we define a group of Cp -valued quasi-characters on IF /F ∗ by Sp := C∗p × Zp × Z/|F∗p | . However, in this case we shall be interested only in characters factoring through the compact group A∗p . So we embed Z into Sp by j 7→ (1, j, j). (Note that the image of this map is dense in {1} × Zp × Z/(q d − 1), in contrast with the discrete embedding Z ,→ S∞ . This should be compared with the fact that Z is discrete in C, but not in the p-adics.) 5 For s = (1, y, i) ∈ Sp and a ∈ A∗p , the decomposition (6) suggests to define as := ωpi (a)⟨a⟩yp . Then we obtain a continuous homomorphism ξs : GS → C∗p as composition of the maps σ7→σ|F κ a7→as (25) GS −−−−→ G∞ −−−−→ A∗p −−−−→ C∗p . 3.3.2. The p-adic L-function. As with ζA , we first define a function by a certain power series and then interpret it as specialization of the Stickelberger series. Definition 3.15. For any 0 6 i 6 q d − 2 and any y ∈ Zp , we define the p-adic L-function as   ∑ ∑ (26) Lp (X, y, ωpi ) :=  ωpi (a)⟨a⟩yp  X n . n>0 a∈A+,n −p Note that Lp (X, y, ωpi ) is an element of Ap [[X]]: as such, it converges on the open unit disc of Cp . We can think of it as a function defined on S+ p := {(x, y, i) ∈ Sp : |x| < 1}. Theorem 3.16. We have (27) ξ−s (ΘFS /F,S )(X) = Lp (X, y, ωpi ) for every s = (y, i) ∈ Zp × Z/(q d − 1). 5This definition of S - the same as in [39, §5.5(b)] - differs from the one in [21], where the factor C∗ is p p missing. We decided to insert this factor in order to emphasize the symmetry with S∞ . IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 16 Proof. Equation (26) can be rewritten as an Euler product ∏ ( )−1 (28) Lp (X, y, ωpi ) = 1 − ωpi (πq )⟨πq ⟩yp X deg(q) . q∈PF −S Thus, as in the proof of Theorem 3.8, we just need to check that the equality ξ−s (Fr−1 y q ) = ωp (πq )⟨πq ⟩p = πq i s holds for every s and q. An element in G∞ is completely determined by its action on Φ[p∞ ]; since Φπq (x) ∈ A[x] is monic and it satisfies Φπq (ε) ≡ εdeg(q) (mod q) for every ε ∈ Φ[p∞ ], we get Φπq (ε) = Frq (ε). Then (5) implies that the restriction of Frq to F is exactly κ−1 (πq ).  Theorem 3.16 implies that the series Lp (X, y, ωpi ) converges on the closed unit disc. Actually, one can show that (26) defines an entire function on Cp , by a reasoning similar to the one of Lemma 3.12. Since we are only interested in the specialization at X = 1, we won’t discuss the matter any further (see [21, Chapter 8] for more). Corollary 3.17. Let j be a natural number congruent to i (mod q d − 1). Then (29) Lp (X, j, ωpi ) = (1 − πpj X d )Z(X, j) ∈ A[X] and, for any y ∈ Zp , we have (30) Lp (X, y, ωpi ) ≡ Z(X, i) (mod p) . Proof. Since j is an integer and i is its reduction modulo q d − 1, we have ξ(−j,−i) (Fr−1 i j j j j −1 q ) = ωp (πq )⟨πq ⟩p = πq = ⟨πq ⟩∞ · θ = ψj (Frq ) · θ j for all places q ∈ / S. Therefore Theorem 3.16 gives an equality of power series in F [[X]] Lp (X, j, ωpi ) = ξ(−j,−i) (ΘFS /F,S )(X) = ψj (ΘFS /F,S )(θj X) . It is convenient to extend the exponentiation in (19) by a(xX,y) := ⟨a⟩y∞ (xX)−v∞ (a) (where x, y are as in (19) and X is a formal variable). Then Theorem 3.8 yields ( ) −(θj X,j) 1 ψj (ΘFS /F,S )(θ X) = (1 − πp j ) · ζA , −j = (1 − πpj X d ) Z(X, j) θj X (the first equality is just a restatement of (21) in terms of Laurent series and the second one follows from (23)). As for (30), it is enough to observe that one has ⟨a⟩yp ≡ 1 (mod p) for any a ∈ A∗p and y ∈ Zp . Hence (26) shows that the variable y is irrelevant modulo p and (29) yields Lp (X, y, ωpi ) ≡ Lp (X, i, ωpi ) ≡ Z(X, i) (mod p) .  Remark 3.18. A more direct proof of (29) can be obtained from the equation ∑ Lp (X, j, ωpi ) = (Sn (j) − πpj Sn−d (j))X n , n>0 which is obvious from (26). However, the devious path we followed might be forgiven consid- ering that it illustrates how (29) and (21) are essentially the same statement. Regarding the special values of Lp (X, i, ωpi ) we have the following Lemma 3.19. If i ≡ 0 (mod q − 1), then Lp (1, y, ωpi ) = 0 for all y ∈ Zp . IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 17 Proof. For any y ∈ Zp and m > 1, take j ∈ N − {0} such that j ≡ i (mod q d − 1) and j ≡ y (mod pm ). Then one has m ⟨a⟩yp ≡ ⟨a⟩y−j j j p ⟨a⟩p ≡ ⟨a⟩p (mod pp ) for any a ∈ A∗p and hence, by (26) and (29), m (31) Lp (X, y, ωpi ) ≡ Lp (X, j, ωpi ) = (1 − πpj X d )Z(X, j) (mod pp ) . Now, since j > 1 and j ≡ i ≡ 0 (mod q − 1), we have Z(1, j) = 0 (see [21, Example 8.13.6]) and the lemma follows taking the limit as m goes to infinity.  We also recall one of the main results of [3]. Theorem 3.20. [3, Theorem E] Let 0 6 i 6 q d − 2 with i ̸≡ 0 (mod q − 1). Then Lp (1, −1, ωpi ) ̸= 0 . Remark 3.21. It would be interesting to investigate further the values of Lp (1, y, ωpi ) for odd i. From equation (30) one immediately has that Z(1, i) ̸≡ 0 (mod p) =⇒ Lp (1, y, ωpi ) ̸= 0 ∀ y ∈ Zp . In general: is it true that for any 0 6 i 6 q d − 2 with i ̸≡ 0 (mod q − 1) and for any y ∈ Zp , we have Lp (1, y, ωpi ) ̸= 0 ? We end this section by providing another formula for Lp (X, y, ωpi ). The Sinnott map s of Theorem 2.2 induces a map sX : Λ[[X]] → Dir(Zp , Ap )[[X]] ∑ ∑ in the obvious way, sending n cn X n ∈ Λ[[X]] into the function y 7→ n s(c̄n )(y)X n (where c̄n is the reduction of cn modulo p). Theorem 3.22. For every y ∈ Zp and i ∈ Z/(q d − 1)Z, we have (32) ep−i ))(y) = Lp (X, −y, ωpi ) . sX (Θ∞ (X, ω Proof. This is just an exercise in changing notations. For y ∈ Zp , let κy : Γ → C∗p be the character γ 7→ κ(γ)y . Any such character can be extended, by linearity and continuity, to a ring homomorphism κy : Fp [[Γ]] → Cp , which is uniquely characterized by the following property: if µλ denotes the measure on Γ attached to λ ∈ Fp [[Γ]], then we have ∫ (33) κy (λ) = κy (γ)dµλ = s(λ)(y) Γ (the last equality is the definition of s, as should be clear from the proof of Theorem 2.2). Let αei : Z[[GS ]][[X]] → Λ[[X]] be the homomorphism induced by composition of GS  G∞ with the ring homomorphism ω epi : W [[G∞ ]] → Λ (as explained in Section 3.1.2). Definition 3.5 then becomes Θ∞ (X, ω epi ) = α ei (ΘFS /F,S )(X). Moreover, letting αi denote the reduction of ei modulo p, (25) yields the equality ξ(y,i) = κy ◦ αi . For proving (32), one just has to check α ( ) Lp (X, −y, ωpi ) = ξ(y,−i) (ΘFS /F,S )(X) = κy α−i (ΘFS /F,S ) (X) = sX (Θ∞ (X, ωep−i ))(y) . The first equality is Theorem 3.16 and the last one is an easy consequence of (33).  IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 18 4. Fitting ideals for Iwasawa modules In this section we consider the algebraic aspect of the theory, i.e., Fitting ideals of Iwasawa modules associated with the p-cyclotomic extension. Here the Stickelberger element will appear as a generator of Fitting ideals of (χ-parts of) class groups; the final link between the algebraic and the analytic side will be provided by the Iwasawa Main Conjecture of Section 5. Let F be an algebraic closure of F and fix a topological generator γ of the Galois group GF := Gal(F/F) (the arithmetic Frobenius). For any field L we denote by Lar the composition FL (i.e., the arithmetic extension of L): if L/F is finite, then Gal(FL/L) ≃ GF . The arith- metic extension F ar is unramified at every prime and disjoint from F (which is a geometric extension), so Gal(F ar /F ) ≃ G∞ × GF . 4.1. Iwasawa modules in the p-cyclotomic extension. For any finite extension L/F , we let Cℓ 0 (L) be the group of classes of degree zero divisors and we denote by XL the projective curve (defined over F) associated with L. Let Tp (L) := Tp (Jac(XL )(F)) be the p-Tate module of the F-points of the Jacobian of the curve XL . A first task is to compute the Fitting ideals of the modules C n := Cℓ 0 (Fnar ){p} , Cn := Cℓ 0 (Fn ){p} and Tp (Fn ) as Iwasawa modules over some algebra containing Zp [Γn ] (the {p} indicates the p-part of the module; since we shall mainly work with p-parts, the {p} will often be omitted). Recall that Tp (Fn ) ≃ Hom(Qp /Zp , C n ) as Zp [∆][Γn ]-modules. Then we shall perform a limit on n in order to provide a Fitting ideal in the Iwasawa algebra Λ. This will be achieved by means of several maps induced by the natural norms and inclusions (see Section 4.4). For any prime v of F , we let Fnar (v) be the set of places of Fnar lying above v and we put Hv,n := Zp [Fnar (v)] . If v is unramified in Fn /F , then Hv,n is a Zp -free module of rank |Fnar (v)| and (which is more relevant) a Zp [Gn ]-free module of rank dv := deg(v). Since we shall work with Fitting ideals we recall one of the equivalent definition of these ideals (the one more suitable for our computations). Definition 4.1. Let M be a finitely generated module over a ring R. The Fitting ideal of M over R, FittR (M ) is the ideal of R generated by the determinants of all the (minors of the) matrices of relations for a fixed set of generators of M . 4.1.1. Notation. We remark that the integer n > 1 will always denote objects related with the n-th level Fn of the p-cyclotomic extension. We shall work at a fixed finite level n at first, and then, in Section 4.4, we let n vary to compute limits. We recall that χ is a character in Hom(∆, W ∗ ) =: ∆ b and we shall denote by χ0 the trivial character. To work with χ-parts we extend our coefficients to W by considering W ⊗Zp M for any module M . We will apply the decomposition (9) to lim W ⊗Zp Cn , lim W ⊗Zp C n or ← ← lim W ⊗Zp Tp (Fn ). To lighten notations we omit the Zp ; all tensor products will be defined ← over Zp unless we specify otherwise. For the same purpose whenever we have a map η defined on a module M we shall still denote by η the induced map idW ⊗ η on the module W ⊗ M . Finally, for any finite group U , we put ∑ n(U ) := h ∈ Z[U ] h∈U IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 19 (this will mainly appear in the results on the χ0 -part). 4.2. Fitting ideals for the Tate module (I): finite level. Let Frv denote the Frobenius of v in Gn : it is the unique Frobenius attached to v if v is unramified, or any lift of the Frobenius Frv ∈ Gn /Iv,n to Gn if v is ramified (this construction is easily seen to be independent from the choice of the lift). In particular Frp = 1 (because p is totally ramified in Fn /F ) and I Fr∞ = 1 as well because ∞ is totally split in Fn∞,n /F . We define the Euler factor at v as ev (X) := 1 − Fr−1 v X dv , where X is a variable which will often be specialized to γ −1 , so we also put ev := ev (γ −1 ) = 1 − Fr−1 v γ −dv . The next result is exactly [24, Lemmas 2.1 and 2.2] for Fn /F . Lemma 4.2. Let v be a place of F , then: (1) if v is unramified, we have FittZp [Gn ][[GF ]] (Hv,n ) = (ev ); (2) FittZp [Gn ][[GF ]] (H∞,n ) = (e∞ , Aug∞,n ) = (1 − γ −1 , Aug∞,n ); (3) FittZp [Gn ][[GF ]] (Hp,n ) = (ep , Augp,n ) = (1 − γ −d , Augp,n ), where Augv,n is the augmentation ideal associated to Iv,n , i.e., Augv,n := (τ − 1 , τ ∈ Iv,n ). Remark 4.3. If the prime v is unramified, then the module Hv,n is cyclic over the ring Zp [Gn ][[GF ]] and one has Hv,n ≃ Zp [Gn ][[GF ]]/ FittZp [Gn ][[GF ]] (Hv,n ) = Zp [Gn ][[GF ]]/(ev ) . If we consider ramified primes then the same holds over the ring Zp [Gn /Iv,n ][[GF ]]. Hence H∞,n ≃ Zp [Gn ][[GF ]]/(e∞ , Aug∞,n ) = Zp [∆/F∗ × Γn ][[GF ]]/(e∞ ) and Hp,n ≃ Zp [Gn ][[GF ]]/(ep , Augp,n ) = Zp [[GF ]]/(ep ) . Let Σ be a finite set of places of F disjoint from S and, for any n, put S n := Fnar (S) (resp. Σn := Fnar (Σ)) for the set of places of Fnar lying above places in S (resp. Σ). Consider the Deligne’s Picard 1-motive MS n ,Σn associated to Fnar , S n and Σn ; it is represented by a group homomorphism Div 0 (S n ) −→ JacΣn (XFn )(F) , where Div 0 (S n ) is the kernel of the degree map Z[S n ] → Z and JacΣn (XFn ) is the extension of the Jacobian of XFn by a torus (for more details on the definition of MS n ,Σn and its properties we refer the reader to [23, Section 2]). We shall be working with the p-part of class groups, hence (by [23, Remark 2.7]) there is no contribution from the toric part of JacΣn (XFn ). Therefore we can neglect the set Σ (i.e., assume it is empty) in what follows and focus simply on our S = {p, ∞}. The multiplication by p map Div 0 (S n ) ⊗Z Z/pm −→ Div 0 (S n ) ⊗Z Z/pm−1 induces a surjective map on the pm -torsion of MS n := MS n ,∅ . Thus one defines the p-adic Tate module of MS n as Tp (MS n ) := lim ←− MS n [pm ] m (see [23, Definitions 2.5 and 2.6]). With this notations, in our setting, the main result of [23] reads as IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 20 Theorem 4.4. (Greither-Popescu [23, Theorem 4.3]) One has (34) FittZp [Gn ][[GF ]] (Tp (MS n )) = (ΘFn /F,S (γ −1 )) := (Θn (γ −1 )) where Θn (X) is the Stickelberger element ∏ (35) Θn (X) = (1 − Fr−1 dv −1 v X ) ∈ Z[Gn ][[X]] v ∈S / (see Definition 3.1, with dv := deg(v)). The relation between MS n and degree zero divisors with support in S n leads to an exact sequence (36) 0 → Tp (Fn ) → Tp (MS n ) → Div 0 (S n ) ⊗Z Zp → 0 , (see [23, after Definition 2.6]). Our first task is to compute the Fitting ideal (over Zp [Gn ][[GF ]]) of Tp (Fn ) and then project into Zp [Gn ] by specializing at γ −1 = 1. To do this we have to study the χ-parts of the module Dn := Div 0 (S n ) ⊗Z Zp using the fact that it is contained in Zp [S n ] = ⊕v∈S Hv,n . In most cases we will be able to compute Fitting ideals using short exact sequences, while for the “trivial” component we will have to look for a resolution 0 → Dn (χ0 ) → X3 → X4 → 0 that will fit in the sequence (36) transforming it in a 4-term exact sequence to which we can apply [24, Lemma 2.4]. 4.2.1. The χ-parts of Dn . As seen in Remark 4.3, we have H∞,n ≃ Zp [Gn /I∞,n ][[GF ]]/(1 − γ −1 ) ≃ Zp [∆/F∗ × Γn ] . Therefore the χ-parts depend on the values of χ on the elements of F∗ = I∞,n and we have   0 if χ is odd (37) (W ⊗ H∞,n )(χ) ≃ .  W [Γn ] if χ is even Since there is no action of ∆ on Hp,n ≃ Zp [[GF ]]/(ep ), we have   0 ̸ χ0 if χ = (38) (W ⊗ Hp,n )(χ) ≃ .  W [[GF ]]/(ep ) if χ = χ0 For any χ ̸= χ0 we can also observe that (W ⊗ Dn )(χ) = ker{eχ (W ⊗ (Hp,n ⊕ H∞,n )) −→ eχ (W ⊗ Zp ) = 0} ≃ (W ⊗ H∞,n )(χ) , so we are left with the trivial component (W ⊗ Dn )(χ0 ). The degree map on Hp,n provides a decomposition Hp,n ≃ (1 − γ −1 )Hp,n ⊕ Zp , where (1 − γ −1 )Hp,n is obviously in the kernel of the degree map on Dn as well. For the trivial component we have eχ0 (W ⊗ Hp,n ) = eχ0 (W ⊗ (1 − γ −1 )Hp,n ) ⊕ W · 1Hp (where 1Hp is the unit element of Hp,n ), and the map eχ0 (W ⊗ (1 − γ −1 )Hp,n ) ⊕ eχ0 (W ⊗ H∞,n ) −→ eχ0 (W ⊗ Dn ) given by (α, β) → (α − deg(β)1Hp , β) ∈ eχ0 (W ⊗ (Hp,n ⊕ H∞,n )) IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 21 is easily seen to be an isomorphism of W [Γn ][[GF ]]-modules. Hence we obtain    0 if χ is odd    (39) (W ⊗ Dn )(χ) ≃ W [Γn ] if χ ̸= χ0 is even .      (1 − γ −1 )W [[GF ]]/(1 − γ −d ) ⊕ W [Γn ] if χ = χ0 4.2.2. Computation of Fitting ideals (I). The previous descriptions of the (W ⊗ Dn )(χ) allow the first computation of Fitting ideals for the Tate modules. Proposition 4.5. We have  −1   (Θn (γ , χ)) if χ is odd FittW [Γn ][[GF ]] ((W ⊗ Tp (Fn ))(χ)) = ( )  −1  Θn (γ −1,χ) 1−γ if χ ̸= χ0 is even and ( ) ∗ Θn (γ −1 , χ0 ) n(Γn ) FittW [Γn ][[GF ]] ((W ⊗ Tp (Fn ))(χ0 ) ) = 1, 1 − γ −1 νd ∗ 1−γ −d (where denotes the Zp -dual and νd := 1−γ −1 ). b Proof. We split the proof in three parts, depending on the type of the character χ ∈ ∆. Case 1: χ is odd. Since eχ Dn = 0, we have an isomorphism (W ⊗ Tp (Fn ))(χ) ≃ (W ⊗ Tp (MS n ))(χ) . Hence, by Theorem 4.4 above, FittW [Γn ][[GF ]] ((W ⊗ Tp (Fn ))(χ)) = (Θn (γ −1 , χ)) (because, by equation (16), eχ Θn (X) = Θn (X, χ)eχ ). Case 2: χ ̸= χ0 is even. In this case (W ⊗ Dn )(χ) ≃ W [Γn ][[GF ]]/(1 − γ −1 ) is a cyclic W [Γn ][[GF ]]-module. We have an exact sequence (W ⊗ Tp (Fn ))(χ) ,→ (W ⊗ Tp (MS n ))(χ)  W [Γn ][[GF ]]/(1 − γ −1 ) to which we can apply [15, Lemma 3] to get FittW [Γn ][[GF ]] ((W ⊗ Tp (Fn ))(χ))(1 − γ −1 ) = (Θn (γ −1 , χ)) . Case 3: χ = χ0 . Consider the resolution for Dn (χ0 ) provided by the sequences (40) Zp [Γn ] ,→ Zp [Γn ][[GF ]]/(e∞ )  Zp [Γn ][[GF ]]/(e∞ , n(I∞,n )) and (41) (1 − γ −1 )Zp [[GF ]]/(ep ),→(1 − γ −1 )Zp [Γn ][[GF ]]/(ep )(1 − γ −1 )Zp [Γn ][[GF ]]/(ep , n(Ip,n )) where the map on the left is given by 1Hv → n(Iv,n ) (v = ∞, p). To check exactness one simply observes that all the modules involved are Zp -free modules and counts ranks. Joining the sequences (40) and (41) with the sequence (36) and tensoring with W (limiting ourselves to the χ0 -part), we find (W ⊗ Tp (Fn ))(χ0 ) ,→ (W ⊗ Tp (MS n ))(χ0 ) → W [Γn ][[GF ]]/(e∞ ) ⊕ W [Γn ][[GF ]]/(νd )  W [Γn ][[GF ]]/(e∞ , n(I∞,n )) ⊕ W [Γn ][[GF ]]/(νd , n(Ip,n )) . We note that the assumptions of [24, Lemma 2.4] hold for the previous sequence (actually they hold before tensoring with W but the computation of Fitting ideals is not affected by that, moreover we use the full ring R = Zp [Γn ][[GF ]] instead of the R′ of the original paper IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 22 but the lemma still holds as the authors mention right before stating it). Indeed Tp (MS n )(χ0 ) is finitely generated and free over Zp (so it has no nontrivial finite submodules) and it is Γn - cohomologically trivial by the proof of [23, Theorem 3.9]: hence it is of projective dimension 1 over Zp [Γn ][[GF ]] by [30, Proposition 2.2 and Lemma 2.3]. The other 3 modules are finitely generated and free over Zp and, obviously, Zp [Γn ][[GF ]]/(e∞ )⊕Zp [Γn ][[GF ]]/(νd ) has projective dimension at most 1. Therefore we can apply [24, Lemma 2.4] which immediately yields the final statement of the proposition.  4.3. Fitting ideals for class groups. There are deep relations between Tp (Fn ) and the modules Cn := Cℓ 0 (Fn ){p} (the ones we are primarily interested in). Indeed, as noted at the beginning of [24, Section 3], the Zp -dual T (Fn )∗ of T (Fn ) verifies (42) (Tp (Fn )∗ )GF ≃ Cℓ 0 (Fn ){p}∨ = Cn∨ , i.e., its GF -coinvariants are isomorphic to the Pontrjagin dual of Cn . Another one is provided by the following Lemma 4.6. We have an isomorphism of Zp [Gn ]-modules Cn ≃ Tp (Fn )/(1 − γ −1 )Tp (Fn ) = Tp (Fn )GF . Proof. We recall that C n is the p-Sylow of Cℓ 0 (Fnar ), hence it is divisible and isomorphic to (Qp /Zp )r for some r 6 gn (where gn is the genus of XFn ). Obviously C n is a Zp [Gn ][[GF ]]- module and we have Tp (Fn ) = lim ←− C n [pm ] ≃ Hom(Qp /Zp , C n ) m (the Galois action on the module on the right is the usual one (σ · f )(y) := σf (σ −1 y) for any f ∈ Hom(Qp /Zp , C n )). By Lang’s theorem (see, for example, [33, Chapter VI, §4]) we have an exact sequence 1−γ −1 0 → Cn → C n −−−−−−→ C n → 0 . Applying the functor Hom(Qp /Zp , ∗) (a similar argument can be found in [2, Lemma 4.1]) one gets 1−γ −1 0 → Tp (Fn ) −−−−−−→ Tp (Fn ) → Ext1 (Qp /Zp , Cn ) → 0 because Hom(Qp /Zp , Cn ) = 0 (Cn is finite) and Ext1 (Qp /Zp , C n ) = 0 (C n is divisible). Now from the usual short exact sequence 0 → Zp → Qp → Qp /Zp → 0 , applying Hom(∗, Cn ), we obtain Hom(Qp , Cn ) = 0 → Hom(Zp , Cn ) ≃ Cn → Ext1 (Qp /Zp , Cn ) → Ext1 (Qp , Cn ) = 0 . Therefore we have an isomorphism Cn ≃ Ext1 (Qp /Zp , Cn ) ≃ Tp (Fn )/(1 − γ −1 )Tp (Fn ) = Tp (Fn )GF .  Remark 4.7. Equation (42) and Lemma 4.6 together yield (Tp (Fn )∗ )GF ≃ Cn∨ ≃ (Tp (Fn )GF )∨ . A general statement of this type appears in [23, Lemma 5.18]. Definition 4.8. Let χ ∈ Hom(∆, W ∗ ) with χ ̸= χ0 and n ∈ N ∪ {∞}, we define the modified Stickelberger series as   Θ (X, χ) if χ is odd  n # Θn (X, χ) := .   Θn (X, χ) if χ is even 1−X IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 23 Consider the projection map πGF : W [Γn ][[GF ]] → W [Γn ] which maps γ to 1. The properties of Fitting ideals and Lemma 4.6 yield Corollary 4.9. For χ ̸= χ0 we have FittW [Γn ] ((W ⊗ Cn )(χ)) = (Θ# n (1, χ)) . While the isomorphism (42) leads to Corollary 4.10. For the trivial character χ0 we have ( ) ∨ Θn (X, χ0 ) n(Γn ) FittW [Γn ] ((W ⊗ Cn )(χ0 ) ⊗W Q(W )) = |X=1 1, 1−X d (where Q(W ) is the quotient field of W ). 4.4. Fitting ideals for Tate modules (II): infinite level. Consider the Iwasawa tower F/F and let φn+1 n : XFn+1 → XFn be the morphism of curves corresponding to the field extension Fn+1 /Fn ; it is a Γn+1 n := Gal(Fn+1 /Fn ) Galois cover totally ramified at p. As before χ denotes an element of Hom(∆, W ∗ ). We have a morphism inn+1 : Tp (Fn ) ,→ Tp (Fn+1 ) (induced by the natural map from C n to C n+1 ) and a map Nnn+1 : Tp (Fn+1 ) → Tp (Fn ) induced by the norm map from C n+1 to C n . Observe that, for any n, Nnn+1 ◦ inn+1 = q d . In this section we shall meet various other maps induced by norms (resp. inclusions) on different modules/objects: by abuse of notations we shall denote all of them by Nnn+1 (resp. inn+1 ), when we need some distinction between them we shall write N (•)n+1 n (resp. i(•)nn+1 ) to denote the map defined on the objects • or Tp (•). 4.4.1. Norm and inclusion maps. We have an inclusion inn+1 : Tp (MS n ) ,→ Tp (MS n+1 ) such n+1 that Tp (MS n+1 )Γn = inn+1 (Tp (MS n )) by [23, Theorem 3.1]. We also have a natural norm map N (M)n+1 n : Tp (MS n+1 ) → Tp (MS n ). Lemma 4.11. The norm map N (M)n+1 n is surjective and its kernel is IΓn+1 n Tp (MS n+1 ) where IΓn+1 n is the augmentation ideal (i.e., generated by {σ − 1 : σ ∈ Γn+1 }). n Proof. By [23, Theorem 3.9] (in particular, its proof) we have that Tp (MS n+1 ) is Γn+1 n -coho- mologically trivial, i.e., b i (Γn+1 , Tp (M H n S n+1 )) = 0 ∀ i . For i = 0 we have that n+1 Tp (MS n+1 )Γn = N (M)n+1 n (Tp (MS n+1 )) , n+1 but, as recalled above, Tp (MS n+1 )Γn = Tp (MS n ), therefore the norm map is surjective. With i = −1 we obtain that the kernel of N (M)n+1 n is given by the augmentation module IΓn+1 n Tp (M S n+1 ).  We have a commutative diagram of short exact sequences (43) 0 / Tp (Fn+1 ) / Tp (M / Dn+1 /0 S n+1 ) N (F )n+1 n N (M)n+1 n N (D)n+1 n    0 / Tp (Fn ) / Tp (M ) / Dn /0 Sn where all vertical maps are induced by norms: in particular note that N (D)n+1 n corresponds to the natural map on divisors Zp [Fn+1 ar (∞)] ⊕ Zp [Fn+1 ar (p)] → Zp [Fnar (∞)] ⊕ Zp [Fnar (p)] . IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 24 Lemma 4.12. Let χ ̸= χ0 and n > 1. Then   0 if χ is odd ker(N (D)n+1 )(χ) = . n  I n+1 (W ⊗ D Γn n+1 )(χ) if χ is even Moreover, the map N (D)nn+1 is surjective. Proof. The last assertion immediately follows from the surjectivity of N (M)n+1n and the snake lemma sequence of diagram (43). Now consider the same diagram but with χ-parts and tensored with W and note that the maps N (M)n+1 n and N (D)n+1 n remain surjective. From the computations in Section 4.2.1 the case χ odd is obvious. When χ ̸= χ0 is even we have eχ (W ⊗Dn+1 ) ≃ W [Γn+1 ] and the norm corresponds to the projection W [Γn+1 ] → W [Γn ] which has kernel IΓn+1 n .  The previous two lemmas lead to similar statements for the map N (F )n+1 n (χ). Proposition 4.13. Let χ ̸= χ0 , then (W ⊗ Tp (Fn ))(χ) is Γn and Γn+1 n -cohomologically trivial and a free W [Γn ]-module. In particular n+1 (W ⊗ Tp (Fn+1 ))(χ)Γn = (W ⊗ Tp (Fn ))(χ) , N (F )n+1 n+1 n (χ) is surjective and ker(N (F )n )(χ) = IΓn+1 n (W ⊗ Tp (Fn+1 ))(χ). Moreover it is also a W [Γn ][[GF ]]-module of projective dimension less than or equal to one. Proof. Consider the short exact sequence 0 → W ⊗ Tp (Fn ) → W ⊗ Tp (MS n ) → W ⊗ Dn → 0 . Since Dn (χ) is 0 or W [Γn ], it is W [Γn ]-free and cohomologically trivial, while W ⊗ Tp (MS n ) is also W [Γn ]-free and cohomologically trivial by [23, Theorem 3.9]. Thus (W ⊗ Tp (Fn ))(χ) is projective over W [Γn ] and cohomologically trivial. Now, since Γn is a p-group, W [Γn ] is a local ring and projective modules coincide with free modules. The cohomological triviality over Γn+1 n is similar and straightforward. The assertion on ker(N (F )n+1 b1 n+1 n )(χ) comes from the triviality of the H . Now take Γn - invariants in the sequence for even characters (for odd ones there is nothing to prove) (W ⊗ Tp (Fn+1 ))(χ) ,→ (W ⊗ Tp (MS n+1 ))(χ)  (W ⊗ Dn+1 )(χ) ≃ W [Γn+1 ] , to get n+1 (W ⊗ Tp (Fn+1 ))(χ)Γn ,→ (W ⊗ Tp (MS n ))(χ)  (W ⊗ Dn )(χ) ≃ W [Γn ] n+1 (using Lemmas 4.11 and 4.12). Thus (W ⊗ Tp (Fn+1 ))(χ)Γn = (W ⊗ Tp (Fn ))(χ) and, since the Hb 0 is trivial, N (F )n+1 (χ) is surjective. n The last statement of the lemma follows from [30, Proposition 2.2 and Lemma 2.3] (see also [17, Proposition 5.3]) because (W ⊗ Tp (Fn ))(χ), being free, has no nontrivial finite W [Γn ]- submodule.  Remark 4.14. For χ = χ0 the modules (W ⊗ T (MS n ))(χ0 ) and W [Γn ] are still Γn -cohomo- logically trivial and we have the short exact sequence (1 − γ −1 )W [[GF ]] (W ⊗ Tp (Fn ))(χ0 ) ,→ (W ⊗ T (MS n ))(χ0 )  ⊕ W [Γn ] . (1 − γ −d ) Since Fn /F is disjoint from F/F, the norm acts on GF as multiplication by [Fn : F ]. For any subextension E/K and for any k ∈ Z, we have ( ) b (1 − γ −1 )W [[GF ]] b k+1 (Gal(E/K), (W ⊗ Tp (Fn ))(χ0 )) k H Gal(E/K), ≃H (1 − γ −d ) IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 25 and, in particular, −1 −1 b 0 (Gal(E/K), (W ⊗ Tp (Fn ))(χ0 )) ≃ (1 − γ )W [[GF ]] /| Gal(E/K)| (1 − γ )W [[GF ]] . H (1 − γ −d ) (1 − γ −d ) Therefore (W ⊗ Tp (Fn ))(χ0 ) is not Γn -cohomologically trivial (and not not necessarily W [Γn ]- free). 4.4.2. Computation of Fitting ideals (II). We define Tp (F)(χ) := lim ←− (W ⊗ Tp (Fn ))(χ) n (the limit is on the norm maps studied above). We recall that Tp (Fn ) = Tp (Jac(XFn )(F)), so Tp (F)(χ) is a Λ[[GF ]]-module, where Λ = W [[Γ]]. Proposition 4.15. For χ ̸= χ0 , Tp (F)(χ) is a finitely generated torsion Λ[[GF ]]-module. Proof. The ideals In (defined in Section 2.2) form an open filtration for Λ and we note that In = lim I m+n (where IΓm+n ←− Γn n denotes the augmentation ideal associated to Γm+n n = m Gal(Fn+m /Fn )). From Proposition 4.13 we have that Tp (F)(χ)/(Λ[[GF ]] ⊗Λ In )Tp (F)(χ) ≃ (W ⊗ Tp (Fn ))(χ) for all n > 1. The module on the right is finitely generated over W [Γn ][[GF ]] = Λ[[GF ]]/(Λ[[GF ]] ⊗Λ In ) , so, by a generalized Nakayama Lemma (see [4, Corollary p. 226]), we have that Tp (F)(χ) is finitely generated as a Λ[[GF ]]-module. Moreover Θ# −1 n (γ , χ)(W ⊗ Tp (Fn ))(χ) = 0 for any n (by Proposition 4.5), hence −1 Θ#∞ (γ , χ)Tp (F)(χ) = 0 , i.e., the module Tp (F)(χ) is torsion.  Therefore the Fitting ideal of the Λ[[GF ]]-module Tp (F)(χ) is well defined and we have the following formula for it (for a similar result see [22, Theorem 2.1], but note the particular case of [22, Remark 2.2 (2)] which fits our setting). Theorem 4.16. For χ ̸= χ0 we have −1 FittΛ[[GF ]] (Tp (F)(χ)) = (Θ# ∞ (γ , χ)) . Proof. By the previous proposition we can find an r ∈ N such that the following diagram commutes (44) 0 / Bn+1 / W [Γn+1 ][[GF ]]r / (W ⊗ Tp (Fn+1 ))(χ) /0 bn+1 n n+1 πn N (F )n+1 n (χ)    0 / Bn / W [Γn ][[GF ]]r / (W ⊗ Tp (Fn ))(χ) /0 (where the central map is the canonical projection). The n+1 n+1 ( )r maps πn and N (F )n (χ) are surjective and that their kernels are IΓn+1 n W [Γn+1 ][[GF ]] and IΓn+1 n (W ⊗Tp (Fn+1 ))(χ). The map between these two kernels is obviously surjective, hence, by the snake lemma sequence, we have that bn+1 n is surjective as well. Taking the inverse limit in diagram (44) (which verifies the Mittag-Leffler condition), we obtain the exact sequence (45) 0 → B∞ := lim ←− Bn → Λ[[GF ]]r → lim ←− (W ⊗ Tp (Fn ))(χ) = Tp (F)(χ) → 0 . n n IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 26 Recall that we can use any β1 , . . . , βr ∈ Bn as rows of a matrix Mβ1 ,...,βr ∈ M atr (W [Γn ]) and FittW [Γn ] ((W ⊗ Tp (Fn ))(χ)) is generated by the det(Mβ1 ,...,βr ). The surjectivity of the maps bn+1 n implies the same property for the induced maps bn : B∞  Bn , i.e., the “relations” at the infinite level are all induced by “relations” already existing at lower levels (the technical arguments of the final parts of [22, Theorem 2.1] are not necessary here because of the presence of just one ramified prime and our previous computations on kernels of norm maps). Using the characterization of the Fitting ideal in Definition 4.1, it is easy to see that the surjectivity of the bn and the sequence (45) yield the desired result, i.e., FittΛ[[GF ]] (Tp (F)(χ)) = (det(B∞ )) = lim ←− (det(Bn )) n = lim ←− FittW [Γn ][[GF ]] ((W ⊗ Tp (Fn ))(χ)) n −1 −1 = lim ←− (Θ# # n (γ , χ)) = (Θ∞ (γ , χ)) .  n 5. Iwasawa main conjecture for the p-cyclotomic extension Consider now W ⊗ Cn = W ⊗ Jac(XFn )(F) = W ⊗ P ic0 (XFn )(F) as a W [Gn ]-module and the natural maps i(C)nn+1 : W ⊗ Cn → W ⊗ Cn+1 and N (C)n+1 n : W ⊗ Cn+1 → W ⊗ Cn . Denote by C the W [[G∞ ]]-module lim W ⊗ Cn (defined, as usual, with respect the norm maps ←− N (C)n+1 n ). The main results of this section are the following Theorem 5.1. Let χ ̸= χ0 , then the module C(χ) := εχ C is a finitely generated torsion Λ-module. Therefore the Fitting ideal FittΛ (C(χ)) is well defined and we have Theorem 5.2 (Iwasawa Main Conjecture). Let χ ̸= χ0 , then FittΛ (C(χ)) = (Θ# ∞ (1, χ)) . Remark 5.3. The theorem above allows us to compute the Fitting ideal of C(χ) as the inverse limit of the Fitting ideals appearing in the (natural) filtration of F given by the fields Fn . A different approach to the same problem is provided in [6, Section 5] where the authors use a filtration of Zdp -extensions (a more general approach and the fact that the limit is independent from the filtration are shown in [7]). In that paper the statement of the Main Conjecture involves characteristic ideals but (for Iwasawa modules) they coincide with Fitting ideals whenever the Fitting is principal (see, for example, [5, Lemma 5.10]). Before going into the proofs of the above theorems, we need a crucial lemma. Lemma 5.4. Let F0 ⊂ K ⊂ E ⊂ F, where E/F is a finite extension and the group G := Gal(E/K) is a p-group. For any field L ⊂ F we let pL be the unique prime of L lying above p, we recall that Cℓ 0 (L) denotes the group of classes of degree zero divisors. We have the following properties: (1) the map iK E : Cℓ (K) → Cℓ (E) is injective; 0 0 |G| (2) there is an equality Cℓ0 (E)G = iKE (Cℓ (K)) + ⟨r pt pE − pt iE (v)⟩, where p := (|G|, d), 0 d K t v is a place of K lying above a prime of A of degree r (prime with p) and which is totally split in E. The second term disappears when we consider χ-parts for nontrivial characters, in particular, for χ ̸= χ0 , we have ((W ⊗ Cℓ0 (E))(χ))G = iK E ((W ⊗ Cℓ (K))(χ)) ; 0 E : Cℓ0 (E) → Cℓ0 (K) is surjective; (3) the norm map NK IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 27 (4) for χ ̸= χ0 , we have ker(NK E (χ)) = I (W ⊗ Cℓ0 (E))(χ). G Proof. For simplicity we write N and i for NK E and iK respectively. E (1) For any field L write PL for the principal divisors of L. Consider the exact sequences (46) 0 → F∗ → E ∗ → PE → 0 and (47) 0 → PE → Div 0 (E) → Cℓ0 (E) → 0 . Taking G-cohomology in (46) one finds 0 → F∗ → K ∗ → PEG → 0 → 0 → H 1 (G, PE ) → 0 (because of Hilbert 90 and because G is a p-group so H i (G, F∗ ) = 0 for any i > 1), so, in particular, PEG = PK and H 1 (G, PE ) = 0. The G-invariants of the sequence (47) then fit into the diagram  (48) PK  / Div 0 (K) / / Cℓ0 (K) i iK E    PEG = PK  / Div 0 (E)G / Cℓ0 (E)G / H 1 (G, PE ) = 0 . The injectivity of the central vertical map and the snake lemma sequence yield the desired injectivity of iK E. (2) We need several steps: (a) The group Div(E)G /i(Div(K)) is cyclic of order [E : K] and is generated by the class of pE . Write Div(K) = ⊕v Zv (v runs through all the primes of K) and Div(E) = ⊕v Hv with Hv = ⊕w|v Zw. If v = ̸ pK is unramified, then Hv = Z[G/Gv ]w, where Gv is the decomposition subgroup of v in G, and obviously ∑ HvG = Zi(v) with i(v) = σw . σ∈G/Gv For the ramified place we have σ(pE ) = pE and |G|pE = i(pK ): this yields the statement. (b) The group Div 0 (E)G /i(Div 0 (K)) is killed by pt := (|G|, d). Take D ∈ Div 0 (E)G , by part (a) we can write D = npE + D′ with D′ ∈ i(Div(K)). Since D has degree zero we have −nd = degE (D′ ) = |G| degK (D′ ) and |G| pt divides |G| ′ n= pt n . Therefore pt D = pt npE + pt D′ = |G|n′ pE + pt D′ = n′ pK + pt D′ ∈ i(Div(K)) . Moreover pt n degK (n′ pK + pt D′ ) = n′ d + pt degK (D′ ) = d + pt degK (D′ ) |G| pt = (nd + |G| degK (D′ )) = 0 , |G| hence pt D ∈ i(Div 0 (K)). (c) There are isomorphisms Cℓ0 (E)G /iKE (Cℓ (K)) ≃ Div (E) /i(Div (K)) ≃ Z/p . 0 0 G 0 t The first isomorpshim is a consequence of the snake lemma sequence of diagram (48). For the second, by part (b) it is enough to build a divisor of exact order pt . Let r be prime with p; by Chebotarev density theorem there exists a (monic) irreducible IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 28 polynomial Q in A such that deg(Q) = r and the prime (Q) is totally split in E. Take a prime v of K dividing (Q) so that, in particular, degK (v) = r. Put e := r |G| pE − d i(v) ; D pt pt e ∈ Div 0 (E)G and its order is pt . it is easy to check that (by construction) D e = 0. (d) If χ ̸= χ0 , then eχ (D) Recall that ∆ = Gal(F0 /F ) has order prime to p. The previous steps can be proved exactly in the same way for the field extension E ∆ /K ∆ , i.e., we have Div 0 (E ∆ )G Cℓ0 (E ∆ )G ∆ ≃ K ∆ (Cℓ0 (K ∆ )) ≃ Z/pt iK E∆ (Div 0 (K ∆ )) iE∆ and a generator is the class of f′ := r |G| p ∆ − d iK∆∆ (ṽ) ∈ Div 0 (E ∆ )G D pt E pt E f′ in Div 0 (E) is (where ṽ is a prime of K ∆ lying below v). Note that the image of D ∆ f′ |G| d ∆ iE E (D ) = r |∆|pE − t iK (ṽ) pt p E and it still has order pt because (|∆|, p) = 1. Therefore the class of iE ∆ f′ E (D ) generates Cℓ (E) /i(Cℓ (K)) and, by construction, ∆ acts trivially on it. Hence for χ ̸= χ0 we 0 G 0 obtain eχ (W ⊗ Cℓ0 (E))G = eχ i(W ⊗ Cℓ0 (K)) and for the trivial character we have ( ) (W ⊗ Cℓ0 (E))G eχ0 ≃ W/pt W . i(W ⊗ Cℓ0 (K)) (3) This is just class field theory. Let v be a place of K which divides ∞ and write B for the ring of elements in K which are regular outside v; since v is of degree 1, we have Cℓ(B) ≃ Cℓ0 (K). Let H(K) be the maximal abelian unramified extension of K in which v is totally split. By class field theory, the Artin map provides an isomorphism Gal(H(K)/K) ≃ Cℓ(B) and, because of the ramification in E/K, we have H(K)∩E = K. Denote by C the integral closure of B in E (i.e., the elements in E which are regular outside any w|v); there is a natural map Cℓ0 (E) → Cℓ(C) ≃ Gal(H(E)/E) which preserves Galois action and is surjective because degE (w) = 1 (H(E) is the analog of H(K), now totally split at w). It only remains to prove that the natural norm map Cℓ(C) → Cℓ(B) is surjective. By construction EH(K) ⊂ H(E), hence the restriction map Res : Gal(H(E)/E) → Gal(EH(K)/E) ≃ Gal(H(K)/K) is surjective. The well known diagram of class field theory ≃ / Gal(H(E)/E) Cℓ(C) N Res   ≃ / Gal(H(K)/K) Cℓ(B) concludes the proof. (4) Consider the sequence (exact by part (3) ) N 0 → (W ⊗ ker(N ))(χ) → (W ⊗ Cℓ0 (E))(χ)−→(W ⊗ Cℓ0 (K))(χ) → 0 , IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 29 which yields |(W ⊗ Cℓ0 (E))(χ)| |(W ⊗ ker(N ))(χ)| = |(W ⊗ Cℓ0 (K))(χ)| (we recall that for any W -module M one has |M | = pu·ℓW (M ) with ℓW the length and u := [W ⊗ Qp : Qp ]). We first assume that G = ⟨δ⟩ is cyclic. Then, for χ ̸= χ0 , the sequence (exact by part (2) ) 1−δ 0 → (W ⊗ Cℓ0 (K))(χ) → (W ⊗ Cℓ0 (E))(χ)−−−−−→(1 − δ)(W ⊗ Cℓ0 (E))(χ) → 0 yields |(1 − δ)(W ⊗ Cℓ0 (E))(χ)| = |(W ⊗ ker(N ))(χ)| (simply by counting cardinalities). Since (1 − δ)(W ⊗ Cℓ0 (E))(χ) ⊆ (W ⊗ ker(N ))(χ), we have the equality between them. For the general case G we use an induction argument on |G|. If |G| = 1 there is nothing to prove (or, if |G| = p, then G is cyclic and we have the proof above). Consider now K ⊂ E ′ ⊂ E, where G1 := Gal(E/E ′ ) is a cyclic group and we put G2 := Gal(E ′ /K). By the inductive step and the cyclic case we have ′ (W ⊗ ker(NK E ))(χ) = IG2 (W ⊗ Cℓ0 (E ′ ))(χ) and (W ⊗ ker(NEE′ ))(χ) = IG1 (W ⊗ Cℓ0 (E))(χ) . ′ E ◦ N E , we have By part (3) all norms are surjective and, since N = NK E′ (W ⊗ ker(N ))(χ) = (NEE′ )−1 IG2 (W ⊗ Cℓ0 (E ′ ))(χ) = IG (W ⊗ Cℓ0 (E))(χ) + (W ⊗ ker(NEE′ ))(χ) = IG (W ⊗ Cℓ0 (E))(χ) , because (W ⊗ ker(NEE′ ))(χ) = IG1 (W ⊗ Cℓ0 (E))(χ) ⊂ IG (W ⊗ Cℓ0 (E))(χ).  Proof of Theorem 5.1. Recall that Cn := Cℓ 0 (Fn ){p}: by Corollary 4.9, we know that for χ ̸= χ0 FittW [Γn ] ((W ⊗ Cn )(χ)) = (Θ# n (1, χ)) . By the previous lemma the kernel of N (C)n+1 n : (W ⊗ Cn+1 )(χ)  (W ⊗ Cn )(χ) is IΓn+1 n (W ⊗ Cn+1 )(χ) and we know that In = lim I m+n . Hence ←− Γn m C(χ)/In C(χ) ≃ (W ⊗ Cn )(χ) as W [Γn ]-modules. The generalized version of Nakayama Lemma implies that C(χ) is a finitely generated Λ-module because (W ⊗ Cn )(χ) is a finitely generated W [Γn ]-module. Now simply recall that Θ# # n (1, χ)((W ⊗ Cn )(χ)) = 0 and that Θ∞ (1, χ) = lim ←− Θ# n (1, χ) to get n ( ) Θ# ∞ (1, χ)C(χ) = lim ←− Θ# n (1, χ) lim ←− (W ⊗ C n )(χ) =0, n n i.e., C(χ) is a torsion Λ-module.  Proof of Theorem 5.2 (IMC). By Theorem 5.1 the Fitting ideal FittΛ (C(χ)) is well defined. The statement is equivalent to the equality ( ) ( ) FittΛ lim ←− (W ⊗ Cn )(χ) = lim←− FittW [Γn ] ((W ⊗ Cn )(χ)) . n n IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 30 Take r ∈ N such that eχ C is generated by r elements, consider the following commutative diagram  Bn+1  / W [Γn+1 ]r / / (W ⊗ Cn+1 )(χ) bn+1 n n+1 πn N (C)n+1 n     Bn  / W [Γn ]r / / (W ⊗ Cn )(χ) (where the central vertical map n+1 ( is the natural)rprojection) and note that the kernels of πn and N (C)n+1 n are respectively IΓn+1n W [Γn+1 ] and IΓn+1n (W ⊗Cn+1 )(χ) (by Lemma 5.4 part (4)). Therefore the induced map between the kernels is surjective and this, together with the surjectivity of πnn+1 , yields the surjectivity of bn+1 n by the snake lemma. Now the diagram above verifies the Mittag-Leffler condition, so, taking the limit, we have an exact sequence (49) B∞ := lim ←− Bn ,→ Λr  C(χ) . n Working as in Theorem 4.16, one sees that the surjectivity of the bn and the sequence (49) yield ( ) FittΛ (C(χ)) = lim ←− FittW [Γ n ] ((W ⊗ C n )(χ)) n ( ) = lim ←− Θ # n (1, χ) = (Θ# ∞ (1, χ)) .  n We end this section with a remark on the module structure of C(χ) which depends on the injectivity of the inclusion maps (i.e., part (1) of Lemma 5.4); the proof is similar to [42, Proposition 13.28] (which depends on the injectivity of [42, Proposition 13.26]). Proposition 5.5. The Λ-module C(χ) has no nontrivial finite Λ-submodule. Proof. Let M be a finite Λ-submodule of C(χ) of order |M | = s (obviously a power of p). It is enough to prove that there is no p-torsion in M , so let α = (αn )n∈N ∈ M be such that pα = 0 (so that pαn = 0 for any n ≫ 0). Fix n and take a γi,j among the generators of Γ which acts trivially on Fn . Denote by L∞ = ∪Lm the Zp -extension topologically generated by γi,j over F0 . The s + 1 elements of M p p 2 p s α , γi,j α , γi,j α , . . . , γi,j α cannot be distinct, hence there exist 0 6 r < t 6 s such that ( ) pr pt pr pt −pr γi,j α = γi,j α , i.e., γi,j 1 − γi,j α=0. pr (pt−r −1) p t−r −1 This yields γi,j α = α: since γi,j and γi,j generate the same Zp -extension we can p r assume from the beginning that there exists an r > 0 such that γi,j α = α. By construction p m Fn ∩ L∞ = F0 and, for any m > r, Gal(Lm+1 /Lm ) (generated by γi,j ) acts trivially on α. Take ν big enough to have pαν = 0 and Lm+1 Fn ⊂ Fν , and consider the tower of extensions Fn ⊂ Lm Fn ⊂ Lm+1 Fn ⊂ Fν . From the surjectivity of the norm maps proved in Lemma 5.4 one has C(χ) = lim ←− (W ⊗ CL )(χ) , [L:F0 ]<∞ so we can compute L Fn inν (αn ) = inν (Nnν (αν )) = inν (NFLnm Fn (NLmm+1 Fn (NLFm+1 ν Fn (αν )))) . IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 31 n L F Since Gal(Lm+1 Fn /Lm Fn ) acts trivially on α, the norm NLmm+1 Fn is just multiplication by p and we get inν (αn ) = 0. But Lemma 5.4 part (1) shows that the maps like i are injective so αn = 0 and, eventually, α = 0 as well.  6. Application to Bernoulli-Goss numbers and p-adic L-functions We define an arithmetic invariant related to our p-adic L-function. Definition 6.1. For any i, define  { }   Inf vp (Lp (1, y, ωpi )) : y ∈ Zp for i ̸≡ 0 (mod q − 1)  (50) mp (i) := { ( ) } .   Inf vp d  Lp (X, y, ωp )|X=1 : y ∈ Zp i for i ≡ 0 (mod q − 1) dX Obviously the value of mp (i) depends only on the class of i modulo q d − 1. Lemma 6.2. We have the following equality { } mp (i) = Inf vp (β(j)) : j > 1, j ≡ i (mod q d − 1) . Proof. Let j ≡ i (mod q d − 1), then, by Corollary 3.17, Lp (X, j, ωpi ) = (1 − πpj X d )Z(X, j) . By Definition 3.13, if j ̸≡ 0 (mod q − 1), we have (51) (1 − πpj )β(j) = (1 − πpj )Z(1, j) = Lp (1, j, ωpi ) , while, if j > 1 with j ≡ 0 (mod q − 1), we have d d (52) Lp (X, j, ωpi )|X=1 = (1 − πpj ) Z(X, j)|X=1 = −(1 − πpj )β(j) dX dX (recall Z(1, j) = 0 in this case). The lemma follows noting that the set {j > 1 , j ≡ i (mod q d − 1) } is dense in Zp .  We can now prove a function field version of the Ferrero-Washington Theorem (see, e.g., [42, Theorem 7.15]), but its statement is limited to nontrivial characters. Theorem 6.3. For any 1 6 i 6 q d − 2, one has Θ# epi ) ̸≡ 0 (mod p). ∞ (1, ω Proof. We consider two cases depending on the type of the character ω epi . Recall that, by Lemma 3.14, β(j) ̸= 0 for any j > 0. Case 1: i ̸≡ 0 (mod q − 1), i.e., ωepi is odd. In this case Θ∞ = Θ∞ . Take j ≡ −i (mod q d − 1). Then (51) shows that Lp (1, j, ωp−i ) is # nonzero, hence, by Theorem 3.22, sX (Θ∞ (X, ω epi ))(−j)|X=1 is nonzero as well. It follows that epi )) ̸= 0 and therefore s(Θ∞ (1, ω epi ) ̸≡ 0 Θ∞ (1, ω (mod p) , since the Sinnott map s has domain Λ/pΛ. Case 2: i ≡ 0 (mod q − 1), i.e., ω epi is even but ̸= χ0 . Θ∞ In this case Θ# ∞ = , hence 1−X d Θ# epi ) = − ∞ (1, ω Θ∞ (X, ωepi )|X=1 . dX Again, take j ≡ −i (mod q d − 1). Then (52) shows d Lp (X, j, ωp−i )|X=1 ̸= 0 dX IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 32 and Theorem 3.22 yields ( ) d d d sX Θ∞ (X, ωep ) (−j) = i epi ))(−j) = sX (Θ∞ (X, ω Lp (X, j, ωp−i ) . dX dX dX From here we get the claim by the same reasoning as in case 1.  As a consequence we find that the pn -torsion of C(χ) looks like a pseudo-null module in the non-noetherian Iwasawa algebra Λ. Corollary 6.4. For any character χ ̸= χ0 , p does not divide FittΛ (C(χ)) and the pn -torsion modules C(χ)[pn ] have at least two relatively prime annihilators. Proof. Easy consequences of the previous theorem and Theorem 5.2  Remarks 6.5. 1. Since we are working in the non-noetherian algebra Λ, the module C(χ)[p∞ ] might be not finitely generated on W . The last statement (recalling pseudo-nullity for noether- ian Iwasawa algebras) might be false if we consider the whole set of p-power torsion points C(χ)[p∞ ]. However a combination of Proposition 5.5 and the previous corollary suggests to investigate the possibility that C(χ)[p∞ ] = 0. 2. In [25, page 4446] the authors provide a formula for the class number growth in subex- tensions of the p-cyclotomic extension and note that the growth can be exponential, i.e., the direct analog of the Ferrero-Washington Theorem (µ = 0) does not hold for function fields. An estimate for mp (i) is provided by the following Lemma 6.6. For any positive integer ∑ m let ℓ(m) be the sum of the digits of∑the q-adic expansion of m (i.e., writing m = mi q i with 0 6 mi 6 q − 1, one has ℓ(m) = mi ). For any 1 6 i 6 q − 2 with i ̸≡ 0 (mod q − 1), one has d i ℓ(i) (53) mp (i) 6 · . d q−1 Proof. By [19, Corollary 2.12], one has ∑ ℓ(j) Sn (j) = aj = 0 if n > . q−1 a∈A+,n Hence ℓ(j) ⌊ q−1 ⌋ ∑ ∑ ∑ β(j) = Sn (j) = 1 + aj n>0 n=1 a∈A+,n (where ⌊∗⌋ means the integral part of ∗). Clearly we have ⌊ ⌋ ℓ(j) deg(β(j)) 6 j q−1 and the result follows from Lemma 6.2.  Remark 6.7. Using the bounds on deg Sn (j) provided in [40, Section 6] (please be aware that the notations in that paper differ from ours, in particular our deg Sn (j) corresponds to −sn (−j) there), it is possible to improve the bound (53) and also to find a lower bound for mp (i). We decided to stop here and just provide (53) as an easy example of what can be achieved. Computations using the bounds of [40] can become quite cumbersome and our techniques (depending on the IMC) are of a completely different nature. IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 33 In the classical setting of p-adic L-functions defined for the cyclotomic Zp -extension of a number field (see, for example, [42, Chapter 5]), there is the following natural problem on the p-adic valuation of values of p-adic L-functions (related to p-adic valuations of generalized Bernoulli numbers) Open Question 6.8. Let χ be an even character in Hom(Gal(Q(µp )/Q), Z∗p ), then is it true that Inf {vp (Lp (y, χ)) : y ∈ Zp } 6 1 ? In the following we still consider non-trivial characters only. Corollary 6.9 (Arithmetic properties of Bernoulli-Goss numbers). Let 1 6 i 6 q d − 2 and define Np (i) := Inf{n > 0 : Θ# epi ) ̸≡ 0 (mod p) } n (1, ω (it is well defined because of Theorem 6.3). Then { } Np (i) 6 Inf vp (β(j)) : j > 1, j ≡ −i (mod q d − 1) = mp (q d − 1 − i) = mp (−i) . Proof. Assume i ̸≡ 0 (mod q − 1): by definition of mp (i) (or by Lemma 6.2), there exists y0 ∈ Zp such that Lp (1, y0 , ωp−i ) ̸≡ 0 (mod pmp (−i)+1 ), while for any y ∈ Zp we have Lp (1, y, ωp−i ) ≡ 0 (mod pmp (−i) ). In Section 2.3 we saw that the map s can be computed by taking the limit on the (induced) maps sn : W/pW [Γn ] → C 0 (Zp , Ap /pn+1 ) . Therefore, for n < mp (−i), we obtain that sn (Θ# epi )) is the zero function, while n (1, ω smp (−i) (Θ# epi ))(−y0 ) ̸= 0 . mp (−i) (1, ω Since the maps sn are not injective in general (see Proposition 2.4) we only obtain an inequality Np (i) = Inf{n > 0 : Θ# n (1, χ) ̸≡ 0 (mod pW [Γn ]) } 6 mp (−i) . The proof for even nontrivial characters (i.e., for i ≡ 0 (mod q − 1), i ̸= 0) is similar.  Remark 6.10. We can define similar arithmetic invariants for the Zp -cyclotomic extension of a number field k. For simplicity we just consider k = Q with p ̸= 2, the generalization is straightforward. Take a character χ in Hom(Gal(Q(µp )/Q), Z∗p ) and let Lp (y, χ) = f ((1 + p)y − 1, χ) , f (T, χ) ∈ Zp [[T ]] be the associated p-adic L-function. Let Qn := Q(µpn+1 ) be the layers of the Zp -cyclotomic extension and let Cℓ(Q∞ ) := lim Cℓ(Qn )⊗Z Zp . As mentioned in the introduction, the Iwasawa ←− Main Conjecture in this setting reads as FittZp [[T ]] (Cℓ(Q∞ )(ωp χ−1 )) = (f (T, χ)) , and we can define mp (χ) := Inf{ vp (Lp (y, χ)) : y ∈ Zp } ∈ N . Finally let ΘQn /Q,p (χ) be the Stickelberger element (see [42, Chapter 6]): again via the Main Conjecture, we have FittZp [Gal(Qn /Q)] (Cℓ(Qn )(ωp χ−1 )) = (ΘQn /Q,p (χ)) . The Ferrero-Washington Theorem implies that Np (χ) := Inf{ n > 0 : ΘQn /Q,p (χ) ̸≡ 0 (mod p) } is well defined. At present it is not clear whether there is any kind of relation between mp (χ−1 ), the p-adic valuations of generalized Bernoulli numbers and Np (χ) in this setting. IMC FOR THE CARLITZ CYCLOTOMIC EXTENSION AND APPLICATIONS 34 References [1] G.W. 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Bruno Anglès: LMNO, Université de Caen BP 5186, 14032 Caen Cedex. France E-mail address: bruno.angles@unicaen.fr Andrea Bandini: Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze, 53/A - 43124 Parma (PR), Italy E-mail address: andrea.bandini@unipr.it Francesc Bars: Departament de Matemàtiques, Facultat de Ciencies, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia E-mail address: francesc@mat.uab.cat Ignazio Longhi: Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren Ai Road, Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou, Jiangsu, 215123, China E-mail address: Ignazio.Longhi@xjtlu.edu.cn