This is the submitted version of the article: Bandini, Andrea; Bars Cortina, Francesc; Coscelli, Edoardo. «Fitting ideals of class groups in Carlitz-Hayes cyclotomic extensions». Journal of Number Theory, (January 2019). DOI 10.1016/j.jnt.2018.12.010 This version is available at https://ddd.uab.cat/record/240659 under the terms of the license FITTING IDEALS OF CLASS GROUPS IN CARLITZ-HAYES CYCLOTOMIC EXTENSIONS ANDREA BANDINI, FRANCESC BARS, AND EDOARDO COSCELLI Dedicated to the memory of David Goss Abstract. We generalize some results of Greither and Popescu to a geometric Galois cover X → Y which appears naturally for example in extensions generated by pn -torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ-function. Keywords: Stickelberger series; L-functions; class groups; function fields; Carlitz-Hayes cyclotomic extensions. MSC 2010: 11R60; 11R23; 11M38; 11R58 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: on one side it deals with the variation of algebraic structures in p-adic towers (e.g. class groups) and, on the other side, it provides a good understanding of special values via interpolation with p-adic L-functions (whose con- struction is one of the major outcome of the theory). The algebraic counterpart of p-adic L-functions is usually represented by (generators of) characteristic ideals or Fitting ideals and the link between the algebraic and analytic side of the theory is the subject of various instances of Iwasawa Main Conjecture (IMC). Let F be a function field of transcendence degree 1 over a finite field F of cardinality q = pr , i.e. the function field of a smooth projective curve Y defined over F. For the particular case of F = F(t) and the cyclotomic Z∞ n p -extension Fp /F generated by the p -powers torsion of the Carlitz module (p any prime of F[t]), the IMC is proved in [1]: the authors use some results of Greither and Popescu on cohomological triviality of p-adic motives and on Fitting ideals (see [7] and [8]) to compute Fitting ideals of class groups of the finite subextensions of Fp , and then check all compatibility conditions needed to apply a limit process (similar results for the same type of Z∞ p -extensions but for more general function fields F can be found in the third author PhD thesis [5]). The main goal of this paper is to describe a more general setting: we consider Galois extensions (or Galois coverings between curves) with at least one totally ramified prime (or almost totally ramified, see Section 2.2) and provide a general formula for the Fitting ideal of (the Pontrjagin dual of) their class groups applying again the result of [7] and [8]. The main application we have in mind is the following: let F be a global function field as above and consider its extension generated by the pn - torsion of a sign normalized rank 1 Drinfeld module (or a Hayes module) Φ. The extension F (Φ[p])/F contains the Hilbert class field HA of F (A is the ring of integers of F with respect to a fixed prime at ∞) and its Galois group acts on all class groups of the extensions F (Φ[pn ]). Therefore we can consider F. Bars supported by MTM2016-75980-P and MDM-2014-0445. 1 FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 2 χ-parts of those class groups for any character χ of Gal(F (Φ[p])/F ) and the computation of their Fitting ideals should lead to special values (and sometimes trivial zeros) of a p-adic L-function or of Goss ζ-function for F . Unfortunately our computations strongly depend on the behavior of the character χ (i.e. on Ker(χ)) and in some cases it is only possible to compute Fitting ideals of duals of class groups: nevertheless the relations/formulas proved in Theorems 3.2 and 3.3 (and in their corollaries) provide the algebraic side of the IMC for this setting (see also [5]) and shed some light on the phenomenon of double zeroes for type 3 characters (see Section 3) which did not appear in the basic setting F = F(t) of [1]. 1.1. Brief summary. We start by considering the following general setting: let X → Y be a finite abelian Galois covering of smooth projective curves over a finite field F as above, with Galois group G and unramified outside a finite set of places S. Equivalently we can consider a finite abelian Galois extensions K/F (where K and F are the function fields F(X) and F(Y ) respectively) with Galois group G. We always assume that F is the field of constants of X, i.e. the covering X → Y is geometric. We generalize some of the results of Greither and Popescu in [8, Sections 2 and 3] (with ` = p) on the Fitting ideal of the Tate module of the Picard 1-motive of X to the case in which there is a totally ramified place v ∈ S (not necessarily F-rational). We use this to compute Fitting ideals of Zp -duals of Tate modules which are easily linked to Pontrjagin duals of class groups of F(X) (see Theorems 2.6 and 2.7). Then, in Section 2.2, we obtain similar results for a mixed cover, i.e. X → Y 0 → Y where X → Y 0 is of p-power degree and totally ramified at some prime and Y 0 → Y is of order prime to p. In Section 3 we specialize to the cyclotomic extension generated by the pn -powers torsion of a Hayes module, where Fn := F (Φ[pn ]) plays the role of F(X) (for some n > 2) and F1 := F (Φ[p]) plays the role of F(Y 0 ) and we assume that the class group of F = F(Y ) has order prime to p. The main outcome is summarized in Theorem 3.3 and Corollary 3.5 where we provide formulas for Fitting ideals for class groups and duals of class groups involving Stickelberger elements (which arise from the main theorems of Greither and Popescu) and some correction factors like 1−γ1 −1 (where γ is a topological generator of Gal(F/F) ) which emphasize the presence of trivial zeroes (sometimes of order 2). For example let W be a ring extension of Zp containing all values of characters defined over Gal(F1 /F ), let ΘFn /F,S,χ be the X details are in Section 2, here S := {p, ∞}), Stickelberger element for the extension Fn /F (more put dp := deg(p) and n(Gal(Fn /F1 )) := σ. If χ is trivial on Gal(F1 /HA ) (i.e. it σ∈Gal(Fn /F1 ) is of type 3 in the terminology of Definition 3.1) and on the Frobenius at p but it is not the trivial character χ0 , then ΘFn /F,S,χ   0 ∨ n(Gal(Fn /F1 )) (1) FittW [Gal(Fn /F1 )] (C` (Fn )(χ) ) = · . (1 − γ −1 )2 |γ=1 dp In the final Section 4 we point out some possible applications of our results to the study of special values of Goss ζ-function. 1.2. Basic notations. Let F be a global function field of positive characteristic p, i.e. a finite extension of a field of transcendence degree 1 over a finite field F := Fq , which we call the constant field of F . We fix a place ∞ of F and let A be the ring of integers at ∞, i.e. the elements of F which are regular outside ∞. We denote by HA the Hilbert class field of A, i.e. the maximal abelian extension of F which is unramified at every prime of A and totally split at ∞. Let h0 (F ) be the class number of F so that [HA : F ] = h0 (F ). We assume that p - h0 (F ), moreover we assume also that the degree of ∞ is deg(∞) := d∞ = 1 so that the constant field of HA is still F. FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 3 Acknowledgements. We (as probably most of the people actually working on function fields) are deeply indebted to the efforts made by David Goss throughout his career to put function fields on the (so to speak) mathematical map. He never missed an opportunity to encourage young mathematicians to explore this beautiful and still quite mysterious world and his legacy goes much beyond his remarkable work. The second author wishes to recall in particular the enthusiasm and energy that David was able to pass on. When he met him for the first time in Barcelona during 2010, he was totally fascinated by his energy, his positivity and his dedication to the area of function field arithmetic. Thank you David. 2. Fitting ideals: totally ramified and almost totally ramified cases We start by fixing a few notations for the main objects we will be interested in and by recalling the general setting already mentioned in the introduction. Let X → Y be a finite abelian Galois covering of smooth projective curves over a finite field F (of order q = pr ) with Galois group G and unramified outside a finite set of places S (i.e. in terms of function fields, G = Gal(F(X)/F(Y )) ). We assume that X → Y is geometric, i.e. F is the field of constants of X, and that S 6= ∅. Let S := X(S) be the set of closed points of the base change X := X ×F F (where F is a fixed algebraic closure of F) lying above points of S, and let Σ be a set of primes of F := F(Y ) disjoint from S. Let Tp (MS,Σ ) be the p-adic realization (or p-adic Tate module) associated with the Picard 1-motive MS,Σ : a detailed description of MS,Σ in terms of divisor classes quite useful for computations is provided in [7, Section 2]. For any set Σ there is a short exact sequence (see [8, Equation (2)]) 0 → Tp (τΣ (F)) → Tp (MS,Σ ) → Tp (MS ) → 0 , and, since we will only consider the p = char(F ) case, the toric part Tp (τΣ (F)) vanishes (see [7, Remark 2.7]). Hence Σ has no concrete influence on the module Tp (MS,Σ ) ' Tp (MS ) we are interested in and we can (and will) assume Σ = ∅. Definition 2.1. Let K := F(X) and define the Stickelberger series associated with K/F (or X → Y ) and S as Y ΘK/F,S (u) := (1 − Fr−1 q u deg(q) −1 ) ∈ Z[G][[u]], q6∈S where the product is taken over places q of F and Frq ∈ G denotes the Frobenius at q. One can actually define a kind of universal Stickelberger series in Z[GS ][[u]], where GS is the Galois group of the maximal abelian extension of F unramified outside S, and find back ΘK/F,S (u) as the natural projection of that series, (see [1, Section 3.1]). Remark 2.2. We recall that for a finite extension K/F the Stickelberger element actually verifies 1 ΘK/F,S (u) ∈ Z[G][u] ∩ Z[G][[u]], (1 − qu)|G| where Z is the integral closure of Z in an algebraic closure of Q (see, e.g. [11, Theorem 15.13]). For the convergence of the universal Stickelberger series for GS as an element in a Tate algebra see [1, Proposition 3.2 and Theorem 3.4]. In this setting the main result of [7] (i.e. [7, Theorem 4.3], see also [8, Lemma 2.3]) reads as follows FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 4 Theorem 2.3 (Greither-Popescu). The Tate module Tp (MS ) is cohomologically trivial over G and free of finite rank over Zp , hence projective over Zp [G]. Moreover the Fitting ideal of Tp (MS ) over Zp [G][[GF ]] is principal and generated by ΘK/F,S (γ −1 ), i.e. (2) FittZp [G][[GF ]] (Tp (MS )) = (ΘK/F,S (γ −1 )) (where γ is the arithmetic Frobenius in GF := Gal(F/F) ). 2.1. The totally ramified case. In this subsection we assume that the cover X → Y is totally ramified at some place v1 ∈ S. We recall the short exact sequence of Zp [G][[GF ]]-modules (3) 0 → Tp (Jac(X)(F)) → Tp (MS ) → L → 0, (see [7, after Definition 2.6]) where Jac(X)(F) is the set of the F-points of the Jacobian of X, Tp (Jac(X)(F)) is the usual Tate module at p and L is the kernel of the degree map Zp [X(S)] → Zp . Since our first goal is to compute the Fitting ideal of Tp (Jac(X)(F)) and Theorem 2.3 takes care of the central element in the sequence, we focus now on the Zp [G][[GF ]]- module L. For a place v ∈ Y denote by Gv the decomposition subgroup of v in G and by Iv its inertia subgroup. Put Hv := Zp [X(v)] (where X(v) denotes the set of points of X ×F F above v): it is a Zp [G][[GF ]]-module and we observe that Zp [X(S)] = ⊕v∈S Hv . Let Frv denote the Frobenius of v in G (if v is unramified, in the ramified case any lift of a Frobenius of v in Gv /Iv will do) and put ev (u) := 1 − Fr−1 v u dv ∈ Z[G][u] (where d is the degree of the place v). v In this setting [8, Lemmas 2.1 and 2.2] read as Lemma 2.4. Hv is a cyclic Zp [G][[GF ]]-module and we have: (i) if v ∈ Y is unramified in X, then FittZp [G][[GF ]] (Hv ) = (ev (γ −1 )) and Hv ' Zp [G][[GF ]]/(ev (γ −1 )) ; (ii) if v ∈ Y is ramified in X, then FittZp [G][[GF ]] (Hv ) = (ev (γ −1 ), τ − 1 : τ ∈ Iv ) and Hv ' Zp [G][[GF ]]/(ev (γ −1 ), τ − 1 : τ ∈ Iv ) ' Zp [G/Iv ][[GF ]]/(ev (γ −1 )) . Lemma 2.5. Let X → Y be a geometric Galois cover with a totally ramified prime v1 ∈ S. Then we have an isomorphism of Zp [G][[GF ]]-modules: ! M (4) L ' (γ − 1)Hv1 ⊕ Hv v∈S 0 where S0 := S − {v1 }. Proof. The primes in X(S) are points in X ×F F and have degree 1 so deg : Hv → Zp is surjective. Since v1 is totally ramified, the degree map provides a decomposition Hv1 = (γ −1)Hv1 +Zp 1Hv1 (where 1Hv1 ∈ Hv1 is such that deg : Zp 1Hv1 → Zp is surjective), moreover the previous lemma yields Hv1 ' Zp [[GF ]]/(1 − γ −dv1 ) as a Zp [G][[GF ]]-module. We have an injective morphism of Zp [G][[GF ]]-modules ! ! M M (γ − 1)Hv1 ⊕ Hv ,→ ((γ − 1)Hv1 + Zp 1Hv1 ) ⊕ Hv v∈S 0 v∈S 0 given by (α, β) 7→ (α − deg(β)1Hv1 , β) . Since the degree map on Hv1 sends 1Hv1 to a unit in Zp , the image of the morphism is inside L, and it is actually equal to L because of the above decomposition for Hv1 (one can also check directly that the Zp -ranks are the same using the identification with Zp [[GF ]]/(1−γ −dv1 ) ).  FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 5 Now we can generalize [8, Theorem 2.6]. Theorem 2.6. Assume X → Y is a finite geometric Galois cover of Galois group G and with a totally ramified point v1 ∈ S of degree dv1 . Then     ∗ −1 n(G)  Y n(Iv ) (5) FittZp [G][[GF ]] (Tp (Jac(X)(F)) ) = (ΘK/F,S (γ )) · 1,  · 1, , 1−γ −dv1 0 ev (γ −1 ) 1−γ −1 v∈S ∗ where denotes the Zp -dual, (a, b) is the fractional Zp [G][[G X F ]]-ideal generated by the elements a and b and, for any finite group N , we put n(N ) := σ ∈ Z[N ]. σ∈N ! M Proof. By Lemma 2.5, L ' (γ − 1)Hv1 ⊕ Hv as a Zp [G][[GF ]]-module. For any v ∈ S v∈S 0 we have the following short exact sequence (compare with [8] after Lemma 2.4) (6) 0 → Hv → Zp [G][[GF ]]/ev (γ −1 ) → Zp [G][[GF ]]/(ev (γ −1 ), n(Iv )) → 0 (where the map on the left sends 1Hv to n(Iv ) ). We now obtain a short exact sequence for the Zp [G][[GF ]]-module (γ − 1)Hv1 (which is not trivial whenever dv1 > 1). Write (γ − 1)Hv1 = (γ − 1)Zp 1Hv1 + · · · + (γ dv1 −1 − γ dv1 −2 )Zp 1Hv1 as a free Zp [G]-module of rank dv1 − 1, and put wi := (γ i − γ i−1 )1Hv1 . Fixing the basis w1 , . . . , wdv1 −1 and applying [7, Proposition 2.1] we compute 1 − udv1 det(Id − γu | (γ − 1)Hv1 ) = 1 + u + u2 + . . . + udv1 −1 = . 1−u Therefore one has a short exact sequence 1 − γ −dv1 1 − γ −dv1     (7) (γ − 1)Hv1 ,→ Zp [G][[GF ]]/  Zp [G][[G F ]]/ , n(G) . 1 − γ −1 1 − γ −1 Putting together equations (3), (6) and (7) we obtain the four term exact sequence 1 − γ −dv1 M   −1 Tp (Jac(X)(F)) ,→ Tp (MS ) → Zp [G][[GF ]]/ev (γ ) ⊕ Zp [G][[GF ]]/ 0 1 − γ −1 v∈S  1 − γ −dv1 M   −1 Zp [G][[GF ]]/(ev (γ ), n(Iv )) ⊕ Zp [G][[GF ]]/ , n(G) . 1 − γ −1 v∈S 0 Denote by X2 , X3 and X4 the second, third and fourth modules appearing in the sequence above. All modules are finitely generated and free over Zp , moreover X3 has projective dimension 0 or 1 over Zp [G][[GF ]], while X2 has projective dimension 1 over Zp [G][[GF ]] because it has no non-trivial finite submodules (enough by [10, Proposition 2.2 and Lemma 2.3]) and is G-cohomologically trivial by Theorem 2.3. With these properties [8, Lemma 2.4] yields FittZp [G][[GF ]] (Tp (Jac(X)(F))∗ ) FittZp [G][[GF ]] (X3 )=FittZp [G][[GF ]] (X2 ) FittZp [G][[GF ]] (X4 ). Since ! 1 − γ −dv1   Y FittZp [G][[GF ]] (X3 ) = · ev (γ −1 ) 1 − γ −1 v∈S 0 and 1 − γ −dv1   Y FittZp [G][[GF ]] (X4 ) = , n(G) · (ev (γ −1 ), n(Iv )) , 1 − γ −1 0 v∈S FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 6 Theorem 2.3 immediately implies equation (5).  To apply this to class groups just note that Tp (Jac(X)(F))∗ ' Hom(Jac(X)(F){p}, Qp /Zp ) where Jac(X)(F){p} denotes the divisible group given by the p-power torsion elements and ∗ indicates the Zp -dual. Consider the projection morphism π GF : Zp [G][[GF ]] → Zp [G] mapping γ to 1. For a finitely generated Zp [G][[GF ]]-module M , we have the equality π GF FittZp [G][[GF ]] (M ) = FittZp [G] (MGF ),  where MGF denotes the GF -coinvariants M/(γ − 1)M . Moreover the GF -coinvariants of Tp (Jac(X)(F))∗ correspond to (Tp (Jac(X)(F))∗ )GF ' Hom(Jac(X)(F)GF , Qp /Zp ) = Hom(Jac(X)(F), Qp /Zp ) = C` 0 (X)∨ , where the final module is the Pontrjagin dual of the p-part of the class group associated with X (see [1, Lemma 4.6 and Remark 4.7]). The following generalizes [8, Theorem 3.2]. Theorem 2.7. Assume X → Y is a geometric Galois cover with a totally ramified point v1 ∈ S. Then FittZp [G] (C` 0 (X)∨ ) = h gW · corG/I G W (ΘK W /F,S−W (1)) , W ⊂ S 0 i , where cor denotes the corestriction map, and for any T ⊂ Q S we write IT for the compositum |Iv | of all inertia groups Iv with v ∈ T , K := K T and gT := v∈T T I ∈ N. |IT | Proof. As explained in [8, p. 232], the Euler relations give us all the generators for the Fitting ideal (using Theorem 2.6), and we have for any subset T of S ! Y Y n(Iv ) ΘK/F,S (γ −1 ) = gT · ev (γ −1 ) · corG/I G T (ΘK T /F,S−T (γ −1 )) . v∈T v∈T Moreover, if v1 ∈/ T we obtain the relations that appear in Theorem 2.6 while, if v1 ∈ T , then n(G) Y n(Iv ) Θ (γ −1 ) = gT · corG/I G (ΘK T /F,S−T (γ −1 )) . −1 ev1 (γ ) ev (γ −1 ) K/F,S T v∈T −v1 Note that ev1 (γ −1 ) = 1 − γ −dv1 and consider the element ev1 (γ −1 ) edv1 := = 1 − γ −1 ∈ Zp [G][[GF ]] , 1−γ −dv1 1−γ −1 which is not a zerodivisor in Zp [G][[GF ]]. Multiplying the equalities above by edv1 we obtain equalities of ideals in Zp [G][[GF ]]. The projection map π GF maps edv1 to zero and, using Theorem 2.6, we obtain the claim (in particular no more generators are needed for the Fitting ideal over Zp [G] when v1 ∈ T ).  2.2. The almost totally ramified case. In this subsection, we assume that X → Y is a finite abelian geometric cover with Galois group G, e ramified at a finite set S and such that it factors through X → Y → Y (with X 6= Y ), where F(X)/F(Y 0 ) is a p-extension totally 0 0 ramified at some place v10 of Y 0 lying above a prime v1 of Y and F(Y 0 )/F(Y ) is a Galois extension of degree coprime with p (i.e. Gal(F(X)/F(Y 0 )) is the p-Sylow subgroup of G) e 1. 1The fact that p - [F(Y 0 ) : F(Y )] is essential. But if there is subextension in F(X)/F(Y 0 ) which has degree prime to p (and is totally ramified) one can move it to F(Y 0 )/F(Y ) by enlarging F(Y 0 ), so the fact that F(X)/F (Y 0 ) is a p-extension is not really restrictive. FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 7 We put G e ' Gal(F(X)/F(Y 0 )) × Gal(F(Y 0 )/F(Y )) := G × H and consider χ ∈ Hom(H, C∗ ) a character of H. All such characters have values in µ|H| so we fix a ring extension W of Zp containing µ|H| , and consider χ as a p-adic character with values in W ∗ . As usual we denote by eχ the idempotent associated with χ, i.e. 1 X eχ := χ(δ −1 )δ ∈ W [H] , |H| δ∈H and, for any Zp [G][[G e F ]]-module M we write M (χ) for the χ-part of M (i.e. the submodule eχ (M ⊗Zp W ) ), which is a W [G][[GF ]]-module inside M ⊗Zp W . The trivial character will be denoted by χ0 as usual. We identify G e with G × H and denote by πH (resp. πG ) the canonical projection W [G] e → W [H] (resp. W [G]e → W [G]). In this setting we can take χ-parts in the exact sequence (3) obtaining (8) 0 → Tp (Jac(X)(F))(χ) → Tp (MS )(χ) → L(χ) → 0 , where L(χ) is the (χ-part of the) kernel of degree map W [X(S)] → W (and we consider the trivial action on W ). Lemma 2.8. Let X → Y be a geometric abelian Galois cover as above. Then we have an isomorphism of W [G][[GF ]]-modules ! M L(χ0 ) ' (γ − 1)Hv1 (χ0 ) ⊕ Hv (χ0 ) v∈S 0 where S 0 = S − {v1 }, and M L(χ) = Hv (χ) f or any χ 6= χ0 . v∈S Proof. Consider the degree map deg : ⊕v∈S Hv → Zp . Taking χ-parts for χ 6= χ0 one immedi- ately has deg(χ) : ⊕v∈S Hv (χ) → 0, hence L(χ) = ⊕v∈S Hv (χ). Now we deal the case χ = χ0 : the hypothesis on the ramification of v1 yields e v ][[GF ]]/(ev (γ −1 )) = Zp [H/πH (Iv )][[GF ]]/(ev (γ −1 )) Hv1 = Zp [G/I 1 1 1 1 and, since χ0 is trivial on πH (Iv1 ) and on Frv1 , we have Hv1 (χ0 ) ' W [[GF ]]/(1 − γ −dv1 ) as a W [G][[GF ]]-module. We have an injective morphism of W [G][[GF ]]-modules ! ! M M (γ − 1)Hv1 (χ0 ) ⊕ Hv (χ0 ) ,→ ((γ − 1)Hv1 (χ0 ) + W 1Hv1 (χ0 ) ) ⊕ Hv (χ0 ) v∈S 0 v∈S 0 defined exactly as in Lemma 2.5 and the proof follows the same path.  Lemma 2.9. Let X → Y be an abelian geometric cover as above with v1 ∈ S a ramified prime which is totally ramified in X → Y 0 . Using the decomposition G e = G × H, for any prime p write Frp = (Frp,G , Frp,H ) ∈ G × H. If χ(πH (Iv )) 6= 1, then Hv (χ) = 0, otherwise Hv (χ) = W [G][[GF ]]/(1 − χ(Fr−1 −1 −dv v,H ) Frv,G γ , πG (τ ) − 1 : τ ∈ Iv ). Moreover, we have an exact sequence Hv (χ) ,→ W [G][[GF ]]/(1−χ(Fr−1 −1 −dv v,H ) Frv,G γ )  W [G][[GF ]]/(1−χ(Fr−1 −1 −dv v,H ) Frv,G γ , n(πG (Iv ))). FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 8 Proof. Recall that −dv e v ][[GF ]]/(1 − Frv γ −dv ). Hv = Zp [G][[G F ]]/(1 − Frv γ , τ − 1 : τ ∈ Iv ) = Zp [G/I e If χ(πH (Iv )) 6= 1, then we immediately have Hv (χ) = 0. Now assume χ(πH (Iv )) = 1. Then eχ (τ − 1) = (πG (τ ) − 1)eχ ∈ W [G] for any τ ∈ Iv , where πG (τ ) − 1 ∈ Z[πG (Iv )]. Thus eχ ((1 − Frv γ −dv )) = (1 − χ(Frv,H ) Frv,G γ −dv )eχ ∈ W [G][GF ]. Therefore, taking χ-parts in the exact sequence 0 → (1 − Frv γ −dv , τ − 1 : τ ∈ Iv ) → Zp [G][[G e F ]] → Hv → 0, we have that Hv (χ) = W [G][[GF ]]/(1 − χ(Fr−1 −1 −dv v,H ) Frv,G γ , πG (τ ) − 1 : τ ∈ Iv ). The exact sequence follows as in (6) using the remarks after [8, Lemma 2.4]: indeed we have −1 −1 −dv a map Hv (χ) → W [G][GF ]/(1 − χ(F rv,H )Fv,G γ ), sending 1Hv to n(πG (Iv )), and all is clear in the sequence except the injectivity of this map. To work with W -modules we observe −1 −1 −dv that 1 − χ(F rv,H )Fv,G γ is a polynomial of degree dv in γ −1 with leading term a unit in −1 −1 −dv W [G], therefore W [G][GF ]/(1 − χ(F rv,H )Fv,G γ ) is W -free of rank dv |G|. Similarly Hv (χ) is W -free of rank dv |G/πG (Iv )| and exactness follows.  Remark 2.10. Note that whenever v is only ramified in Y 0 → Y , i.e. πG (Iv ) = 0, we have n(πG (Iv )) = 1 and the exact sequence mentioned in the previous lemma reduces to the isomorphism Hv (χ) ' W [G][[GF ]]/(1 − χ(Fr−1 −1 −dv v,H ) Frv,G γ ). The decomposition of modules in χ-parts obviously reflects on their Fitting ideals as well, hence we define Definition 2.11. For any χ ∈ Hom(H, W ∗ ), the χ-Stickelberger series for K/F is ΘK/F,S,χ (u) := χ(ΘK/F,S )(u) ∈ W [G][[u]] . The usual fundamental relations for idempotents yield (9) eχ ΘK/F,S (u) = ΘK/F,S,χ (u)eχ and X ΘK/F,S (u) = ΘK/F,S,χ (u)eχ . χ Following the proof of Theorem 2.6 and using Lemma 2.9 we obtain Theorem 2.12. Assume X → Y is an abelian geometric cover as above. If χ 6= χ0 , then ΘK/F,S,χ (γ −1 ) FittW [G][[GF ]] (Tp (Jac(X)(F))(χ)∗ ) = Y (1 − χ(Frv,H )−1 Fr−1 v,G γ −dv ) (1) (10) v∈Sχ ! Y n(πG (Iv )) · 1, , (1) 1 − χ(Frv,H )−1 Fr−1 v,G γ −dv v∈Sχ −Sχ (1) where ∗ now denotes the W -dual, Sχ := {v ∈ S : χ(πH (Iv )) = 1} and Sχ := {v ∈ Sχ : πG (Iv ) = 0}. FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 9 If χ = χ0 , then (11) ΘK/F,S,χ0 (γ −1 ) FittW [G][[GF ]] (Tp (Jac(X)(F))(χ0 )∗ ) = Y (1 − Fr−1 v,G γ −dv ) (1) v∈Sχ0  v6=v1  ! n(G)  Y n(πG (Iv )) · 1, · 1, . (1 − Fr−1 −dv −dv ) v,G γ 1−γ 1 (1) 1−γ −1 v∈S−Sχ0 v6=v1 Proof. Assume first χ 6= χ0 . From Lemmas 2.8 and 2.9 (and recalling Remark 2.10) we have the four term exact sequence M Tp (Jac(X)(F))(χ) ,→ Tp (MS )(χ) → W [G][[GF ]]/(1 − χ(Fr−1 −1 −dv v,H ) Frv,G γ ) v∈Sχ  M W [G][[GF ]]/((1 − χ(Fr−1 −1 −dv v,H ) Frv,G γ ), n(πG (Iv ))). (1) v∈Sχ −Sχ Denote by X2 , X3 and X4 the second, third and fourth modules appearing in the sequence above. All modules are finitely generated and free over W , moreover X3 has projective dimension 0 or 1 over W [G][[GF ]], while X2 has projective dimension 1 over W [G][[GF ]] because it has no non-trivial finite submodules and is G-cohomologically trivial by (an easy application of) Theorem 2.3. Then [8, Lemma 2.4] yields FittW [G][[GF ]] (Tp (Jac(X)(F))(χ)∗ ) FittW [G][[GF ]] (X3 )=FittW [G][[GF ]] (X2 ) FittW [G][[GF ]] (X4 ). Since   Y FittW [G][[GF ]] (X3 ) =  (1 − χ(Fr−1 −1 −dv  v,H ) Frv,G γ ) v∈Sχ and Y FittZp [G][[GF ]] (X4 ) = (1 − χ(Fr−1 −1 −dv v,H ) Frv,G γ , n(πG (Iv ))) , (1) v∈Sχ −Sχ Theorem 2.3 immediately implies equation (10). Assume now χ = χ0 : obviously Sχ0 = S. Since Hv1 (χ0 ) = W [[GF ]]/(1 − γ −dv1 ), with the same argument of the proof of Theorem 2.6, we find an analog of the exact sequence (7) that now reads as 1 − γ −dv1 1 − γ −dv1     (γ − 1)Hv1 (χ0 ) ,→ W [G][[GF ]]/  W [G][[GF ]]/ , n(G) . 1 − γ −1 1 − γ −1 As above we obtain the following four term exact sequence 1 − γ −dv1 M W [G][[GF ]]   Tp (Jac(X)(F))(χ0 ) ,→ Tp (MS )(χ0 ) → W [G][[GF ]]/ 1 − γ −1 v∈S (1 − Fr−1 v,G γ −dv ) v6=v1  1 − γ −dv1   M W [G][[GF ]]/ , n(G) W [G][[GF ]]/(1 − Fr−1 v,G γ −dv , n(πG (Iv ))) 1 − γ −1 (1) v∈S−Sχ0 v6=v1 FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 10 (because of χ0 we could actually use Zp in place of W here, we keep W for coherence with the other formulas). Denote by Y3 and Y4 the third and fourth modules appearing in this last sequence. Another application of [8, Lemma 2.4], the facts that   1 − γ −dv1   Y −1 −dv   FittW [G][[GF ]] (Y3 ) = ·  (1 − Frv,G γ ) 1 − γ −1   v∈S v6=v1 and 1 − γ −dv1   Y FittZp [G][[GF ]] (Y4 ) = , n(G) · (1 − Fr−1 v,G γ −dv , n(πG (Iv ))) , 1 − γ −1 (1) v∈S−Sχ0 v6=v1 and Theorem 2.3 immediately imply equation (11).  Remark 2.13. The first factor appearing in the Fitting ideals of Theorem 2.12 hints at the possibility of finding extra zeroes for the Stickelberger element when specializing γ to 1. As predictable the number of such zeroes depends on the behavior of ramified primes and, in (1) particular, on the cardinality of the set Sχ . We shall see an explicit example in the next section. Similarly we can obtain an analog of Theorem 2.7 but we can safely leave this to the reader (there is nothing new in that computation but formulas get a bit involved), and move to the arithmetic applications for the Carlitz-Hayes cyclotomic extensions. 3. Fitting ideals in Carlitz-Hayes cyclotomic extension We consider a well-known abelian cover arising from class field theory. Let Φ be a sign normalized rank 1 Drinfeld module, i.e. a ring homomorphism Φ : A → HA {τ }, where HA {τ } is the ring of skew-polynomials in the variable τ with coefficients in HA (for more details see [6, Chapter 7]). For any ideal a of A, denote by Φ[a] the a-torsion of Φ: we recall that HA {τ } is right-euclidean so, if we let Φa be the unique monic generator of the ideal generated by (Φ(a) : a ∈ a), we have Φ[a] := {x ∈ F sep : Φa (x) = 0} , where F sep is a separable closure of F and we interpret Φa as a polynomial via τ (x) := x#F . It is well-known that F (a) := HA (Φ[a]) is an abelian extension of F where the only ramified primes are the ones dividing a and ∞. The inertia group of ∞ coincides with its decomposition group and is isomorphic to F∗ (recall that we assume deg(∞) = d∞ = 1). Moreover, when a = pn is a power of a prime p, F (a)/HA is totally ramified at the place p (see [6, Chapter 7]). The infinite Galois extension Fp := ∪n F (pn ) has properties similar to the ones of ∪n k(µpn ) for a number field k, hence it will be called the p-cyclotomic extension of F . To apply the results of the previous section we consider F0 := F = F(Y ) F (p) := F1 = F(Y 0 ) Fn := F (pn ) = F(Xn ) n > 1, we use Xn in place of X (and later on Gn , I∞,n , Ip,n and so on) to keep track of the layer of the cyclotomic extension we are dealing with. To fit the previous hypotheses, we assume from now on that h0 (F ) = [HA : F ] is prime with p so that [F1 : F ] = (q dp − 1)h0 (F ) is prime with p as well. We put W as a ring extension of Zp containing µ|Gal(F1 /F )| and recall that Gn is a p-group. The rest of the setting of Section 2 translates into Gal(Fn /F ) = G e n ' Gal(Fn /F1 ) × Gal(F1 /F ) =: Gn × H. I The set of ramified primes is S = {p, ∞} with I∞ := I∞,n ' F∗ ,→ H and Fnp,n = HA for any n. FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 11 By Lemma 2.4 we have that Hn,∞ ' Zp [Gn × H/F∗ ][[GF ]]/(1 − γ −1 ) (note that we can choose Fr∞ = 1 because the decomposition and inertia groups of ∞ coincide) and Hn,p ' Zp [Gal(HA /F )][[GF ]]/(1 − Frp γ −dp ). Definition 3.1. Let χ be a character of H, i.e. χ ∈ Hom(H, W ∗ ). We distinguish three types of characters • χ is said to be of type 1 if χ(I∞ ) 6= 1; • χ is said to be of type 2 if χ(I∞ ) = 1 and χ(Gal(F1 /HA )) 6= 1; • χ is said to be of type 3 if χ(Gal(F1 /HA )) = 1. Among the characters of type 3, there is the trivial one which will be denoted as usual by χ0 . From Lemma 2.9 we immediately obtain  0 if χ is of type 1 or 2 Hn,p (χ) ' W [[GF ]]/(1 − χ(Fr−1 p )γ −dp ) otherwise and  0 if χ is of type 1 Hn,∞ (χ) ' W [Gn ] otherwise. Characters of type 1 and 2 have a behavior similar to nontrivial characters for the basic case of the rational function field F(t) (detailed in [1]): indeed for those characters we can extend the results of the previous section to include the computation of the Fitting ideals of class groups (not only of its dual). We denote by Tp (Fn ) the Tate module of the F-rational points of the Jacobian of Xn (which here plays the role of Tp (Jac(X)(F)) of the previous section). Theorem 3.2. Let χ be a character of type 1 or 2. We have −1    ΘFn /F,S,χ (γ ) ! if χ is of type 1  FittW [Gn ][[GF ]] (Tp (Fn )(χ)) = ΘFn /F,S,χ (γ −1 ) if χ is of type 2. 1 − γ −1   Proof. The exact sequence (8) here reads as (12) 0 → Tp (Fn )(χ) → Tp (Mn,Sn )(χ) → Ln (χ) → 0, (where Sn is the set of primes of Fn lying above primes in S) and Theorem 2.3 yields (taking χ-parts) F ittW [Gn ][[GF ]] (Tp (Mn,Sn )(χ)) = (ΘFn /F,S,χ (γ −1 )). By Lemma 2.8 and the computations above we have  0 if χ is of type 1 Ln (χ) = W [Gn ] if χ is of type 2. Now the statement for type 1 characters is obvious. For characters of type 2 apply [4, Lemma 3] to the sequence (12) to obtain (1 − γ −1 ) FittW [Gn ][[GF ]] (Tp (Fn )(χ)) = FittW [Gn ][[GF ]] (Tp (Mn,Sn )(χ)) and conclude the proof.  For all characters, as a consequence of Theorem 2.12, we have FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 12 Theorem 3.3. With notations and hypotheses as above, FittW [Gn ][[GF ]] (Tp (Fn )(χ)∗ ) is equal to: ΘFn /F,S,χ (γ −1 ) !  χ of type 1 ΘFn /F,S,χ (γ −1 ) χ of type 2 1 − γ −1 ΘFn /F,S,χ (γ −1 )   n(Gn ) · 1, χ of type 3, χ(Frp,H ) 6= 1 (1 − γ −1 ) −1 −dp  1 − χ(Frp,H  ) γ . ΘFn /F,S,χ (γ −1 ) n(Gn ) · 1, χ 6= χ0 of type 3, χ(Frp,H ) = 1 (1 − γ −1 )  1 − γ −dp ΘFn /F,S,χ (γ −1 ) n(Gn )  −1 · 1, −dp χ = χ0 1−γ 1−γ −1 1−γ Proof. Just specialize the formulas of Theorem 2.12 to the various cases. Note that S = {p, ∞} so the Sχ = {v ∈ S : χ(πH (Iv )) = 1} of that theorem is empty if χ is of type 1, it contains only ∞ if χ is of type 2 and is equal to S if χ is of type 3. Moreover since I∞ ,→ H we have (1) Sχ = {∞} whenever χ is not of type 1. Finally p plays the role of v1 , we already mentioned that Fr∞ = 1 and, since p is totally ramified in Fn /F1 , we have Frp,Gn = 1 as well.  Remark 3.4. For characters of type 1 or 2 the Fitting ideals of the Tate module and of its W ∗ dual coincide (those are the only cases in which we can compute both). 3.1. Fitting ideals of class groups. Let Cn := C` 0 (Fn ){p} be the p-torsion of the class groups of degree zero divisors of Fn , which is a Zp [Gn × H]-module. It is well-known that Cn can be obtained as the GF -coinvariants of the Tate module Tp (Fn ) (see, for example, [1, Lemma 4.6] or [5, Lemma 2.4.1]), therefore thanks to the properties of Fitting ideals we can compute the Fitting ideals of class groups (or of their duals) simply by specializing the previous formulas with γ 7→ 1. For the dual of the class groups as noted in [8, Section3] (see also [1, Remark 4.7]), we know that Cn∨ := Hom(Cn , Qp /Zp ) ' Tp (Fn )∗ /(1 − γ −1 )Tp (Fn )∗ therefore we have W [G ][[GF ]] FittW [Gn ] (Cn (χ)∨ ) = πW [Gnn] (FittW [Gn ][[GF ]] (Tp (Fn )(χ)∗ )). We do not write down the class group case because we only have formulas for the characters of type 1 and 2 and they coincide with the ones for the dual. Corollary 3.5. With notations and hypotheses as above, FittW [Gn ] (Cn (χ)∨ ) is equal to  ΘFn /F,S,χ (1) ! χ of type 1 ΘFn /F,S,χ (γ −1 ) χ of type 2 1 − γ −1 |γ=1 ΘFn /F,S,χ (γ −1 )   n(Gn ) · 1, χ of type 3, χ(Frp,H ) 6= 1 (1 − γ −1 ) |γ=1 1 − χ(Frp,H )−1 −1 ΘFn /F,S,χ (γ )   n(Gn ) · χ 6= χ0 of type 3, χ(Frp,H ) = 1 (1 − γ −1 )2 |γ=1 dp −1 ΘFn /F,S,χ (γ )   n(Gn ) · 1, χ = χ0 1 − γ −1 |γ=1 dp FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 13 Proof. Just specialize the formulas of Theorem 3.3. Note the particular case of nontrivial characters of type 3 with χ(Frp,H ) = 1: Theorem 3.3 yields generators (for the fractional ideal) ΘFn /F,S,χ (γ −1 ) ΘFn /F,S,χ (γ −1 ) n(Gn ) and · . (1 − γ −1 ) |γ=1 (1 − γ −1 )2 |γ=1 dp Since we already know Fitting ideals are well defined, the first one has to be 0, i.e. the Stickelberger element has a zero of order (at least) 2 in 1 in this setting.  3.2. Limits of Fitting ideals. For characters of type 1 and 2 (two of the three cases in which we have a principal Fitting ideal) it is possible to consider the inverse limit of the Cn with respect to the maps induced by the natural norm maps NFm /Fn : Cm → Cn (for any m > n) and compute the Fitting ideal of that limit as the inverse limit of the Fitting ideals of Corollary 3.5 (without taking duals of course). This provides an element in the Iwasawa algebra Λ := W [[Gal(Fp /F1 )]] which can be used as the algebraic counterpart of a p-adic L-function in the Iwasawa Main Conjecture for this setting. Details for the case of the Carlitz module (i.e. F = F(t) ) are in [1], the generalization to the case presented here can be found in the third author PhD thesis [5]. Here we would like to deal briefly with limits for Fitting ideals computed for characters of type 3: a motivation comes from the usual application to Iwasawa theory, unfortunately, since we can only work with duals, it is not immediately clear what kind of arithmetic meaning can be associated with these limits (i.e. which is the Iwasawa module they are related to). Directly from the definition of Stickelberger series we have πnn+m (ΘFn+m /F,S,χ (u)) = ΘFn /F,S,χ (u). So they are compatible with respect to the projection maps defining the Iwasawa algebra Λ as an inverse limit of the group rings W [Gn ]. Thus we can define (13) ΘFp /F,S,χ (u) := lim ←− ΘFn /F,S,χ (u) ∈ Λ[[u]], n and its specialization (14) ΘFp /F,S,χ (γ −1 ) := lim ←− ΘFn /F,S,χ (γ −1 ) ∈ Λ[[GF ]]. n Regarding the other element appearing as a generator of the Fitting ideals we have πnn+m (n(Gn+m )) = [Fn+m : Fn ]n(Gn ). Hence for any m > 0 πnn+m (FittW [Gn+m ] (Cn+m (χ)∨ )) ⊆ FittW [Gn ] (Cn (χ)∨ ) for any χ of type 3 (note that we actually have equality for characters of type 1 and 2) and this is enough to have a compatible inverse system of ideals and to define their inverse limit inside Λ[[GF ]] (a similar procedure has been used in [2] and [3] for characteristic ideals). Definition 3.6. With the above notations we define the pro-Fitting ideal of the dual of the χ-part of the class groups as g Λ (C ∨ (χ)) := lim FittW [G ] (Cn (χ)∨ ). Fitt ←− n n Remark 3.7. For characters of type 1 and 2 we have g Λ (C ∨ (χ)) = lim FittW [G ] (Cn (χ)) Fitt ←− n n as well. Moreover, thanks to the presence of a totally ramified prime, one can also show that lim ←− FittW [Gn ] (Cn (χ)) = FittΛ (lim ←− Cn (χ)) n n FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 14 (see [1, Sections 4.4 and 5] or [5, Sections 2.3 and 2.4]). As mentioned above, this provides a link between the Fitting ideal of the inverse limit of the class groups of subextensions of Fp (i.e. the class group of Fp ) and a Stickelberger element. Such formula is one of the incarnations of Iwasawa Main Conjecture for the function field setting, its relation with more analytic objects (like Goss ζ-function) will be briefly explained in the next section. Corollary 3.8. For characters of type 3 one has  !  ΘFp /F,S,χ (γ −1 ) if χ(Frp,H ) 6= 1   (1 − γ −1 ) |γ=1     g Λ (C ∨ (χ)) = Fitt (0) ! if χ 6= χ0 , χ(Frp,H ) = 1 .    −1 ΘFp /F,S,χ (γ ) if χ = χ0   1 − γ −1   |γ=1 Proof. Just use limits on the ideals of Corollary 3.5 and note that lim ←− n(Gn ) = 0.  n Remark 3.9. Via the maps λn : W [Gn ]∨ → W [Gn ] X λn (ϕ) := ϕ(g) g∈Gn one can obtain a self duality for Iwasawa rings (and for their limit, i.e. for the Iwasawa algebra). Moreover such maps provide commutative diagrams with projections and core- strictions (see, e.g., [9, Appendix A]). It might be interesting to study direct limits as well for our Fitting ideals with respect to corestriction maps. We can immediately remark that n (n(G )) = n(G corn+1 n n+1 ), the corestriction map on Stickelberger elements is natural and al- ready appeared in Theorem 2.7 and, between duals of class groups, the natural maps induced by norms go in the direction of a direct limit (i.e. (Nnn+1 )∨ : Cn (χ)∨ → Cn+1 (χ)∨ ). One has to check compatibility of all maps involved and figure out the behavior of Fitting ideals with respect to such a limit, i.e. does computation of Fitting ideals commute with the direct limit in this setting (as it does with inverse limit as mentioned in Remark 3.7) ? The main reason for not dealing with these issues here is that it is not clear what kind of arithmetic in- formation (if any) one can obtain from such a procedure, but the appearance of Stickelberger elements (with their several relations with L-functions in general) could be a motivation for investigating these direct limits in some future work. 4. Interpolation of Goss ζ-function via Stckelberger elements In this final section we would like to briefly explain how Stickelberger element (and Stick- elberger series in general) can be used to interpolate the key analytic object for the function field setting, i.e. the Goss ζ-function whose central role in the theory is just one among the many key contributions David Goss provided to the subject. Here we still deal with the cyclotomic extension Fp := ∪n>1 Fn . We start by recalling the definition of the Goss ζ-function in our (slightly simplified) setting where we assume that p does not divide the order of the class group of F (details for a general function field are provided in [6, Chapter 8]). Let F∞ denote the completion of F at the prime at infinity and let C∞ be the completion of a fixed algebraic closure of F∞ . We fix a sign function ∗ → F∗ and a positive uniformizer π sgn : F∞ ∞ ∈ F∞ , i.e. with sgn(π∞ ) = 1. The 1-units of F∞ will be denoted by U∞ 1 . Since the sign function has image in F∗ (which has order prime with p), our hypothesis on the class group implies that the group IF /P + (integral ideals of F modulo principal ideals generated by a positive element) has order prime with p as well. Let FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 15 h := |IF /P + |, then, for any ideal a of A, one has ah = (α) for some positive α ∈ A. Denote by hαi∞ ∈ U∞ 1 the 1-unit associated to α and define hai := hαi1/h 1 ∞ ∈ U∞ as the unique 1-unit whose h-power is hαi∞ (it is still in U∞ 1 because of our hypothesis on h and Hensel’s Lemma). We drop the index ∞ because in this section we only deal with interpolation at the prime at infinity, for interpolation at different primes see [1, Section 3.3] and [5, Section 1.8]. Note that the hypothesis d∞ = 1 makes this definition independent from the choice of the uniformizer π∞ (and, obviously, also from the choice of one of its d∞ -th roots). Definition 4.1. For s = (x, y) ∈ S∞ := C∗∞ × Zp define the exponential of an ideal a by as := xdeg a haiy . The Goss ζ-function for F is X ζA (s) := a−s , a6=0 where the sum runs through the set of all the non-zero integral ideals of A. One of the more relevant properties of Goss ζ-function (which trivially converges for |x|∞ > 1) is that it can be extended analytically to the whole space S∞ by using some inequalities arising from the Riemann-Roch Theorem (see [6, Section 8.9]). Let GS := Gal(FS /F ) be the Galois group of the maximal abelian extension of F unramified outside S, then Gal(Fp /F ) is a quotient of GS . Let WS be the subgroup of GS generated by all the Frobenius Frv (v 6∈ S) and let MS be the fixed field of the topological closure of WS . Since all primes not in S split completely in MS , Chebotarev density theorem yields MS = F and GS = WS . Therefore to define a continuous character on GS (and, a fortiori, on Gal(Fp /F ) ) it suffices to describe its values on Frv for all v 6∈ S. For any y ∈ Zp , consider the C∗∞ -valued character Ψy : Gal(Fp /F ) → C∗∞ Ψy (Frq ) := hqi−y and extend it in a natural way to a map Λ[[u]] → C∞ [[u]] (still denoted by Ψy by a little abuse of notations). Theorem 4.2. For any s = (x, y) ∈ S∞ , Ψy (ΘFp /F,S )(x) = (1 − ps )ζA (−s) . Proof. Directly from Definition 4.1 one can see that there exists an Euler product formula for ζA which reads as Y Y −1 (15) ζA (s) = (1 − q−s )−1 = 1 − hqi−y x− deg(q) q6=∞ q6=∞ (because −s = (x−1 , −y) ). Now from Definition 2.1 and equation (13) we have Y −1 Ψy (ΘFp /F,S )(u) = 1 − Ψy (Fr−1 q )udeg(q) q6∈S (16) Y −1 = 1 − hqiy udeg(q) . q6∈S FITTING IDEALS OF CARLITZ-HAYES CYCLOTOMIC TYPE EXTENSIONS 16 Specializing u 7→ x (and comparing with (15)) we obtain Y −1 Ψy (ΘFp /F,S )(x) = 1 − hqiy xdeg(q) ∈S q6Y = (1 − qs )−1 q6∈S (1 − qs )−1 = (1 − ps )ζA (−s). Y = (1 − ps )  q6=∞ Remark 4.3. The character Ψy can actually be interpreted as a character on ideles via the reciprocity map (composed with some projection map) recFp : IF /F ∗  Gal(Fp /F ). This interpretation has been introduced in [1] for F(t) (and generalized in [5]) and exploited not only for characters with values in C∞ but also for those with values in Cq for any prime q, leading to similar interpolation formulas for q-adic L-functions. References [1] B. Anglés, A. Bandini, F. Bars and I. Longhi, Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications, arXiv:1412.5957 [math.NT] (2015), submitted. [2] A. Bandini, F. Bars and I. Longhi, Characteristic ideals and Iwasawa theory, New York J. Math. 20 (2014), 759–778. [3] A. Bandini, F. Bars and I. Longhi, Characteristic ideals and Selmer groups, J. Number Theory 157 (2015), 530–546. [4] P. Cornachhia and C. Greither, Fitting ideals of class group of real fields with prime power conductor, J. Number Theory 73 (1998), 459–471. [5] E. Coscelli, Stickelberger series and Iwasawa Main Conjecture for function fields, PhD thesis, University of Milan (2018). [6] D. Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik 35, Springer-Verlag, Berlin, 1996. [7] C. Greither and C.D. Popescu, The Galois module structure of `-adic realizations of Picard 1-motives and applications, Int. Math. Res. Not. (2012), no. 5, 986–1036. [8] C. Greither and C.D. Popescu, Fitting ideals of `-adic realizations of Picard 1-motives and class groups of global function fields, J. Reine Angew. Math. 675 (2013), 223–247. [9] K. Kidwell, On the structure of Selmer groups of p-ordinary modular forms over Zp -extensions, J. Number Theory 187 (2018), 296–331. [10] C.D. Popescu, On the Coates-Sinnott conjecture, Math. Nachr. 282 (2009), no. 10, 1370–1390. [11] M. Rosen, Number theory in function fields, GTM 210, Springer-Verlag, New York, 2002. Andrea Bandini: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy E-mail address: andrea.bandini@unipi.it Francesc Bars: Departament de Matemàtiques, Facultat de Ciencies, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia E-mail address: francesc@mat.uab.cat Edoardo Coscelli: Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Via Cesare Saldini, 50, 20133 Milano (MI), Italy E-mail address: edoardo.coscelli@unimi.it