ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDS 1 ...

ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDSANDREA BANDINI, FRANCESC BARS , IGNAZIO LONGHIAbstract. We consider Z N p-extensions F of a global function field F and study various aspectsof Iwasawa theory with emphasis on the two main themes already (and still) developedin the number fields case as well. When dealing with the Selmer group of an abelian varietyA defined over F , we provide all the ingredients to formulate an Iwasawa Main Conjecturerelating the Fitting ideal and the p-adic L-function associated to A and F. We do the same,with characteristic ideals and p-adic L-functions, in the case of class groups (using knownresults on characteristic ideals and Stickelberger elements for Z d p-extensions). The final sectionprovides more details for the cyclotomic Z N p-extension arising from the torsion of theCarlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers bya Coates-Wiles homomorphism.2010 MSB: Primary: 11R23. Secondary: 11R58, 11G05, 11G10, 14G10, 11R60.Keywords: Iwasawa Main Conjecture, global function fields, L-functions, Selmer groups, class groups,Bernoulli-Carlitz numbers1. IntroductionThe main theme of number theory (and, in particular, of arithmetic geometry) is probablythe study of representations of the Galois group Gal(Q/Q) - or, more generally, of the absoluteGalois group G F := Gal(F sep /F ) of some global field F . A basic philosophy (basically, partof the yoga of motives) is that any object of arithmetic interest is associated with a p-adicrealization, which is a p-adic representation ρ of G F with precise concrete properties (andto any p-adic representation with such properties should correspond an arithmetic object).Moreover from this p-adic representation one defines the L-function associated to the arithmeticobject. Notice that the image of ρ is isomorphic to a compact subgroup of GL n (Z p )for some n, hence it is a p-adic Lie group and the representation factors through Gal(F ′ /F ),where F ′ contains subextensions F and F ′ such that F/F ′ is a pro-p extension and F ′ /F andF ′ /F are finite.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 toolscurrently available for the knowledge (and interpretation) of zeta values associated to anarithmetic object when F is a number field [26]. This theory constructs some sort of elements,called p-adic L-functions, which provide a good understanding of both the zeta values andthe arithmetic properties of the arithmetic object. In particular, the various forms of IwasawaMain Conjecture provide a link between the zeta side and the arithmetic side.The prototype is given by the study of class groups in the cyclotomic extensions Q(ζ p n)/Q.In this case the arithmetic side corresponds to a torsion Λ-module X, where Λ is an Iwasawaalgebra related to Gal(Q(ζ p ∞)/Q) and X measures the p-part of a limit of cl(Q(ζ p n)). Asfor the zeta side, it is represented by a p-adic version of the Riemann zeta function, that is,an element ξ ∈ Λ interpolating the zeta values. One finds that ξ generates the characteristicideal of X.SUPPORTED BY MTM2009-10359SUPPORTED BY NSC 099-2811-M-002-0961

IWASAWA THEORY OVER FUNCTION FIELDS 3generators) can be considered as a worthy candidate for an algebraic L-function in this setting.In section 4 we deal with the analytic counterpart, giving a brief description of the p-adic L-functions which have been defined (by various authors) for abelian varieties and the extensionsF/F . Sections 3 and 4 should provide the ingredients for the formulation of an Iwasawa MainConjecture in this setting. In section 5 we move to the problem of class groups. We usesome techniques of an (almost) unpublished work of Kueh, Lai and Tan to show that thecharacteristic ideals of the class groups of Z d p-subextensions of a cyclotomic Z N p -extension aregenerated by some Stickelberger element. Such a result can be extended to the whole Z N p -extension via a limit process because, at least under a certain assumption, the characteristicideals behave well with respect to the inverse limit (as Stickelberger elements do). Thisprovides a new approach to the Iwasawa Main Conjecture for class groups. At the end ofsection 5 we briefly recall some results on what is known about class groups and characteristicp zeta values. Section 6 is perhaps the closest to the spirit of function field arithmetic. Forsimplicity we deal only with the Carlitz module. We study the Galois module of cyclotomicunits by means of Coleman power series and show how it fits in an Iwasawa Main Conjecture.Finally we compute the image of cyclotomic units by Coates-Wiles homomorphisms: onegets special values of the Carlitz-Goss zeta function, a result which might provide some hintstowards its interpolation.1.2. Some notations. Given a field L, L will denote an algebraic closure and L sep a separableclosure; we shall use the shortening G L := Gal(L sep /L). When L is (an algebraic extensionof) a global field, L v will be its completion at the place v, O v the ring of integers of L v andF v the residue field. We are going to deal only with global fields of positive characteristic: soF L shall denote the constant field of L.As mentioned before, let F be a global field of characteristic p > 0, with field of constantsF F of cardinality q. We also fix algebraic closures F and F v for any place v of F , togetherwith embeddings F ↩→ F v , so to get restriction maps G Fv ↩→ G F . All algebraic extensions ofF (resp. F v ) will be assumed to be contained in F (resp. F v ).Script letters will denote infinite extensions of F . In particular, F shall always denote aGalois extension of F , ramified only at a finite set of places S and such that Γ := Gal(F/H) isa free Z p -module, with H/F a finite subextension (to ease notations, in some sections we willjust put H = F ); the associated Iwasawa algebra is Λ := Z p [[Γ]]. We also put ˜Γ := Gal(F/F )and ˜Λ := Z p [[˜Γ]].The Pontrjagin dual of an abelian group A shall be denoted as A ∨ .Remark 1.1. Class field theory shows that, in contrast with the number field case, in thecharacteristic p setting Gal(F/F ) (and hence Γ) can be very large indeed. Actually, it is wellknown that for every place v the group of 1-units Ov,1 ∗ ⊂ F v ∗ (which is identified with the inertiasubgroup of the maximal abelian extension unramified outside v) is isomorphic to a countableproduct of copies of Z p : hence there is no bound on the dimension of Γ. Furthermore, the only-extension of F which arises somewhat naturally is the arithmetic one F arit , i.e., thecompositum of F with the maximal p-extension of F F . This justifies our choice to concentrateon the case of a Γ of infinite rank: F shall mostly be the maximal abelian extension unramifiedoutside S (often imposing some additional condition to make it disjoint from F arit ).We also recall that a Z p -extension of F can be ramified at infinitely many places [17,Remark 4]: hence our condition on S is a quite meaningful restriction.Z finitep2. Control theorems for abelian varieties2.1. Selmer groups. Let A/F be an abelian variety, let A[p n ] be the group scheme of p n -torsion points and put A[p ∞ ] := lim→A[p n ]. Since we work in characteristic p we define theSelmer groups via flat cohomology of group schemes. For any finite algebraic extension L/F let

IWASAWA THEORY OVER FUNCTION FIELDS 4X L := Spec L and for any place v of L let L v be the completion of L at v and X Lv := Spec L v .Consider the local Kummer embeddingκ Lv: A(L v ) ⊗ Q p /Z p ↩→ lim−→nH 1 fl (X L v, A[p n ]) =: H 1 fl (X L v, A[p ∞ ]) .Definition 2.1. The p part of the Selmer group of A over L is defined as{Sel A (L) p := Ker Hfl 1 (X L, A[p ∞ ]) → ∏ }Hfl 1 (X L v, A[p ∞ ])/Im κ Lvvwhere the map is the product of the natural restrictions at all primes v of L. For an infinitealgebraic extension L/F we define, as usual, the Selmer group Sel A (L) p via the direct limitof the Sel A (L) p for all the finite subextensions L of L.In this section we let F d /F be a Z d p-extension (d < ∞) with Galois group Γ d and associatedIwasawa algebra Λ d . Our goal is to describe the structure of Sel A (F d ) p (actually of itsPontrjagin dual) as a Λ d -module. The main step is a control theorem proved in [2] for the caseof elliptic curves and in [44] in general, which will enable us to prove that S(F d ) := Sel A (F d ) ∨ pis a finitely generated (in some cases torsion) Λ d -module. The proof of the control theoremrequires semi-stable reduction for A at the places which ramify in F d /F : this is not a restrictivehypothesis thanks to the following (see [35, Lemma 2.1])Lemma 2.2. Let F ′ /F be a finite Galois extension. Let Fd ′ := F dF ′ and Λ ′ d := Z p[[Gal(Fd ′ /F ′ )]].Put A ′ for the base change of A to F ′ . If S ′ := Sel A ′(Fd ′ )∨ p is a finitely generated (torsion)Λ ′ d -module then S is a finitely generated (torsion) Λ d-module.Proof. From the natural embeddings Sel A (L) p ↩→ Hfl 1 (X L, A[p ∞ ]) (any L) one gets a diagrambetween dualsHfl 1 (X F d ′ , A[p∞ ]) ∨ S ′H 1 fl (X F d, A[p ∞ ]) ∨ SH 1 (Gal(F ′ d /F d), A[p ∞ ](F ′ d ))∨ S/Im S ′(where in the lower left corner one has the dual of a Galois cohomology group and the whole leftside comes from the dual of the Hochschild-Serre spectral sequence). Obviously Fd ′ /F d is finite(since F ′ /F is) and A[p ∞ ](Fd ′ ) is cofinitely generated, hence H1 (Gal(Fd ′ /F d), A[p ∞ ](Fd ′ ))∨ andS/Im S ′ are finite as well. Therefore S is a finitely generated (torsion) Λ ′ d-module and thelemma follows from the fact that Gal(Fd ′ /F ′ ) is open in Γ d .□2.2. Elliptic curves. Let E/F be an elliptic curve, non-isotrivial (i.e., j(E) ∉ F F ) and havinggood ordinary or split multiplicative reduction at all the places which ramify in F d /F(assuming there is no ramified prime of supersingular reduction one just needs a finite extensionof F to achieve this). We remark that for such curves E[p ∞ ](F sep ) is finite (it is an easyconsequence of the fact that the p n -torsion points for n ≫ 0 generate inseparable extensionsof F , see for example [5, Proposition 3.8]).For any finite subextension F ⊆ L ⊆ F d we put Γ L := Gal(F d /L) and consider the naturalrestriction mapa L : Sel E (L) p −→ Sel E (F d ) Γ Lp .The following theorem summarizes results of [2] and [44].

IWASAWA THEORY OVER FUNCTION FIELDS 5Theorem 2.3. In the above setting assume that F d /F is unramified outside a finite set ofplaces of F and that E has good ordinary or split multiplicative reduction at all ramified places.Then Ker a L is finite (of order bounded independently of L) and Coker a L is a cofinitelygenerated Z p -module (of bounded corank if d = 1). Moreover if all places of bad reduction forE are unramified in F d /F then Coker a L is finite as well (of bounded order if d = 1).Proof. Let F w be the completion of F d at w and, to shorten notations, let⎧⎫⎨G(X L ) := Im⎩ H1 fl (X L, E[p ∞ ]) −→∏⎬Hfl 1 (X L v, E[p ∞ ])/Im κ Lv⎭v∈M L(analogous definition for G(X Fd ) ).Consider the diagram Sel E (L) p H 1 fl (X L, E[p ∞ ]) G(X L )a LSel E (F d ) Γ Lp b L H 1 fl (X F d, E[p ∞ ]) Γ L G(X Fd ) Γ L.c LSince X Fd /X L is a Galois covering the Hochschild-Serre spectral sequence (see [33, III.2.21a),b) and III.1.17 d)]) yieldsKer b L = H 1 (Γ L , E[p ∞ ](F d )) and Coker b L ⊆ H 2 (Γ L , E[p ∞ ](F d ))(where the H i ’s are Galois cohomology groups). Since E[p ∞ ](F d ) is finite it is easy to seethat|Ker b L | ≤ |E[p ∞ ](F d )| d and |Coker b L | ≤ |E[p ∞ ](F d )| d(d−1)2([2, Lemma 4.1]). By the snake lemma the inequality on the left is enough to prove the firststatement of the theorem, i.e.,|Ker a L | ≤ |Ker b L | ≤ |E[p ∞ ](F d )| dwhich is finite and bounded independently of L (actually one can also use the upper bound|E[p ∞ ](F sep )| d which makes it independent of F d as well).We are left with Ker c L : for any place w of F d dividing v defineThend w : H 1 fl (X L v, E[p ∞ ])/Im κ Lv −→ H 1 fl (X F w, E[p ∞ ])/Im κ w .Ker c L ↩→∏ ⋂Ker d wv∈M L(note also that Ker d w really depends on v and not on w). If v totally splits in F d /L then d wis obviously an isomorphism. Therefore from now on we study the Ker d w ’s only for primeswhich are not totally split. Moreover, because of the following diagram coming from theKummer exact sequenceHfl 1 (X L v, E[p ∞ ])/Im κ Lv Hfl 1 (X L v, E)w|vd wH 1 fl (X F w, E[p ∞ ])/Im κ Fw h w H 1 fl (X F w, E) ,one has an injectionwhich allows us to focus on H 1 (Γ Lv , E(F w )).Ker d w ↩→ Ker h w ≃ H 1 (Γ Lv , E(F w ))

IWASAWA THEORY OVER FUNCTION FIELDS 62.2.1. Places of good reduction. If v is unramified let L unrv be the maximal unramified extensionof L v . Then using the inflation map and [34, Proposition I.3.8] one hasH 1 (Γ Lv , E(F w )) ↩→ H 1 (Gal(L unrv /L v ), E(L unrv )) = 0 .Let Ê be the formal group associated with E and, for any place v, let E v be the reducedcurve. From the exact sequenceÊ(O Lv) ↩→ E(L v ) ↠ E v (F v )and the surjectivity of E(L v ) ↠ E v (F v ) (see [33, Exercise I.4.13]), one getsH 1 (Γ Lv , Ê(O w)) ↩→ H 1 (Γ Lv , E(F w )) → H 1 (Γ Lv , E v (F w )) .Using the Tate local duality (see [34, Theorem III.7.8 and Appendix C]) and a careful studyof the p n -torsion points in inseparable extensions of L v , Tan proves that H 1 (Γ Lv , Ê(O w)) isisomorphic to the Pontrjagin dual of E v [p ∞ ](F v ) (see [44, Theorem 2]). Hence|H 1 (Γ Lv , E(F w ))| ≤ |H 1 (Γ Lv , E v (F w ))| |E v [p ∞ ](F v )| .Finally let L ′ v be the maximal unramified extension of L v contained in F w (so that, in particular,F w = F L ′ v) and let Γ L ′ v:= Gal(F w /L ′ v) be the inertia subgroup of Γ Lv . The Hochschild-Serre sequence reads asNow Γ Lv /Γ L ′ vThereforeand eventuallyH 1 (Γ Lv /Γ L ′ v, E v (F L ′ v)) ↩→H 1 (Γ Lv , E v (F w ))↓H 1 (Γ L ′ v, E v (F w )) Γ Lv /Γ L ′ v → H 2 (Γ Lv /Γ L ′ v, E v (F L ′ v)) .can be trivial, finite cyclic or Z p and in any case Lang’s Theorem yieldsH i (Γ Lv /Γ L ′ v, E v (F L ′ v)) = 0 i = 1, 2 .H 1 (Γ Lv , E v (F w )) ≃ H 1 (Γ L ′ v, E v (F L ′ v)) Gal(L′ v/L v) ≃ H 1 (Γ L ′ v, E v (F v ))|H 1 (Γ Lv , E(F w ))| ≤ |H 1 (Γ L ′ v, E v (F v ))| |E v [p ∞ ](F v )|≤ |E v [p ∞ ](F v )| d(L′ v)+1 ≤ |E v [p ∞ ](F v )| d+1where d(L ′ v) := rank Zp Γ L ′ v≤ d and the last bound (independent from F d ) comes from [2,Lemma 4.1].We are left with the finitely many primes of bad reduction.2.2.2. Places of bad reduction. By our hypothesis at these primes we have the Tate curveexact sequenceq Z E,v ↩→ F ∗ w ↠ E(F w ) .For any subfield K of F w /L v one has a Galois equivariant isomorphismK ∗ /O ∗ Kq Z E,v ≃ T K(coming from E 0 (F w ) ↩→ E(F w ) ↠ T Fw ) where T K is a finite cyclic group of order −ord K (j(E))arising from the group of connected components (see, for example, [2, Lemma 4.9 and Remark4.10]). ThereforeH 1 (Γ Lv , E(F w )) = lim−→Kwhere (T K ) p is the p-part of T K and d(K) = rank Zp Γ K .H 1 (Γ K , E(K)) ↩→ lim−→KH 1 (Γ K , T K ) ≃ lim−→K(T K ) d(K)p

IWASAWA THEORY OVER FUNCTION FIELDS 7If v is unramified then all T K ’s are isomorphic to T Lv|H 1 (Γ Lv , E(F w ))| = |(T Lv ) p | = |(T Fν ) p |and d(K) = d(L v ) = 1 hencewhere ν is the prime of F lying below v (note that the bound is again independent of F d ).If v is ramified then taking Galois cohomology in the Tate curve exact sequence, one findsKer h w = H 1 (Γ Lv , E(F w )) ↩→ H 2 (Γ Lv , q Z E,v)where the injectivity comes from Hilbert 90.Since Γ Lv acts trivially on qE,v Z , one finds thatKer h w ↩→ H 2 (Γ Lv , qE,v) Z ≃ H 2 (Γ Lv , Z) ≃ (Γ abL v) ∨ ≃ (Q p /Z p ) d(Lv) ↩→ (Q p /Z p ) d ,i.e., Ker h w is a cofinitely generated Z p -module.This completes the proof for the general case. If all ramified primes are of good reduction,then the Ker d w ’s are finite so Coker a L is finite as well. In particular its order is boundedby∏∏|E v [p ∞ ](F Lv )| d+1 × p ordp(ordv(j(E))) × |Coker b L | .v ram, goodv inert, badIf d = 1 the last term is trivial and in a Z p -extension the (finitely many) ramified and inert ofbad reduction places admit only a finite number of places above them. If d ≥ 2 such boundis not independent of L because the number of terms in the products is unbounded. In thecase of ramified primes of bad reduction the bound for the corank is similar.□Both Sel E (F d ) p and its Pontrjagin dual are modules over the ring Λ d in a natural way. Aneasy consequence of the previous theorem and of Nakayama’s Lemma (see [1]) is the following(see for example [2, Corollaries 4.8 and 4.13])Corollary 2.4. In the setting of the previous theorem, let S(F d ) be the Pontrjagin dual ofSel E (F d ) p . Then S(F d ) is a finitely generated Λ d -module. Moreover if all ramified primesare of good reduction and Sel E (F ) p is finite, then S(F d ) is Λ d -torsion.Remark 2.5.1. We recall that, thanks to Lemma 2.2, the last corollary holds when there are noramified primes of supersingular reduction for E.2. The ramified primes of split multiplicative reduction are the only obstacle to thefiniteness of Coker a L and this somehow reflects the number field situation as describedin [31, section II.6], where the authors defined an extended Mordell-Weil group whoserank is rank E(F ) + N (where N is the number of primes of split multiplicativereduction and dividing p, i.e., totally ramified in the cyclotomic Z p -extension theywork with) to deal with the phenomenon of exceptional zeroes.3. A different way of having finite kernels and cokernels (and then, at least in somecases, torsion modules S(F d ) ) consists in a modified version of the Selmer groups.Examples with trivial or no conditions at all at the ramified primes of bad reductionare described in [2, Theorem 4.12].4. The available constructions of a p-adic L-function associated to Z N p -extensions requirethe presence of a totally ramified prime p of split multiplicative reduction for E.Thus the theorem applies to that setting but, unfortunately, it only provides finitelygenerated Λ d -modules S(F d ).2.3. Higher dimensional abelian varieties. We go back to the general case of an abelianvariety A/F . For any finite subextension L/F of F d we put Γ L := Gal(F d /L) and considerthe natural restriction mapa L : Sel A (L) p −→ Sel A (F d ) Γ Lp .The following theorem summarizes results of [44].

IWASAWA THEORY OVER FUNCTION FIELDS 8Theorem 2.6. In the above setting assume that F d /F is unramified outside a finite set ofplaces of F and that A has good ordinary or split multiplicative reduction at all ramified places.Then Ker a L is finite (of bounded order if d = 1) and Coker a L is a cofinitely generated Z p -module. Moreover if all places of bad reduction for A are unramified in F d /F then Coker a Lis finite as well (of bounded order if d = 1).Proof. We use the same notations and diagrams as in Theorem 2.3, substituting the abelianvariety A for the elliptic curve E.The Hochschild-Serre spectral sequence yieldsKer b L = H 1 (Γ L , A[p ∞ ](F d )) and Coker b L ⊆ H 2 (Γ L , A[p ∞ ](F d )) .Let L 0 ⊆ F d be the extension generated by A[p ∞ ](F d ). The extension L 0 /L is everywhereunramified (for the places of good reduction see [44, Lemma 2.5.1 (b)], for the other places notethat the p n -torsion points come from the p n -th roots of the periods provided by the Mumfordparametrization so they generate an inseparable extension while F d /F is separable): henceGal(L 0 /L) ≃ ∆×Z e p where ∆ is finite and e = 0 or 1. Let γ be a topological generator of Z e p inGal(L 0 /L) (if e = 0 then γ = 1) and let L 1 be its fixed field. Then A[p ∞ ](F d ) = A[p ∞ ](L 1 )is finite and we can apply [3, Lemma 3.4] (with b the maximum between |A[p ∞ ](L 1 )| and|A[p ∞ ](F d )/(A[p ∞ ](F d )) div |) to get|Ker b L | ≤ b d and |Coker b L | ≤ b d(d−1)2 .By the snake lemma the inequality on the left is enough to prove that Ker a L is finite (forthe bounded order in the case d = 1 see [44, Corollary 3.2.4]).The bounds for the Ker d w ’s are a direct generalization of the ones provided for the case ofthe elliptic curve so we give just a few details. Recall the embeddingKer d w ↩→ Ker h w ≃ H 1 (Γ Lv , E(F w )) .2.3.1. Places of good reduction. If v is unramified thenH 1 (Γ Lv , A(F w )) ↩→ H 1 (Gal(L unrv /L v ), A(L unrv )) = 0 .If v is ramified one has an exact sequence (as above)By [44, Theorem 2]H 1 (Γ Lv , Â(O F w)) ↩→ H 1 (Γ Lv , A(F w )) → H 1 (Γ Lv , A v (F Fw )) .H 1 (Γ Lv , Â(O F w)) ≃ B v [p ∞ ](F Lv )(where B is the dual variety of A) and the last group has the same order of A v [p ∞ ](F Lv ).Using Lang’s theorem as in 2.2.1 one finds|H 1 (Γ Lv , A(F w ))| ≤ |A v [p ∞ ](F Lv )| d+1 .2.3.2. Places of bad reduction. If v is unramified let π 0,v (A) be the group of connected componentsof the Néron model of A at v. Then, again by [34, Proposition I.3.8],H 1 (Γ Lv , A(F w )) ↩→ H 1 (Gal(L unrv/L v ), A(L unrv)) ≃ H 1 (Gal(L unrv /L v ), π 0,v (A))and the last group has order bounded by |π 0,v (A) Gal(Lunr v /L v) |.If v is ramified one just uses Mumford’s parametrization with a period lattice Ω v ⊂ L v ×· · · × L v (genus A times) to prove that H 1 (Γ Lv , A(F w )) is cofinitely generated as in 2.2.2. □We end this section with the analogue of Corollary 2.4.Corollary 2.7. In the setting of the previous theorem, let S(F d ) be the Pontrjagin dual ofSel A (F d ) p . Then S(F d ) is a finitely generated Λ d -module. Moreover if all ramified primesare of good reduction and Sel A (F ) p is finite, then S(F d ) is Λ d -torsion.

IWASAWA THEORY OVER FUNCTION FIELDS 9Remark 2.8. In [35, Theorem 1.7], by means of crystalline and syntomic cohomology, Ochiaiand Trihan prove a stronger result. Indeed they can show that the dual of the Selmer groupis always torsion, with no restriction on the abelian variety A/F , but only in the case of thearithmetic extension F arit /F , which lies outside the scope of the present paper. Moreoverin the case of a (not necessarily commutative) pro-p-extension containing F arit , they provethat the dual of the Selmer group is finitely generated (for a precise statement, see [35], inparticular Theorem 1.9).3. Λ-modules and Fitting idealsWe need a few more notations.For any Z d p-extension F d as above let Γ(F d ) := Gal(F d /F ) and Λ(F d ) := Z p [[Γ(F d )]] (theIwasawa algebra) with augmentation ideal I F d(or simply Γ d , Λ d and I d if the extension F dis clearly fixed).For any d > e and any Z d−ep -extension F d /F e we put Γ(F d /F e ) := Gal(F d /F e ), Λ(F d /F e ) :=Z p [[Γ(F d /F e )]] and I F dF eas the augmentation ideal of Λ(F d /F e ), i.e., the kernel of the canonicalprojection π F dF e: Λ(F d ) → Λ(F e ) (whenever possible all these will be abbreviated to Γ d e , Λ d e ,Ie d and πe d respectively).Recall that Λ(F d ) is (noncanonically) isomorphic to Z p [[T 1 , . . . , T d ]]. A finitely generatedtorsion Λ(F d )-module is said to be pseudo-null if its annihilator ideal has height at least 2.If M is a finitely generated torsion Λ(F d )-module then there is a pseudo-isomorphism (i.e., amorphism with pseudo-null kernel and cokernel)M ∼ Λ(Fd )n⊕i=1Λ(F d )/(g e ii )where the g i ’s are irreducible elements of Λ(F d ) (determined up to an element of Λ(F d ) ∗ ) andn and the e i ’s are uniquely determined by M (see e.g. [7, VII.4.4 Theorem 5]).Definition 3.1. In the above setting the characteristic ideal of M is{0 if M is not torsionCh Λ(Fd )(M) :=( ∏ ni=1 ge ii ) otherwise .Let Z be a finitely generated Λ(F d )-module and letϕΛ(F d ) a −→Λ(F d ) b ↠ Zbe a presentation where the map ϕ can be represented by a b × a matrix Φ with entries inΛ(F d ) .Definition 3.2. In the above setting the Fitting ideal of Z is⎧0 if a < b⎪⎨the ideal generated by all theF itt Λ(Fd )(Z) :=determinants of the b × b if a ≥ b⎪⎩minors of the matrix ΦLet F/F be a Z N p -extension with Galois group Γ. Our goal is to define an ideal in Λ :=Z p [[Γ]] associated with S the Pontrjagin dual of Sel A (F) p . For this we consider all the Z d p-extensions F d /F (d ∈ N) contained in F (which we call Z p -finite extensions). Then F = ∪F dand Λ = lim←Λ(F d ) := lim←Z p [[Gal(F d /F )]]. The classical characteristic ideal does not behavewell (in general) with respect to inverse limits (because the inverse limit of pseudo-null modulesis not necessarily pseudo-null). For the Fitting ideal, using the basic properties described inthe Appendix of [32], we have the following.

IWASAWA THEORY OVER FUNCTION FIELDS 10Lemma 3.3. Let F d ⊂ F e be an inclusion of multiple Z p -extensions, e > d. Assume thatA[p ∞ ](F) = 0 or that F itt Λ(Fd )(S(F d )) is principal. Thenπ FeF d(F itt Λ(Fe)(S(F e ))) ⊆ F itt Λ(Fd )(S(F d )) .Proof. Consider the natural map a e d : Sel A(F d ) p → Sel A (F e ) Γe dpand dualize to getwhere (as in Theorem 2.6)S(F e )/I e d S(F e) → S(F d ) ↠ (Ker a e d )∨Ker a e d ↩→ H1 (Γ e d , A[p∞ ](F e ))is finite. If A[p ∞ ](F) = 0 then (Ker a e d )∨ = 0 andIf (Ker a e d )∨ ≠ 0 one hasπ e d (F itt Λ e(S(F e ))) = F itt Λd (S(F e )/I e d S(F e)) ⊆ F itt Λd (S(F d )) .F itt Λd (S(F e )/I e d S(F e))F itt Λd ((Ker a e d )∨ ) ⊆ F itt Λd (S(F d )) .The Fitting ideal of a finitely generated torsion module contains a power of its annihilator,so let σ 1 , σ 2 be two relatively prime elements of F itt Λd ((Ker a e d )∨ ) and θ d a generator ofF itt Λd (S(F d )). Then θ d divides σ 1 α and σ 2 α for any α ∈ F itt Λd (S(F e )/I e d S(F e)) (it holds,in the obvious sense, even for θ d = 0). Henceπ e d (F itt Λ e(S(F e ))) = F itt Λd (S(F e )/I e d S(F e)) ⊆ F itt Λd (S(F d )) .Remark 3.4. In the case A = E an elliptic curve, the hypothesis E[p ∞ ](F) = 0 is satisfiedif j(E) ∉ (F ∗ ) p , i.e., when the curve is admissible (in the sense of [5]); otherwise j(E) ∈(F ∗ ) pn − (F ∗ ) pn+1 and one can work over the field F pn . The other hypothesis is satisfied ingeneral by elementary Λ(F d )-modules or by modules having a presentation with the samenumber of generators and relations.Let π Fd be the canonical projection from Λ to Λ(F d ) with kernel I Fd . Then the previouslemma shows that, as F d varies, the (π Fd ) −1 (F itt Λ(Fd )(S(F d ))) form an inverse system ofideals in Λ.Definition 3.5. Assume that A[p ∞ ](F) = 0 or that F itt Λ(Fd )(S(F d )) is principal for any F d .Define˜F itt Λ (S(F)) := lim←−(π Fd ) −1 (F itt Λ(Fd )(S(F d )))Fdto be the pro-Fitting ideal of S(F) (the Pontrjagin dual of Sel E (F) p ).Proposition 3.6. Assume that A[p ∞ ](F) = 0 or that F itt Λ(Fd )(S(F d )) is principal forany F d . If corank Zp Sel A (F 1 ) Γ(F 1)p ≥ 1 for any Z p -extension F 1 /F contained in F, then˜F itt Λ (S(F)) ⊂ I (where I is the augmentation ideal of Λ).Proof. Recall that I F dis the augmentation ideal of Λ(F d ), that is, the kernel of π F d: Λ(F d ) →Z p . By hypothesis F itt Zp ((Sel A (F 1 ) Γ(F 1)p ) ∨ ) = 0. Thus, since Z p = Λ(F 1 )/I F 1and (Sel A (F 1 ) Γ(F 1)p ) ∨ =S(F 1 )/I F 1S(F 1 ),0 = F itt Zp ((Sel A (F 1 ) Γ(F 1)p ) ∨ ) = π F 1(F itt Λ(F1 )(S(F 1 ))) ,i.e., F itt Λ(F1 )(S(F 1 )) ⊂ Ker π F 1= I F 1.For any Z d p-extension F d take a Z p -extension F 1 contained in F d . Then, by Lemma 3.3,π F dF 1(F itt Λ(Fd )(S(F d ))) ⊆ F itt Λ(F1 )(S(F 1 )) ⊂ I F 1.□

Note that π F d= π F 1◦ π F dF 1. Thereforei.e., F itt Λ(Fd )(S(F d )) ⊂ I F dIWASAWA THEORY OVER FUNCTION FIELDS 11F itt Λ(Fd )(S(F d )) ⊂ I F d⇐⇒ π F dF 1(F itt Λ(Fd )(S(F d ))) ⊂ I F 1,for any Z p -finite extension F d . Finally˜F itt Λ (S(F)) := ⋂ F d(π Fd ) −1 (F itt Λ(Fd )(S(F d ))) ⊂ ⋂ F d(π Fd ) −1 (I F d) ⊂ I .Remark 3.7. From the exact sequenceKer a F1 ↩→ Sel A (F ) pa F1−−→Sel A (F 1 ) Γ(F 1)p ↠ Coker a F1and the fact that Ker a F1 is finite one immediately finds out that the hypothesis on corank Zp Sel A (F 1 ) Γ(F 1)pis satisfied if rank Z A(F ) ≥ 1 or corank Zp Coker a F1 ≥ 1. As already noted, when there isa totally ramified prime of split multiplicative reduction, the second option is very likely tohappen. In the number field case, when F is the cyclotomic Z p -extension and, in some cases,Sel E (F) ∨ p is known to be a torsion module, this is equivalent to saying that T divides a generatorof the characteristic ideal of Sel E (F) ∨ p (i.e., there is an exceptional zero). Note thatall the available constructions of p-adic L-function for our setting require a ramified place ofsplit multiplicative reduction and they are all known to belong to I.4. Modular abelian varieties of GL 2 -typeThe previous sections show how to define the algebraic (p-adic) L-function associated withF/F and an abelian variety A/F under quite general conditions. On the analytic side thereis, of course, the complex Hasse-Weil L-function L(A/F, s), so the problem becomes to relateit to some element in an Iwasawa algebra. In this section we will sketch how this can be doneat least in some cases; in order to keep the paper to a reasonable length, the treatment herewill be very brief.We say that the abelian variety A/F is of GL 2 -type if there is a number field K such that[K : Q] = dim A and K embeds into End F (A) ⊗ Q. In particular, this implies that for anyl ≠ p the Tate module T l A yields a representation of G F in GL 2 (K ⊗ Q l ). The analogousdefinition for A/Q can be found in [40], where it is proved that Serre’s conjecture impliesthat every simple abelian variety of GL 2 -type is isogenous to a simple factor of a modularJacobian. We are going to see that a similar result holds at least partially in our functionfield setting.4.1. Automorphic forms. Let A F denote the ring of adeles of F . By automorphic form forGL 2 we shall mean a function f : GL 2 (A F ) −→ C which factors through GL 2 (F )\GL 2 (A F )/K,where K is some open compact subgroup of GL 2 (A F ); furthermore, f is cuspidal if it satisfiessome additional technical condition (essentially, the annihilation of some Fourier coefficients).A classical procedure associates with such an f a Dirichlet sum L(f, s): see e.g. [49, ChaptersII and III].The C-vector spaces of automorphic and cuspidal forms provide representations of GL 2 (A F ).Besides, they have a natural Q-structure: in particular, the decomposition of the space ofcuspidal forms in irreducible representations of GL 2 (A F ) holds over Q (and hence over anyalgebraically closed field of characteristic zero); see e.g. the discussion in [39, page 218]. Wealso recall that every irreducible automorphic representation π of GL 2 (A F ) is a restrictedtensor product ⊗ ′ vπ v , v varying over the places of F : the π v ’s are representations of GL 2 (F v )and they are called local factors of π.□

IWASAWA THEORY OVER FUNCTION FIELDS 12Let W F denote the Weil group of F : it is the subgroup of G F consisting of elements whoserestriction to F q is an integer power of the Frobenius. By a fundamental result of Jacquetand Langlands [23, Theorem 12.2], a two-dimensional representation of W F corresponds toa cuspidal representation if the associated L-function and its twists by characters of W F areentire functions bounded in vertical strips (see also [49]).Let A/F be an abelian variety of GL 2 -type. Recall that L(A/F, s) is the L-function associatedwith the compatible system of l-adic representations of G F arising from the Tate modulesT l A, as l varies among primes different from p. Theorems of Grothendieck and Deligne showthat under certain assumptions L(A/F, s) and all its twists are polynomials in q −s satisfyingthe conditions of [23, Theorem 12.2] (see [14, §9] for precise statements). In particular allelliptic curves are obviously of GL 2 -type and one finds that L(A/F, s) = L(f, s) for some cuspform f when A is a non-isotrivial elliptic curve.4.2. Drinfeld modular curves. From now on we fix a place ∞.The main source for this section is Drinfeld’s original paper [15]. Here we just recall thatfor any divisor n of F with support disjoint from ∞ there exists a projective curve M(n)(the Drinfeld modular curve) and that these curves form a projective system. Hence one canconsider the Galois representationH := lim→H 1 et(M(n) × F sep , Q l ).Besides, the moduli interpretation of the curves M(n) allows to define an action of GL 2 (A f )on H (where A f denotes the adeles of F without the component at ∞). Let Π ∞ be theset of those cuspidal representations having the special representation of GL 2 (F ∞ ) (i.e., theSteinberg representation) as local factor at ∞. Drinfeld’s reciprocity law [15, Theorem 2](which realizes part of the Langlands correspondence for GL 2 over F ) attaches to any π ∈ Π ∞a compatible system of two-dimensional Galois representations σ(π) l : G F −→ GL 2 (Q l ) byestablishing an isomorphism of GL 2 (A f ) × G F -modules(4.1) H ≃ ⊕(⊗ ′ v≠∞ π v) ⊗ σ(π) l .π∈Π ∞As σ(π) l one obtains all l-adic representations of G F satisfying certain properties: for aprecise list see [39] 1 . Here we just remark the following requirement: the restriction of σ(π) lto G F∞ has to be the special l-adic Galois representation sp ∞ . For example, the representationoriginated from the Tate module T l E of an elliptic curve E/F satisfies this condition if andonly if E has split multiplicative reduction at ∞.The Galois representations appearing in (4.1) are exactly those arising from the Tate moduleof the Jacobian of some M(n). We call modular those abelian varieties isogenous to somefactor of Jac(M(n)). Hence we see that a necessary condition for abelian varieties of GL 2 -typeto be modular is that their reduction at ∞ is a split torus.The paper [16] provides a careful construction of Jacobians of Drinfeld modular curves bymeans of rigid analytic geometry.4.3. The p-adic L-functions. For any ring R let Meas(P 1 (F v ), R) denote the R-valuedmeasures on the topological space P 1 (F v ) (that is, finitely additive functions on compact opensubsets of P 1 (F v )) and Meas 0 (P 1 (F v ), R) the subset of measures of total mass 0. A keyingredient in the proof of (4.1) is the identification of the space of R-valued cusp forms withdirect sums of certain subspaces of Meas 0 (P 1 (F ∞ ), R) (for more precise statements, see [39,§2] and [16, §4]). Therefore we can associate with any modular abelian variety A some measure1 Actually, [39] provides a thorough introduction to all this subject.

IWASAWA THEORY OVER FUNCTION FIELDS 13µ A on P 1 (F ∞ ); this fact can be exploited to construct elements (our p-adic L-functions) inIwasawa algebras in the following way.Let K be a quadratic algebra over F : an embedding of K into the F -algebra of 2 × 2matrices M 2 (F ) gives rise to an action of the group G := (K ⊗ F ∞ ) ∗ /F ∗ ∞ on the P GL 2 (F ∞ )-homogeneous space P 1 (F ∞ ). Class field theory permits to relate G to a subgroup Γ of ˜Γ =Gal(F/F ), where F is a certain extension of F (depending on K) ramified only above ∞.Then the pull-back of µ A to G yields a measure on Γ; this is enough because Meas(Γ, R) iscanonically identified with R ⊗ Λ (and Meas 0 (Γ, R) with the augmentation ideal). Variousinstances of the construction just sketched are explained in [30] for the case when A is anelliptic curve: here one can take R = Z. Similar ideas were used in Pál’s thesis [37], wherethere is also an interpolation formula relating the element in Z[[Γ]] so obtained to special valuesof the complex L-function. One should also mention [38] for another construction of p-adicL-function, providing an interpolation property for one of the cases studied in [30]. Noticethat in all this cases the p-adic L-function is, more or less tautologically, in the augmentationideal.A different approach had been previously suggested by Tan [43]: starting with cuspidalautomorphic forms, he defines elements in certain group algebras and proves an interpolationformula [43, Proposition 2]. Furthermore, if the cusp form is “well-behaved” his modularelements form a projective system and originate an element in an Iwasawa algebra of thekind considered in the present paper: in particular, this holds for non-isotrivial elliptic curveshaving split multiplicative reduction. In the case of an elliptic curve over F q (T ) Teitelbaum[46] re-expressed Tan’s work in terms of modular symbols (along the lines of [31]); in [21] itis shown how this last method can be related to the “quadratic algebra” techniques sketchedabove.A unified treatment of all of this will appear in [4].Thus for a modular abelian variety A/F we can define both a Fitting ideal and a p-adicL-function: it is natural to expect that an Iwasawa Main Conjecture should hold, i.e., thatthe Fitting ideal should be generated by the p-adic L-function.Remark 4.1. In the cases considered in this paper (involving a modular abelian varietyand a geometric extension of the function field) the Iwasawa Main Conjecture is still wideopen. However, recently there has been some interesting progress in two related settings:both results are going to appear in [29].First, one can take A to be an isotrivial abelian variety (notice that [43, page 308] definesmodular elements also for an isotrivial elliptic curve). Ki-Seng Tan has observed that thenthe Main Conjecture can be reduced to the one for class groups, which is already known tohold (as it will be explained in the next section).Second, one can take as F the maximal arithmetic pro-p-extension of F , i.e., F = F arit =F F (p)F, where F(p)Fis the subfield of F F defined by Gal(F (p)F/F F ) ≃ Z p (notice that this is thesetting of [35, Theorem 1.7]). In this case Trihan has obtained a proof of the Iwasawa MainConjecture, by techniques of syntomic cohomology. He needs no assumption on the abelianvariety A/F : his p-adic L-function is defined by means of cohomology and it interpolates theHasse-Weil L-function.5. Class groupsFor any finite extension L/F , A(L) will denote the p-part of the group of degree zero divisorclasses of L; for any F ′ intermediate between F and F, we put A(F ′ ) := lim←A(L) as L runsamong finite subextensions of F ′ /F (the limit being taken with respect to norm maps). Thestudy of similar objects and their relations with zeta functions is an old subject and was the

IWASAWA THEORY OVER FUNCTION FIELDS 15respect to norm maps). For two finite extensions L ⊃ L ′ ⊃ F , the degree maps deg L anddeg L ′ fit into the commutative diagram (with exact rows)(5.1) A(L) M(L)deg L Z pN L L ′A(L ′ ) N L L ′ M(L ′ )deg L ′ Z p ,where N L L ′ denotes the norm and the vertical map on the right is multiplication by [F L : F L ′](the degree of the extension between the fields of constants). For an infinite extension L/Fcontained in F, taking projective limits (and recalling Assumption 5.1 above), one gets anexact sequence(5.2) A(L) M(L)deg L Z p .Remark 5.2. If one allows non-geometric extensions then the deg L map above becomes thezero map exactly when the Z p -extension F arit is contained in L.It is well known that M(F d ) is a finitely generated torsion Λ(F d )-module (see e.g. [19,Theorem 1]), so the same holds for A(F d ) as well. Moreover take any Z d p-extension F d of Fcontained in F: since our extension F/F is totally ramified at the prime p, for any F d−1 ⊂ F done has(5.3) M(F d )/I F dF d−1M(F d ) ≃ M(F d−1 )(see for example [48, Lemma 13.15]). As in Section 3, to ease notations we will often erasethe F from the indices (for example I F dF d−1will be denoted by Id−1 d ), hoping that no confusionwill arise. Consider the following diagram(5.4) A(F d ) M(F d ) deg Z pγ−1γ−1A(F d ) γ−1 M(F d ) deg Z p(where 〈γ〉 = Gal(F d /F d−1 ) =: Γ d d−1; note also that the vertical map on the right is 0) andits snake lemma sequence(5.5) A(F d ) Γd d−1 M(F d ) Γd d−1deg Z pdegZ p M(F d )/Id−1 d M(F d) A(F d )/Id−1 d A(F d) .For d ≥ 2 the Λ d -module Z p is pseudo-null, hence (5.2) yields Ch Λd (M(F d )) = Ch Λd (A(F d )),and, using (5.3) and (5.5), one finds (for d ≥ 3)(5.6)Ch Λd−1 (A(F d )/I d d−1 A(F d)) = Ch Λd−1 (M(F d )/I d d−1 M(F d))= Ch Λd−1 (M(F d−1 )) = Ch Λd−1 (A(F d−1 ))(where all the modules involved are Λ d−1 -torsion modules).Let(5.7) N(F d ) ↩→ A(F d )ι−→ E(F d ) ↠ R(F d )

IWASAWA THEORY OVER FUNCTION FIELDS 16be the exact sequence coming from the structure theorem for Λ d -modules (see section 3) whereE(F d ) := ⊕ iΛ d /(f i,d )is an elementary module and N(F d ), R(F d ) are pseudo-null. Let Ch Λd (A(F d )) = (Π i f i,d ) bethe characteristic ideal of A(F d ): we want to compare Ch Λd−1 (A(F d−1 )) with π F dF d−1(Ch Λd (A(F d )))for some F d−1 ⊂ F d and show that these characteristic ideals form an inverse system (in Λ).Consider the module B(F d ) := N(F d ) ⊕ R(F d ). We need the following hypothesis.Assumption 5.3. There is a choice of the pseudo-isomorphism ι in (5.7) and a splitting ofthe projection Γ d ↠ Γ d−1 so thati) Γ d = 〈γ d 〉 ⊕ Γ d−1 ;ii) B(F d ) is a finitely generated torsion Z p [[Γ d−1 ]]-module.As explained in [20] (see the remarks just before Lemma 3), for any F d and ι one can finda subfield F d−1 so that Assumption 5.3 holds.In order to ease notations, we put γ = γ d , so that Γ d d−1 = 〈γ〉.Lemma 5.4. With the above notations, one hasCh Λd−1 (A(F d )/I d d−1 A(F d)) = π d d−1(ChΛd (A(F d )) ) · Ch Λd−1 (A(F d ) Γd d−1 ) .Proof. We split the previous sequence in two byN(F d ) ↩→ A(F d ) ↠ C(F d ) , C(F d ) ↩→ E(F d ) ↠ R(F d )and consider the snake lemma sequences coming from the following diagrams(5.8) N(F d ) A(F d ) C(F d ) C(F d ) E(F d ) R(F d )γ−1N(F d ) γ−1γ−1γ−1A(F d ) C(F d ) C(F d ) i.e.,(5.9) N(F d ) Γd d−1 A(F d ) Γd d−1γ−1γ−1 E(F d ) R(F d ) , C(F d ) Γd d−1C(F d )/Id−1 d C(F d) A(F d )/Id−1 d A(F d) N(F d )/Id−1 d N(F d)and(5.10) C(F d ) Γd d−1 E(F d ) Γd d−1 R(F d ) Γd d−1R(F d )/Id−1 d R(F d) E(F d )/Id−1 d E(F d) C(F d )/Id−1 d C(F d) .From (5.7) we get an exact sequenceA(F d )/I d d−1 A(F d) −→ ⊕ iΛ d /(γ − 1, f i,d ) −→ R(F d )/I d d−1 R(F d)where the last term is a torsion(Λ d−1 -module. So is A(F d−1 ) for d ≥ 3 and, by (5.6),Ch Λd−1 (A(F d−1 )) = Ch Λd−1 A(Fd )/Id−1 d A(F d) ) . It follows that none of the f i,d ’s belongto Id−1 d . Therefore:

IWASAWA THEORY OVER FUNCTION FIELDS 171. the map γ − 1 : E(F d ) −→ E(F d ) has trivial kernel, i.e., E(F d ) Γd d−1 = 0 so thatC(F d ) Γd d−1 = 0 as well;2. the characteristic ideal of the Λ d−1 -module E(F d )/I d d−1 E(F d) is generated by theproduct of the f i,d ’s modulo I d d−1 , hence it is obviously equal to πd d−1 (Ch Λ d(A(F d ))).Moreover, from the fact that N(F d ) and R(F d ) are finitely generated torsion Λ d−1 -modules3 and the multiplicativity of characteristic ideals, looking at the left (resp. right) verticalsequence of the first (resp. second) diagram in (5.8), one findsCh Λd−1 (N(F d ) Γd d−1 ) = ChΛd−1 (N(F d )/I d d−1 N(F d))andCh Λd−1 (R(F d ) Γd d−1 ) = ChΛd−1 (R(F d )/Id−1 d R(F d)) .Hence from (5.9) one hasCh Λd−1 (A(F d )/I d d−1 A(F d)) = Ch Λd−1(C(Fd )/I d d−1 C(F d) ) · Ch Λd−1 (N(F d ) Γd d−1 )= Ch Λd−1(C(Fd )/I d d−1 C(F d) ) · Ch Λd−1 (A(F d ) Γd d−1 )(where the last line comes from the isomorphism A(F d ) Γd d−1 ≃ N(Fd ) Γd d−1 ). The sequence(5.10) provides the equalityCh Λd−1 (C(F d )/I d d−1 C(F d)) = Ch Λd−1 (E(F d )/I d d−1 E(F d))= π d d−1 (Ch Λ d(A(F d ))) .Therefore one concludes that(5.11) Ch Λd−1 (A(F d )/Id−1 d A(F d)) = πd−1d (ChΛd (A(F d )) ) · Ch Λd−1 (A(F d ) Γd d−1 ) .□Our next step is to prove that A(F d ) Γd d−1 = 0 (note that it would be enough to prove thatit is pseudo-null as a Λ d−1 -module). For this we need first a few lemmas.Lemma 5.5. Let G be a finite group and endow Z p with the trivial G-action. Then for anyG-module M we haveH i (G, M ⊗ Z p ) = H i (G, M) ⊗ Z pfor all i ≥ 0.This result should be well-known. Since we were not able to find a suitable reference, hereis a sketch of the proof.Proof. Let X be a G-module which has no torsion as an abelian group and put Y := X ⊗ Q.It is not hard to prove that Y G ⊗ Z p = (Y ⊗ Z p ) G and it follows that the same holds for X,since X G ⊗ Z p is a saturated submodule of X ⊗ Z p . Applying this to the standard complex bymeans of which the H i (G, M) are defined, one can prove the equality in the case M has notorsion as an abelian group. The general case follows because any G-module is the quotientof two such modules.□Up to now we have mainly considered M(L) as an Iwasawa module (for various L), nowwe focus on its interpretation as a group of divisor classes. Let L be a finite extensions of Fand recall that we defined M(L) = (Div(L)/P L ) ⊗ Z p . From the exact sequenceF ∗ L ↩→ L ∗ ↠ P Land the fact that |F ∗ L | is prime with p, one finds an isomorphism between L∗ ⊗Z p and P L ⊗Z p .Hence we can (and will) identify the two.holds.3 It might be worth to notice that this is the only point where we use the hypothesis that Assumption 5.3

IWASAWA THEORY OVER FUNCTION FIELDS 18Lemma 5.6. For any finite Galois extension L/F , the mapis surjective.Proof. The sequenceDiv(L) Gal(L/F ) ⊗ Z p −→ M(L) Gal(L/F )(5.12) L ∗ ⊗ Z p ↩→ Div(L) ⊗ Z p ↠ M(L)is exact because Z p is flat and |F ∗ L| is prime with p. The claim follows by taking the Gal(L/F )-cohomology of (5.12) and applying Lemma 5.5 and Hilbert 90.□For any finite subextension L of F/F let p L be the unique prime lying above p. In thefollowing lemma, we identify p L with its image in Div(L) ⊗ Z p . Moreover for any elementx ∈ M(F) we let x L denote its image in M(L) via the canonical norm map.Lemma 5.7. Let x ∈ M(F) Γ : then, for any L as above, x L is represented by a Γ-invariantdivisor supported in p L .Proof. For any L, let y L be the image of p L in M(L). Since Z p y L is a closed subset of M(L),to prove the lemma it is enough to show that ( x L + p n M(L) ) ∩ Z p y L ≠ ∅ for any n.For any finite Galois extension K/L we have the mapsι K L : Div(L) ⊗ Z p −→ Div(K) ⊗ Z pandNL K : Div(K) ⊗ Z p −→ Div(L) ⊗ Z prespectively induced by the inclusion and the norm. For any divisor whose support is unramifedin K/L we haveNL K (ι K L (D)) = [K : L]D .Also, Lemma 5.5 yields(Div(K) ⊗ Z p ) Gal(K/L) = Div(K) Gal(K/L) ⊗ Z p = ι K L (Div(L) ⊗ Z p )(since in a Gal(K/L)-invariant divisor all places of K above a same place of L occur with thesame multiplicity).Choose n and let K ⊂ F be such that [K : L] ≥ p n . By Lemma 5.6, there exists aGal(K/L)-invariant E K ∈ Div(K) ⊗ Z p having image x K . Write E K = D K + a K p K , wherea K ∈ Z p and D K has support disjoint from p K . Then D K is Galois invariant, so D K = ι K L (D L)and (using Assumption 5.1)N K L (E K ) = [K : L]D L + a K p L .Projecting into M(L) we get x L ∈ a K y L + p n M(L) .Corollary 5.8. A(F d ) Γd d−1 = 0.Proof. Taking Γ d d−1 -invariants in (5.2) (with L = F d), one finds a similar sequence□(5.13) A(F d ) Γd d−1 M(F d ) Γd d−1deg Fd Z p .Lemma 5.7 holds, with exactly the same proof, also replacing F and Γ with F d and Γ d . Thereforeany x = (x L ) L ∈ M(F d ) Γd d−1 can be represented by a sequence (aL p L ) L . FurthermoreNL K(a Kp K ) = a L p L implies that the value a L is independent of L: call it a. Thendeg Fd(x) = lim ( a L deg L (p L ) ) = a deg F (p) .Hence x ∈ Ker(deg Fd) = A(F d ) Γd d−1 only if a = 0. □

IWASAWA THEORY OVER FUNCTION FIELDS 19Remark 5.9. The image of the degree map appearing in (5.13) is (deg p)Z p , so deg Fdalwaysprovides an isomorphism between M(F d ) Γd d−1 and Zp . Moreover, if p does not divide deg p,one has surjectivity as well. In this case, looking back at the sequence (5.5), one finds a shortexact sequenceA(F d )/I d d−1 A(F d) M(F d )/I d d−1 M(F d) deg Z p .From (5.1), by taking the limit with L and L ′ varying respectively among subextensions ofF d and F d−1 , one obtains a commutative diagramM(F d )/I d d−1 M(F d)deg Z pNM(F d−1 )deg Fd−1where the map N is the isomorphism induced by the norm, i.e., the one appearing in (5.3).This and the exact sequence (5.2) for L = F d−1 , show that A(F d )/I d d−1 A(F d) ≃ A(F d−1 ) (forany d ≥ 1).From (5.11) one finally obtains(5.14) Ch Λd−1 (A(F d−1 )) = Ch Λd−1 (A(F d )/I d d−1 A(F d)) = π d d−1 (Ch Λ d(A(F d ))) .We remark that this equation holds for any Z p -extension F d /F d−1 satisfying Assumption5.3. If the filtration {F d : d ∈ N} verifies that Assumption at any level d, then the inverseimages of the Ch Λ(Fd )(A(F d )) in Λ (with respect to the canonical projections π Fd : Λ →Λ(F d ) ) form an inverse system and we can defineDefinition 5.10. The pro-characteristic ideal of A(F) is Z p˜Ch Λ (A(F)) := lim←−Fd(π Fd ) −1 (Ch Λ(Fd )(A(F d ))) .Remark 5.11. Two questions naturally arise from the above definition:a. is there a filtration verifying Assumption 5.3 at any level d?b. (assuming a has a positive answer) is the limit independent from the chosen filtration?In the next section we are going to show (in particular in (5.17) and Corollary 5.16) that thereis an element θ ∈ Λ, independent of the filtration and such that, for all F d , its image in Λ(F d )generates Ch Λ(Fd )(A(F d )). Hence Question b has a positive answer (and presumably so doesQuestion a) and we only needed (5.14) as a first step and a natural analogue of (5.16).Nevertheless we believe that these questions have some interest on their own and it would benice to have a direct construction of a “good” filtration {F d : d ∈ N} based on a generalizationof [20, Lemma 2]. Since our goal here is the Main Conjecture we do not pursue this subjectfurther, but we hope to get back to it in a future paper.We also observe that Assumption 5.3 was used only in one passage in the proof of Lemma5.4, as we evidentiated in a footnote. It might be easier to show that in that passage onedoes not need the finitely generated hypothesis: if so, Definition 5.10 would makes sense forall filtrations {F d } d . We are currently considering this approach.Remark 5.12. We could have used Fitting ideals, just as we did in section 3, to provide amore straightforward construction (there would have been no need for preparatory lemmas).But, since the goal is a Main Conjecture, the characteristic ideals, being principal, provide abetter formulation. We indeed expect equality between Fitting and characteristic ideals in allthe cases studied in this paper but, at present, are forced to distinguish between them (butsee Remark 5.17).

IWASAWA THEORY OVER FUNCTION FIELDS 205.3. Stickelberger elements. We shall briefly describe a relation between the characteristicideal of the previous section and Stickelberger elements. The main results on those elementsare due to A. Weil, P. Deligne and J. Tate and for all the details the reader can consult [45,Ch. V]. Let S be a finite set of places of F containing all places where the extension F/Framifies; since we are interested in the case where F is substantially bigger than the arithmeticextension, we assume S ≠ ∅. We consider also another non-empty finite set T of places of Fsuch that S ∩ T = ∅. For any place outside S let F r v be the Frobenius of v in Γ = Gal(F/F ).Let(5.15) Θ F/F,S,T (u) := ∏ v∈T(1 − F r v q deg(v) u deg(v) ) ∏ v∉S(1 − F r v u deg(v) ) −1 .For any n ∈ N there are only finitely many places of F with degree n: hence we canexpand (5.15) and consider Θ F/F,S,T (u) as a power series ∑ c n u n ∈ Z[Γ][[u]]. Moreover, itis clear that for any continuous character ψ : Γ −→ C ∗ the image ψ(Θ F/F,S,T (q −s )) is theL-function of ψ, relative to S and modified at T . For any subextension F ⊂ L ⊂ F, let: Z[Γ][[u]] → Z[Gal(L/F )][[u]] be the natural projection and defineπ F LΘ L/F,S,T (u) := π F L (Θ F/F,S,T (u)).For L/F finite it is known (essentially by Weil’s work) that Θ L/F,S,T (q −s ) is an element in thepolynomial ring C[q −s ] (see [45, Ch. V, Proposition 2.15] for a proof): hence Θ L/F,S,T (u) ∈Z[Gal(L/F )][u]. It follows that the coefficients c n of Θ F/F,S,T (u) tend to zero inTherefore we can definelim←Z[Gal(L/F )] =: Z[[Γ]] ⊂ Λ .θ F/F,S,T := Θ F/F,S,T (1) ∈ Λ .We also observe that the factors (1 − F r v q deg(v) u deg(v) ) in (5.15) are units in the ring Λ[[u]].Hence the ideal generated by θ F/F,S,T is independent of the auxiliary set T and we can definethe Stickelberger element∏θ F/F,S := θ F/F,S,T (1 − F r v q deg(v) ) −1 .We also define, for F ⊂ L ⊂ F,v∈Tθ L/F,S,T := π F L (θ F/F,S,T ) = Θ L/F,S,T (1).It is clear that these form a projective system: in particular, for any Z p -extension F d /F d−1the relation(5.16) π d d−1 (θ F d /F,S,T ) = θ Fd−1 /F,S,Tclearly recalls the one satisfied by characteristic ideals (equation (5.14)). Also, to define θ L/F,Sthere is no need of F: one can take for a finite extension L/F the analogue of product (5.15)and reason as above.Theorem 5.13 (Tate, Deligne). For any finite extension L/F one has that |F ∗ L |θ L/F,S is inthe annihilator ideal of the class group of L (considered as a Z[Gal(L/F )]-module).Proof. This is [45, Ch. V, Théorème 1.2].Remark 5.14. Another proof of this result was given by Hayes [22], by means of Drinfeldmodules.Corollary 5.15. Let F d /F be a Z d p-extension as before and S = {p}, the unique (totally)ramified prime in F/F : then1. θ Fd /F,SA(F d ) = 0;□

IWASAWA THEORY OVER FUNCTION FIELDS 212. if θ Fd /F,S is irreducible in Λ(F d ), then Ch Λ(Fd )(A(F d )) = (θ Fd /F,S) m for some m ≥ 1;3. if θ Fd /F,S is irreducible in Λ(F d ) for all F d ’s, then ˜Ch Λ (A(F)) = (θ F/F,S ) m for somem ≥ 1.Proof. For 1 one just notes that |F ∗ L| is prime with p. Part 2 follows from the structuretheorem for torsion Λ(F d )-modules. Part 3 follows from 2 by taking limits (as in Definition5.10) and noting that the m is constant through the F d ’s because of equations (5.14) and(5.16). □The exponent in 2 and 3 of the corollary above is actually m = 1. A proof of this fact isbased on the following technical result of [28] (generalized in [9, Theorem A.1]). Once F d isfixed it is always possible to find a Z c p-extension of F containing F d , call it L d , such that:a. the extension L d /F is ramified at all primes of a finite set ˜S containing S (moreover˜S can be chosen arbitrarily large);b. the Stickelberger element θ Ld /F, ˜S is irreducible in the Iwasawa algebra Λ(L d);c. there is a Z p -extension L ′ of F contained in L d which is ramified at all primes of ˜S andsuch that the Stickelberger element θ Lis monomial, i.e., congruent to u(σ − ′ 1)r/F, ˜Smodulo (σ − 1) r+1 (where σ is a topological generator of Gal(L ′ /F ) and u ∈ Z ∗ p ).With condition b and an iteration of equation (5.14) one proves thatCh Λ(Fd )(A(F d )) = (θ Fd /F,S) m for some m ≥ 1 .The monomiality condition c (using L ′ as a first layer in a tower of Z p -extensions) leads tom = 1 (see [27, section 4] or [9, section A.1] which uses the possibility of varying the set˜S, provided by a, more directly). We remark that the proof only uses the irreducibility ofθ Ld, i.e.,/F, ˜S(5.17) Ch Λ(Fd )(A(F d )) = (θ Fd /F,S)holds in general for any F d .Corollary 5.16 (Iwasawa Main Conjecture). In the previous setting we have˜Ch Λ (A(F)) = (θ F/F,p ) .Proof. From the main result of [27], one has thatand we take the limit in both sides.Ch Λ(Fd )(A(F d )) = (θ Fd /F,p)Remark 5.17. The equality between characteristic ideals and ideals generated by Stickelbergerelements has been proved by K.-L. Kueh, K. F. Lai and K.-S. Tan ([27]) and by D.Burns ([8] and the Appendix coauthored with K.F. Lai and K-S. Tan [9]) in a more generalsituation. The Z d p-extension they consider has to be unramified outside a finite set S of primesof F (but there is no need for the primes to be totally ramified). Moreover they require thatnone of the primes in S is totally split (otherwise θ Fd /F,S = 0). The strategy of the proofis basically the same but, of course, many technical details are simplified by our choice ofhaving just one (totally) ramified prime (just compare, for example, Lemma 5.7 with [27,Lemma 3.3 and 3.4]). Moreover, going back to the Fitting vs. characteristic ideal situation,it is worth noticing that Burns proves that the first cohomology group of certain complexes(strictly related to class groups, see [8, Proposition 4.4] and [9, section A.1]) are of projectivedimension 1 ([8, Proposition 4.1]). In this case the Fitting and characteristic ideals are knownto be equal to the inverse of the Knudsen-Mumford determinant (the ideal by which all theresults of [8] are formulated).□

IWASAWA THEORY OVER FUNCTION FIELDS 225.4. Characteristic p L-functions. One of the most fascinating aspects of function fieldarithmetic is the existence, next to complex and p-adic L-functions, of their characteristic pavatars. For a thorough introduction the reader is referred to [18, Chapter 8]: here we justprovide a minimal background.Recall our fixed place ∞ and let C ∞ denote the completion of an algebraic closure of F ∞ .Already Carlitz had studied a characteristic p version of the Riemann zeta function, definedon N and taking values in C ∞ (we will say more about it in section 6.6). More recently Gosshad the intuition that, like complex and p-adic L-functions have as their natural domainsrespectively the complex and the p-adic (quasi-)characters of the Weil group, so one couldconsider C ∞ -valued characters. In particular, the analogue of the complex plane as domainfor the characteristic p L-functions is S ∞ := C ∗ ∞ × Z p , that can be seen as a group of C ∞ -valued homomorphisms on F ∗ ∞, just as for s ∈ C one defines x ↦→ x s on R + . The additivegroup Z embeds discretely in S ∞ . Similarly to the classical case, one can define L(ρ, s) forρ a compatible system of v-adic representation of G F (v varying among places different from∞) by Euler products converging on some “half-plane” of S ∞ .The theory of zeta values in characteristic p is still quite mysterious and at the moment wecan at best speculate that there are links with the Iwasawa theoretical questions consideredin this paper 4 . To the best of our knowledge, the main results available in this direction arethe following. Let F (p)/F be the extension obtained from the p-torsion of a Drinfeld-Hayesmodule (in the simplest case, F (p) is the F 1 we are going to introduce in section 6.1). Gossand Sinnott have studied the isotypic components of A(F (p)) and shown that they are nonzeroif and only if p divides certain characteristic p zeta values: see [18, Theorem 8.14.4] fora precise statement. Note that the proof given in [18], based on a comparison between thereductions of a p-adic and a characteristic p L-function respectively mod p and mod p ([18,Theorem 8.13.3]), makes use of Crew’s result. Okada [36] obtained a result of similar flavorfor the class group of the ring of “integers” of F (p) when F is the rational function field,and Shu [42] extended it to any F ; since Okada’s result is strictly related with the subject ofsection 6.6 below, we will say more about it there.6. Cyclotomy by the Carlitz module6.1. Setting. From now on we take F = F q (T ) and let ∞ be the usual place at infinity, so thatthe ring of elements regular outside ∞ is A := F q [T ]: this allows a number of simplifications,leaving intact the main aspects of the theory. The “cyclotomic” theory of function fields isobtained via Drinfeld-Hayes modules: in the setting of the rational function field the onlyone is the Carlitz module Φ: A → A{τ}, T ↦→ Φ T := T + τ (here τ denotes the operatorx ↦→ x q and, if R is an F p -algebra, R{τ} is the ring of skew polynomials with coefficients inR: multiplication in R{τ} is given by composition).We also fix a prime p ⊂ A and let π ∈ A be its monic generator. In order to underline thefact that A and its completion at p play the role of Z and Z p in the Drinfeld-Hayes cyclotomictheory, we will often use the alternative notation A p for the ring of local integers O p ⊂ F p .Let C p be the completion of an algebraic closure of F p .As usual, if I is an ideal of A, Φ[I] will denote the I-torsion of Φ (i.e., the common zeroesof all Φ a , a ∈ I). One checks immediately that if ι is the unique monic generator of I thenΦ ι (x) =∏(x − u) .u∈Φ[I]4 [Note added in proof] This field is in rapid evolution. After this paper was written, L. Taelman introducedsome important new ideas: see [L. Taelman, Special L-values of Drinfeld modules. To appear in Annals ofMath. 175 (2012), 369–391] and [L. Taelman, A Herbrand-Ribet theorem for function fields. To appear inInvent. Math., Online First DOI: 10.1007/s00222-011-0346-3].

IWASAWA THEORY OVER FUNCTION FIELDS 23We putF n := F (Φ[p n ])andK n := F p (Φ[p n ]) .As stated in §1.2, we think of the F n ’s as subfields of C p , so that the K n ’s are their topologicalclosures. We shall denote the ring of A-integers in F n by B n and its closure in K n by O n , andwrite U n for the 1-units in O n . Let F := ∪F n and ˜Γ := Gal(F/F ).Consider the ring of formal skew power series A p {{τ}}: it is a complete local ring, withmaximal ideal πA p +A p {{τ}}τ. It is easy to see that Φ extends to a continuous homomorphismΦ: A p → A p {{τ}} (i.e., a formal Drinfeld module) and this allows to define a “cyclotomic”character χ: ˜Γ −→ ∼ A ∗ p. More precisely, let T p Φ := lim Φ[p n ] (the limit is taken with respect to←x ↦→ Φ π (x)) be the Tate module of Φ. The ring A p acts on T p Φ via Φ, i.e., a·(u) n := (Φ a (u n )) n ,and the character χ is defined by σu =: χ(σ) · u, i.e., χ(σ) is the unique element in A ∗ p suchthat Φ χ(σ) (u n ) = σu n for all n. From this it follows immediately that ˜Γ = ∆ × Γ, where∆ ≃ F ∗ p is a finite group of order prime to p and Γ is the inverse image of the 1-units.Since Φ has rank 1, T p Φ is a free A p -module of rank 1. As in [6], we fix a generatorω = (ω n ) n≥1 : this means that the sequence {ω n } satisfiesΦ π n(ω n ) = 0 ≠ Φ π n−1(ω n ) and Φ π (ω n+1 ) = ω n .By definition K n = F p (ω n ). By Hayes’s theory, the minimal polynomial of ω n over Fis Eisenstein: it follows that the extensions F n /F and K n /F p are totally ramified, ω n is auniformizer for the field K n , O n = A p [[ω n ]] = A p [ω n ]. The extension F n /F is unramified atall other finite places: this can be seen directly by observing that Φ π n has constant coefficientπ n . Furthermore F n /F is tamely ramified at ∞ with inertia group I ∞ (F n /F ) ≃ F ∗ q.The similarity with the classical properties of Q(ζ p n)/Q is striking.The formula N Fn+1 /F n(ω n+1 ) = ω n shows that the ω n ’s form a compatible system underthe norm maps (the proof is extremely easy; it can be found in [6, Lemma 2]). This and theobservation that [F n+1 : F n ] = q deg(p) for n ≥ 1 imply(6.1) lim←K ∗ n = ω Z × F ∗ p × lim←U n .Note that lim←U n is a ˜Λ-module.6.2. Coleman’s theory. A more complete discussion and proofs of results in this sectioncan be found in [6, §3]. Let R be a subring of C p : then, as usual, R((x)) := R[[x]](x −1 ) is thering of formal Laurent series with coefficients in R. Moreover, following [11] we define R[[x]] 1and R((x)) 1 as the subrings consisting of those (Laurent) power series which converge on thepunctured open ballB ′ := B(0, 1) − {0} ⊂ C p .The rings R[[x]] 1 and R((x)) 1 are endowed with a structure of topological R-algebras, inducedby the family of seminorms {‖ · ‖ r }, where r varies in |C p | ∩ (0, 1) and ‖f‖ r := sup{|f(z)| :|z| = r}.All essential ideas for the following two theorems are due to Coleman [11].Theorem 6.1. There exists a unique continuous homomorphismsuch thatN : F p ((x)) ∗ 1 → F p ((x)) ∗ 1∏f(x + u) = (N f) ◦ Φ π .u∈Φ[p]

IWASAWA THEORY OVER FUNCTION FIELDS 24Theorem 6.2. The evaluation map ev : f ↦→ {f(ω n )} gives an isomorphism(A p ((x)) ∗ ) N =id ≃ lim←K ∗ nwhere the inverse limit is taken with respect to the norm maps.We shall write Col u for the power series in A p ((x)) ∗ associated to u ∈ lim←K ∗ n by Coleman’sisomorphism of Theorem 6.2.Remark 6.3. An easily obtained family of N -invariant power series is the following. Leta ∈ A ∗ p: then∏Φ a (x + u) =∏(Φ a (x) + Φ a (u)) = Φ π (Φ a (x))u∈Φ[p]u∈Φ[p](since Φ a permutes elements in Φ[p]) and from Φ π Φ a = Φ a Φ π in A p {{τ}} it follows that Φ a (x)is invariant under the Coleman norm operator N . (As observed in [6, page 797], this justamounts to replacing ω with a · ω as generator of the Tate module.)Following [11], we define an action of Γ on F p [[x]] 1 by (σ ∗ f)(x) := f(Φ χ(σ) (x)). ThenCol σu = (σ ∗ Col u ), as one sees from(6.2) (σ ∗ Col u )(ω n ) = Col u (Φ χ(σ) (ω n )) = Col u (σω n ) = σ(Col u (ω n )) = σ(u n ).6.3. The Coates-Wiles homomorphisms. We introduce some operators on power series.Let dlog : F p ((x)) ∗ 1 → F p((x)) 1 be the logarithmic derivative, i.e., dlog (g) := g′g. Also, forany j ∈ N let ∆ j : F p ((x)) → F p ((x)) be the jth Hasse-Teichmüller derivative, defined by theformula( ∞)∑∞∑( )∆ j c n x n n + j:=cj n+j x nn=0d jn=0(i.e., ∆ j “is” the differential operator 1 j!). A number of properties of the Hasse-Teichmüllerdx jderivatives can be found in [24]; here we just recall that the operators ∆ j are F p -linear andthat∞∑(6.3) f(x) = ∆ j (f) |x=0 x j .j=0The last operator we need to introduce is composition with the Carlitz exponential e C (x) =x + . . . , i.e., f ↦→ f(e C (x)).Definition 6.4. For any integer k ≥ 1, define the kth Coates-Wiles homomorphism δ k : lim←O ∗ n →F p byδ k (u) := ∆ k−1((dlog Colu )(e C (x)) ) |x=0 = ( ∆ k−1 ((dlog Col u ) ◦ e C ) ) (0) .Notice that by (6.3) this is equivalent to putting∞∑(6.4) (dlog Col u )(e C (x)) = δ k (u)x k−1 .k=1Lemma 6.5. The Coates-Wiles homomorphisms satisfyδ k (σu) = χ(σ) k δ k (u).Proof. Recall that ddx Φ a(x) = a for any a ∈ A p . Then from (6.2) it followsdlog Col σu = dlog (Col u ◦ Φ χ(σ) ) = χ(σ)(dlog Col u ) ◦ Φ χ(σ) ,

since dlog (f ◦ g) = g ′ ( f ′fby (6.4),IWASAWA THEORY OVER FUNCTION FIELDS 25◦ g). Composing with e C and using Φ a (e C (x)) = e C (ax), one gets,(dlog Col σu )(e C (x)) = χ(σ)(dlog Col u )(e C (χ(σ)x)) = χ(σ)The result follows.6.4. Cyclotomic units.∞∑δ k (u)χ(σ) k−1 x k−1 .Definition 6.6. The group C n of cyclotomic units in F n is the intersection of B ∗ n with thesubgroup of F ∗ n generated by σ(ω n ), σ ∈ Gal(F n /F ).By the explicit description of the Galois action via Φ, one sees immediately that this is thesame as B ∗ n ∩ 〈Φ a (ω n )〉 a∈A−p .Lemma 6.7. Let ∑ c σ σ be an element in Z[Gal(F n /F )]: then∏σ(ω n ) cσ ∈ C n ⇐⇒ ∑ c σ = 0 .σ∈Gal(F n/F )Proof. Obvious from the observation that ω n is a uniformizer for the place above p and a unitat every other finite place of F n .□Let C n and C 1 n denote the closure respectively of C n ∩ O ∗ n and of C 1 n := C n ∩ U n .Let a ∈ A ∗ p. By Remark 6.3 (Φ a (ω n )) n is a norm compatible system: hence one can definea homomorphismΥ: Z[˜Γ] −→ lim K ∗←n∑cσ σ ↦→ ∏ (σ(ωn ) cσ) n = ∏ (Φχ(σ) (ω n ) cσ) nLet lim ̂ K ∗←n be the p-adic completion of lim K ∗←n. By (6.1) one gets the isomorphism lim ̂ K ∗←n ≃ω Zp × lim←U n .Lemma 6.8. The restriction of Υ to Z[Γ] can be extended to Υ: Λ → ̂ lim←K ∗ nProof. If a ∈ A p is a 1-unit, then(6.5) Φ a (ω n ) = ω n u nwith u n ∈ U n . Since by definition Γ = χ −1 (1 + πA p ), it follows that Υ sends Z[Γ] intoω Z × lim←U n . To complete the proof it suffices to check that Υ is continuous with respect tothe natural topologies on Λ = lim(Z/p n Z)[Gal(F n /F )] and lim ̂ K ∗← ←n. But a ≡ a ′ mod π n in A pimplies Φ a (ω j ) = Φ a ′(ω j ) for any j ≤ n; and the result follows from the continuity of χ. □Proposition 6.9. Let I ⊂ Λ denote the augmentation ideal: then Υ induces a surjectivehomomorphism of Λ-modules I −→ lim←C 1 n. The kernel has empty interior.Proof. From Lemma 6.7 and (6.5) it is clear that Υ(α) ∈ lim←C 1 n if and only if α ∈ I. Thismap is surjective because I is compact and already the restriction to the augmentation idealof Z[Γ] is onto Cn 1 for all n. A straightforward computation shows that it is a homomorphismof Λ-algebras: for γ ∈ Γ(∑ ) ( (∏ ) ) (∏ )(6.6) γΥ cσ σ = γ Φχ(σ) (ω n ) cσ = Φχ(σγ) (ω n ) cσnnbecause γ(Φ a (ω n )) = Φ a (γ(ω n )) = Φ a (Φ χ(γ) (ω n )).k=1□

IWASAWA THEORY OVER FUNCTION FIELDS 26For the statement about the kernel, let A + ⊂ A be the subset of monic polynomials andconsider∏any function A + −→ Z, a ↦→ n a , such that n a = 0 for almost all a. We claim thata∈A + Φ a(x) na = 1 only if n a = 0 for all a. To see it, let u a denote a generator of the cyclicA-module Φ[(a)]. Then x − u a divides Φ b (x) if and only if b ∈ (a): hence the multiplicitym a of u a as root of ∏ Φ a (x) na = 1 is exactly ∑ b∈(a)∩A + n b. For b ∈ A, let ε(b) denote thenumber of primes of A dividing b (counted with multiplicities): then a simple combinatorialargument shows thatn a = ∑(−1) ∑ ε(b) n abc .b∈A + c∈A +It follows that m a = 0 for all a ∈ A + if and only if n a = 0 for all a.As in §5.3, for v ≠ p, ∞ let F r v ∈ ˜Γ be its Frobenius. By [18, Proposition 7.5.4] one findsthat χ(F r v ) is the monic generator of the ideal in A corresponding to the place v: hence byChebotarev χ −1 (A + ) is dense in ˜Γ. Thus the isomorphism of Theorem 6.2 shows that wehave proved that Υ: I −→ lim←O ∗ n is injective on a dense subset: the kernel must have emptyinterior.Remark 6.10. Since I ⊗ Z Z[∆] = ⊕ δ∈∆ Iδ, formula (6.6) shows that Υ can be extended to ahomomorphism of ˜Λ-modules I ⊗ Z Z[∆] −→ lim←C n .Proposition 6.11. We have: lim←O ∗ n/ lim←C n ≃ lim←U n / lim←C 1 n .Proof. Consider the commutative diagram1 −−−−→ lim C 1←n −−−−→ lim U n −−−−→ lim U n / lim C 1← ← ←n −−−−→ 1⏐↓ α ⏐1 ↓ α ⏐2↓ α 31 −−−−→ lim←C n −−−−→ lim←O ∗ n −−−−→ lim←O ∗ n/ lim←C n −−−−→ 1 .All vertical maps are injective and by (6.1) the cokernel of α 2 is F ∗ p. For δ ∈ ∆ one hasδω n = Φ χ(δ) (ω n ) = χ(δ)ω n u n for some u n ∈ U n . By the injectivity part of the proof ofProposition 6.9, C n = Cn 1 × Υ(Z[∆]) and it follows that the cokernel of α 1 is also F ∗ p. □6.5. Cyclotomic units and class groups. Let F n + ⊂ F n be the fixed field of the inertiagroup I ∞ (F n /F ). The extension F n + /F is totally split at ∞ and ramified only above theprime p. We shall denote the ring of A-integers of F n+ by B n + . Also, define E n and En 1 to bethe closure respectively of Bn ∗ ∩ On ∗ and Bn ∗ ∩ U n .We need to introduce a slight modification of the groups A(L) of section 5. For any finiteextension L/F , A ∞ (L) will be the p-part of the class group of A-integers of L, so that, byclass field theory, A ∞ (L) ≃ Gal(H(L)/L), where H(L) is the maximal abelian unramifiedp-extension of L which is totally split at places dividing ∞. We shall use the shorteningA n := A ∞ (F n + ).Also, let X n := Gal(M(F n + )/F n + ), where M(L) is the maximal abelian p-extension of Lunramified outside p and totally split above ∞. As in the number field case, one has an exactsequence(6.7) 1 −−−−→ E 1 n/C 1 n −−−−→ U n /C 1 n −−−−→ X n −−−−→ A n −−−−→ 1coming from the followingProposition 6.12. There is an isomorphism of Galois modulesU n /E 1 n ≃ Gal(M(F + n )/H(F + n )) .□

IWASAWA THEORY OVER FUNCTION FIELDS 27Proof. This is a consequence of class field theory in characteristic p > 0, as the analogousstatement in the number field case: just recall that the role of archimedean places is nowplayed by the valuations above ∞. Under the class field theoretic identification of ideleclasses I F+n/(F n + ) ∗ with a dense subgroup of Gal((F n + ) ab /F n + ), one finds a surjection∏OP ∗ ↠ Gal(M(F n + )/H(F n + ))P|pwhose kernel contains the closure of∏P|pO ∗ P ∩ (F + n ) ∗ ∏ w∤p∏(F n,w) + ∗ = ι p ((B n + ) ∗ )Ow∗ w|∞(where ι p denotes the diagonal inclusion). Reasoning as in [48, Lemma 13.5] one proves theproposition.□Taking the projective limit of the sequence (6.7), we get(6.8) 1 −−−−→ E 1 ∞/C 1 ∞ −−−−→ U ∞ /C 1 ∞ −−−−→ X ∞ −−−−→ A ∞ −−−−→ 1 .Lemma 6.13. The sequence (6.8) is exact.Proof. Taking the projective limit of the short exact sequencewe obtain1 −−−−→ E 1 n −−−−→ U n −−−−→ Gal(M(F + n )/H(F + n )) −−−−→ 11 −−−−→ E 1 ∞ −−−−→ U ∞ −−−−→ Gal(M(F + )/H(F + )) −−−−→ lim← 1 E 1 n ,where M(F + ) and H(F + ) are the maximal abelian p-extensions of F + totally split above ∞and unramified respectively outside the place above p and everywhere.To prove the lemma it is enough to show that lim← 1 (E 1 n) = 1. By a well-known result inhomological algebra, the functor lim← 1 is trivial on projective systems satisfying the Mittag-Leffler condition. We recall that an inverse system (B n , d n ) enjoys such property if for any nthe images of the transition maps B n+m → B n are the same for large m. So we are reducedto check that this holds for the En’s 1 with the norm maps.Observe first that En 1 is a finitely generated Λ n -module, thus noetherian because so isΛ n . Consider now ∩ k Image(N n+k,n ), where N n+k,n : En+k 1 → E1 n is the norm map. Thisintersection is a Λ n -submodule of En, 1 non-trivial because it contains the cyclotomic units.By noetherianity it is finitely generated, hence there exists l such that Image(N n+k,n ) is thesame for k ≥ l. Therefore (En) 1 satisfies the Mittag-Leffler property.□The exact sequence (6.8) lies at the heart of Iwasawa theory. Its terms are all Λ-modulesand, in section 5.2, we have shown how to associate a characteristic ideal to A ∞ and its closerelation with Stickelberger elements. In a similar way, i.e., working on Z d p-subextensions,one might approach a description of X ∞ , while, for the first two terms of the sequence, thefiltration of the F n + ’s seems more natural (as the previous sections show).Assume for example that the class number of F is prime with p, then it is easy to see thatA n = 1 for all n. Moreover, using the fact that, by a theorem of Galovich and Rosen, theindex of the cyclotomic units is equal to the class number (see [41, Theorem 16.12]), one canprove that En/C 1 n 1 = 1 as well. These provide isomorphismsandU n /C 1 n ≃ X nU ∞ /C 1 ∞ ≃ X ∞ .

IWASAWA THEORY OVER FUNCTION FIELDS 28In general one expects a relation (at least at the level of Z d p-subextensions, then a limitprocedure should apply) between the pro-characteristic ideal of A ∞ and the (yet to be defined)analogous ideal for E 1 ∞/C 1 ∞ (the Stickelberger element might be a first hint for the study of thisrelation). Consequently (because of the multiplicativity of characteristic ideals) an equalityof (yet to be defined) characteristic ideals of X ∞ and of U ∞ /C 1 ∞ is expected as well. Anyof those two equalities can be considered as an instance of Iwasawa Main Conjecture for thesetting we are working in.6.6. Bernoulli-Carlitz numbers. We go back to the subject of characteristic p L-function.Let A + ⊂ A be the subset of monic polynomials. The Carlitz zeta function is definedζ A (k) := ∑ 1a ka∈A +for k ∈ N.Recall that the Carlitz module corresponds to a lattice ξA ⊂ C ∞ and can be constructedvia the Carlitz exponential e C (z) := z ∏ a∈A ′(1 − zξ−1 a −1 ) (where A ′ denotes A − {0}).Rearranging summands in the equality1e C (z) = dlog (e C(z)) = ∑ a∈A1z − ξa = 1 z − ∑ a∈A ′(and using A ′ = F ∗ q × A + ) one gets the well-known formula(6.9)1e C (z) = 1 ∞ z + ∑n=1∞ ∑k=1ζ A (n(q − 1))ξ n(q−1) z n(q−1)−1 .z k−1(ξa) kFrom §6.4 it follows that for any a, b ∈ A − p, the function Φa(x)Φ b (x)is an N -invariant powerseries, associated with(6.10) c(a, b) := Φ ( )a(ω)Φ b (ω) = Φa (ω n )∈ lim O ∗Φ b (ω n ) ←n .Theorem 6.14. The kth Coates-Wiles homomorphism applied to c(a, b) is equal to:{0 if k ≢ 0 mod q − 1δ k (c(a, b)) =(a k − b k ) ζ A(k).if k ≡ 0 mod q − 1.ξ kWe remark that the condition k = n(q − 1) here is the analogue of k being an even integerin the classical setting (since q − 1 = |F ∗ q| just as 2 = |Z ∗ |).Proof. Observe that (6.10) amounts to giving the Coleman power series Col c(a,b) .Let λ be the Carlitz logarithm: i.e., λ ∈ F {{τ}} is the element uniquely determined bye C ◦ λ = 1. Then Φ a (x) = e C (aλ(x)) and by (6.10) and (6.9) one getsdlog Col c(a,b) (x) =Since λ(e C (x)) = x, we getaΦ a (x) −(6.11) (dlog Col c(a,b) )(e C (x)) = ∑ n≥1nbΦ b (x) = ∑ (a n(q−1) − b n(q−1) ) ζ A(n(q − 1))ξ n(q−1) λ(x) n(q−1)−1 .n≥1(a n(q−1) − b n(q−1) ) ζ A(n(q − 1))ξ n(q−1)x n(q−1)−1and the theorem follows comparing (6.11) with (6.4).□

IWASAWA THEORY OVER FUNCTION FIELDS 29Remark 6.15. As already known to Carlitz, ζ A (k)ξ −k is in F when q − 1 divides k. Notethat by a theorem of Wade, ξ ∈ F ∞ is transcendental over F . Furthermore, Jing Yu [50]proved that ζ A (k) for all k ∈ N and ζ A (k)ξ −k for k “odd” (i.e., not divisible by q − 1) aretranscendental over F .Theorem 6.14 can be restated in terms of the Bernoulli-Carlitz numbers BC k [18, Definition9.2.1]. They can be defined by1e C (z) = ∑ n≥0BC nΠ(n) zn−1(where Π(n) is a function field analogue of the classical factorial n!); in particular BC n = 0when n ≢ 0 (mod q − 1). Then Theorem 6.14 becomes(6.12) δ k (c(a, b)) = (a k − b k ) BC kΠ(k) .Theorem 6.14 and formula (6.12) can be seen as extending a result by Okada, who in [36]obtained the ratios BC kΠ(k) (for k = 1, . . . , qdeg(p) − 2) as images of cyclotomic units under theKummer homomorphisms (which are essentially a less refined version of the Coates-Wileshomomorphisms). From here one proves that the non-triviality of an isotypic component ofA 1 implies the divisibility of the corresponding “even” Bernoulli-Carlitz number by p: werefer to [18, §8.20] for an account. As already mentioned, Shu [42] generalized Okada’s workto any F (but with the assumption deg(∞) = 1): it might be interesting to extend Theorem6.14 to a “Coates-Wiles homomorphism” version of her results.6.7. Interpolation? In the classical setting of cyclotomic number fields, the analogue of theformula in Theorem 6.14 can be used as a key step in the construction of the Kubota-Leopoldtzeta function (see e.g. [10]). Hence it is natural to wonder if something like it holds in ourfunction field case. For now we have no answer and can only offer some vague speculation.As mentioned in section 5.4, Goss found a way to extend the domain of ζ A from N to S ∞ .He also considered the analogue of the p-adic domain and defined it to be C ∗ p × S p , withS p := Z p × Z/(q deg(p) − 1) (observe that C ∗ p × S p is the C p -valued dual of Fp ∗ ). Then functionslike ζ A enjoy also a p-adic life: for example, letting π v ∈ A + be a uniformizer for a place v,ζ A,p is defined on C ∗ p × S p byζ A,p (s) := ∏(1 − πv −s ) −1 ,v∤p∞at least where the product converges.The ring Z embeds discretely in S ∞ and has dense image in 1 × S p . So Theorem 6.14seems to suggest interpolation of ζ A,p on 1 × S p . Another clue in this direction is the fact thatS p is the “dual” of Γ, just as Z p is the “dual” of Gal(Q(ζ p ∞)/Q). (A strengthening of thisinterpretation has been recently provided by the main result of [25].)Acknowledgments. We thank Sangtae Jeong, King Fai Lai, Jing Long Hoelscher, Ki-SengTan, Dinesh Thakur, Fabien Trihan for useful conversations and Bruno Anglès for pointingout a mistake in the first version of this paper. F. Bars and I. Longhi thank CRM for providinga nice environment to complete work on this paper.References[1] P.N. Balister and S. Howson, Note on Nakayama’s lemma for compact Λ-modules. Asian J. Math. 1 (1997),224–229.[2] A. Bandini and I. Longhi, Control theorems for elliptic curves over function fields. Int. J. Number Theory5 (2009), 229–256.

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