CHARACTERISTIC IDEALS AND SELMER GROUPS ANDREA BANDINI, FRANCESC BARS, AND IGNAZIO LONGHI Abstract. Let A be an abelian variety defined over a global field F of positive characteristic p and let F /F be a ZNp -extension, unramified outside a finite set of places of F . Assuming that all ramified places are totally ramified, we define an algebraic element associated to the Pontrjagin dual of the p-primary Selmer group of A in order to formulate an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra Zp [[Gal(F /F )]] (which we also prove for a constant abelian variety). To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a Zdp -extension Fd /F and for any Zd−1 -extension contained in Fd , and then use a limit process. p 1. Introduction Let F be a global function field of characteristic p and F/F a ZN p -extension unramified outside a finite set of places. We take an abelian variety A defined over F and let SA be a finite set of places of F containing exactly the ramified primes and the primes of bad reduction for A. For any infinite Galois extension L/F , let Λ(L) := Zp [[Gal(L/F )]] be the associated Iwasawa algebra: we recall that, if Gal(L/F ) ≃ Zdp , then Λ(L) ≃ Zp [[t1 , .., td ]] is a Krull domain. We are interested in the definition of a characteristic ideal in Λ(F) for the Pontrjagin dual of the Selmer group SelA (F)p . For any extension v of some place of F to the algebraic closure F and for any finite extension E/F , we denote by Ev the completion of E with respect to v and we put Fd,v := ∪Ev , where the union is taken over all finite subextensions. We define the p-part of the Selmer group of A over E as ) ( Y Sel(E) := SelA (E)p = Ker Hf1l (XE , A[p∞ ]) −→ Hf1l (XEv , A)[p∞ ] v e:1 (where the map is the product of the natural restrictions at all places v of E, XE := Spec(E) and Hf1l denotes flat cohomology). For infinite algebraic extensions we define the Selmer groups by taking direct limits on all the finite subextensions as usual. For any algebraic extension K/F , let S(K) denote the Pontrjagin dual of Sel(K) (other Pontrjagin duals will be indicated by the symbol ∨ ). If R is a noetherian Krull domain and M a finitely generated torsion R-module, the structure theorem for M provides an exact sequence n M (1.1) 0 −→ P −→ M −→ R/pei i R −→ Q −→ 0 i=1 where the pi ’s are height 1 prime ideals of R and P and Q are pseudo-null R-modules (i.e., torsion modules with annihilator of height at least 2). With this sequence one defines the characteristic ideal of M as n Y ChR (M ) := pei i i=1 (if M is not torsion, we put ChR (M ) = 0 and if M is pseudo-null, then ChR (M ) = (1) ). SUPPORTED BY MTM2009-10359 1 CHARACTERISTIC IDEALS AND SELMER GROUPS 2 In commutative Iwasawa theory characteristic ideals provide the algebraic counterpart for the p-adic L-functions associated to Iwasawa modules (such as duals of Selmer groups). We fix a Zp -basis {γi }i∈N for Γ and for any d > 0 we let Fd ⊂ F be the fixed field of {γi }i>d . The correspondence γi − 1 ↔ ti provides an isomorphism between the subring Λ(Fd ) of Λ(F) and Zp [[t1 , . . . , td ]]: by a slight abuse of notation, the two shall be identified in this paper. In particular, for any d > 1 we have Λ(Fd ) = Λ(Fd−1 )[[td ]] (we remark that the isomorphism is uniquely determined once the γi have been fixed, but we allow complete freedom in their initial choice). Our goal is to define an element in Λ(F) associated to S(F) via a limit of the ChΛ(Fd ) (S(Fd )). d d Let πd−1 be the canonical projection Λ(Fd ) → Λ(Fd−1 ) with kernel Id−1 = (td ) and put d Γd−1 := Gal(Fd /Fd−1 ). In order to define an algebraic element for the non-noetherian d (Ch Iwasawa algebra Λ(F), we need to study the relation between πd−1 Λ(Fd ) (S(Fd ))) and ChΛ(Fd−1 ) (S(Fd−1 )). Some techniques to deal with this type of descent to ensure that the limit does not depend on the filtration, have been described in [3], in particular we shall need the following (which is [3, Proposition 2.10]) CharId1 descentEq Proposition 1.1. Let R be a Krull domain and put Λ = R[[t]]. Let π : Λ → R be the natural projection and let M be a finitely generated torsion Λ-module with t-torsion Mt . Then
(1.2) ChR (Mt ) π ChΛ (M ) = ChR (M/tM ) . In Section 2 we show that if S(Fe ) is Λ(Fe )-torsion, then S(Fd ) is Λ(Fd )-torsion for any d > e and then use Proposition 1.1 to provide a general relation d d d ChΛ(Fd−1 ) (S(Fd )Γd−1 )πd−1 (ChΛ(Fd ) (S(Fd ))) = ChΛ(Fd−1 ) (S(Fd−1 )) · θd−1 d (see (2.7) where the extra factor θd−1 is more explicit). Then we move to the “cyclotomic” setting, i.e., extensions in which all ramified primes are assumed to be totally ramified (at least from a certain point on). An example are the extensions obtained from F by adding the f n -torsion points of a normalized rank 1 Drinfeld module over F . In this setting, adapting some techniques and results of K.-S. Tan ([13]), we obtain (see Corollary 3.5) IntroThm Theorem 1.2. Assume all ramified primes in F/F are totally ramified, then for any d ≫ 0 d πd−1 (ChΛ(Fd ) (S(Fd ))) = (pν ) · ChΛ(Fd−1 ) (S(Fd−1 )) . Moreover if no unramified places v ∈ SA splits completely in F, then ν = 0. This allows us to define the inverse limit of these characteristic ideals and, as an application, we use a deep result of [7] to show that such limit is the algebraic counterpart of a p-adic L-function in the non-noetherian Iwasawa Main Conjecture for constant abelian varieties (see Theorem 3.7). SelGrSec 2. General descent for Selmer groups To be able to define characteristic ideals we need the following SelStruct Theorem 2.1. Assume that A has good ordinary or split multiplicative reduction at all primes of the finite set SA . Then, for any d and any Zdp -extension L/F contained in F, the group S(L) is a finitely generated Λ(L)-module. Proof. In this form the theorem is due to Tan ([12]). See also [2, Section 2] and the references there.  In order to obtain a nontrivial relation from Proposition 1.1 between the characteristic ideals, we need something more than just Theorem 2.1, so we make the following AssSel Assumption 2.2. There exists an e > 0 such that S(Fe ) is a torsion Λ(Fe )-module. CHARACTERISTIC IDEALS AND SELMER GROUPS 3 Remarks 2.3. 1. This hypothesis is satisfied in many cases: for example when Fe contains the arithmetic Zp -extension of F (proof in [13, Theorem 2], extending [10, Theorem 1.7]) or when Sel(F ) is finite and A has good ordinary reduction at all places which ramify in Fe /F (easy consequence of [12, Theorem 4]). d (Ch 2. Our goal is an equation relating πd−1 Λ(Fd ) S(Fd )) and the characteristic ideal of S(Fd−1 ). If the above assumption is not satisfied for any e, then all characteristic ideals are 0 and there is nothing to prove. Consider the diagram DiagSel (2.1) Sel(Fd−1 )   // H 1 (Xd−1 , A[p∞ ]) fl  add−1 d Sel(Fd )Γd−1   bdd−1 // H 1 (Xd , A[p∞ ])Γdd−1 fl  // // G(Xd−1 )  cdd−1 // G(Xd )Γdd−1 where Xd := Spec(Fd ) and G(Xd ) is the image of the product of the restriction maps Y Hf1l (Xd , A[p∞ ]) −→ Hf1l (Xd,w , A)[p∞ ] , w with w running over all places of Fd . KercTor e:kerckerh Lemma 2.4. For any d > 2, the module Ker cdd−1 is a cofinitely generated torsion Λ(Fd−1 )module. Proof. For any place v of Fd−1 we fix an extension to F, which by a slight abuse of notation we still denote by v, so that the set of places of Fd above v will be the Galois orbit Γdd−1 · v. We have an obvious injection     Q   Y Y rw 1 ∞ 1 ∞ d Hf l (Xd,w , A)[p ] (2.2) Ker cd−1 ֒→ Ker Hf l (Xd−1,v , A)[p ]−−−−−→     v w∈Γd ·v d−1 (where the rw are the natural restrictions). By the Hochschild-Serre spectral sequence, we get kerrwh1 (2.3) Ker rw ≃ H 1 (Γdd−1,w , A(Fd,w ))[p∞ ] where Γdd−1,w = Γdd−1,v is the decomposition group of w in Γdd−1 . So we can regard Ker as a submodule of M H 1 (Γdd−1,w , A(Fd,w ))[p∞ ] . Hv (Fd ) := Q rw w∈Γdd−1 ·v Let Λ(Fi,v ) := Zp [[Gal(Fi,v /Fv )]] be the Iwasawa algebra associated to the decomposition group of v and note that each Ker rw is a Λ(Fd−1,v )-module. Moreover, we get an action of Gal(Fd /F ) on Hv (Fd ) by permutation of the w’s and an isomorphism e:lemmatan (2.4) Hv (Fd ) ≃ Λ(Fd−1 ) ⊗Λ(Fd−1,v ) H 1 (Γdd−1,v , A(Fd,w ))[p∞ ] , provided that hv (Fd ) := H 1 (Γdd−1,v , A(Fd,v )[p∞ ]) is finitely generated over Λ(Fd−1,v ) (more details can be found in [13, Section 3]). Therefore we can neglect all primes which totally split in Fd /Fd−1 . First assume that the place v is unramified in Fd /F . Then Fd−1,v = Fv 6= Fd,v and one has, by [8, Proposition I.3.8], hv (Fd ) ≃ H 1 (Γdd−1,v , π0 (A0,v )) , where A0,v is the closed fiber of the Néron model of A over Fv and π0 (A0,v ) its set of connected components. It follows that hv (Fd ) is trivial when v does not lie above SA and that it is finite of CHARACTERISTIC IDEALS AND SELMER GROUPS 4 order bounded by (the p-part of) |π0 (A0,v )| for the unramified places of bad reduction. Hence (2.4) shows that Hv (Fd )∨ is a finitely generated torsion Λ(Fd−1 )-module for unramified v. For the ramified case the exact sequence A(Fd,w )[p]  // A(Fd,w )  p // // pA(Fd,w ) yields a surjection H 1 (Γdd−1,v , A(Fd,w )[p]) // // H 1 (Γd d−1,v , A(Fd,w ))[p] . The first module is obviously finite, so H 1 (Γdd−1,v , A(Fd,w ))[p] is finite as well: this implies that hv (Fd ) has finite Zp -corank. As a finitely generated Zp -module, hv (Fd )∨ must be Zp [[Γd−1,v ]]torsion and (2.4) shows once again that Hv (Fd )∨ is finitely generated and torsion over Λ(Fd−1 ).  Remark 2.5. One can go deeper in the details and compute those kernels according to the reduction of A at v and the behaviour of v in Fd /F . We will do this in Section 3 but only for the particular case of a totally ramified extension (with the statement of a Main Conjecture as a final goal). See [13] for a more general analysis. Here is the result we need to apply Proposition 1.1. FinGenSel d S(F ) Proposition 2.6. Let e be as in Assumption 2.2. For any d > e, the module S(Fd )/Id−1 d is a finitely generated torsion Λ(Fd−1 )-module and S(Fd ) is a finitely generated torsion Λ(Fd )module. Moreover, if d > max{2, e}, d ChΛ(Fd−1 ) (S(Fd )/Id−1 S(Fd )) = ChΛ(Fd−1 ) (S(Fd−1 )) ChΛ(Fd−1 ) ((Coker add−1 )∨ ) . Proof. It suffices to prove the first statement for d = e + 1, since then a standard argument (detailed e.g. in [6, page 207]) shows that S(Fe+1 ) is Λ(Fe+1 )-torsion and we can iterate the process. From diagram (2.1) one gets a sequence DualSeqSel (2.5) ∨ (Coker ae+1 e )  // (Sel(Fe+1 )Γe+1 e )∨ −→ S(Fe ) // // (Ker ae+1 )∨ . e By the Hochschild-Serre spectral sequence, it follows  Coker be+1 e // H 2 (Γe+1 , A[p∞ ](Fe+1 )) = 0 e  (because Γe+1 has p-cohomological dimension 1). Therefore there is a surjective map e Ker ce+1 e // // Coker ae+1 e ∨ and, by Lemma 2.4, (Coker ae+1 e ) is Λ(Fe )-torsion. Hence Assumption 2.2 and sequence (2.5) yield that e+1 (Sel(Fe+1 )Γe )∨ ≃ S(Fe+1 )/Iee+1 S(Fe+1 ) is Λ(Fe )-torsion. To conclude note that (for any d) the duals of d Ker add−1 ֒→ Ker bdd−1 ≃ H 1 (Γdd−1 , A[p∞ ](Fd )) ≃ A[p∞ ](Fd )/Id−1 A[p∞ ](Fd ) are finitely generated Zp -modules (hence Λ(Fd−1 )-pseudo-null for any d > 3). Taking characteristic ideals in the sequence (2.5), for large enough d, one finds d ChΛ(Fd−1 ) (S(Fd )/Id−1 S(Fd )) = ChΛ(Fd−1 ) (S(Fd−1 )) ChΛ(Fd−1 ) ((Coker add−1 )∨ ) .  CHARACTERISTIC IDEALS AND SELMER GROUPS 5 The first consequence of Propositions 2.6 and 1.1 is d d d ChΛ(Fd−1 ) (S(Fd )Γd−1 )πd−1 (ChΛ(Fd ) (S(Fd ))) = ChΛ(Fd−1 ) (S(Fd )/Id−1 S(Fd )) . CharIdSel1 (2.6) CharIdSel2 Moreover, if d > 3, equation (2.6) turns into (2.7) d d ChΛ(Fd−1 ) (S(Fd )Γd−1 )πd−1 (ChΛ(Fd ) (S(Fd ))) = ChΛ(Fd−1 ) (S(Fd−1 ))ChΛ(Fd−1 ) ((Coker add−1 )∨ ) . d Therefore, whenever we can prove that S(Fd )Γd−1 is a pseudo-null Λ(Fd−1 )-module, we get CharIdSel3 d πd−1 (ChΛ(Fd ) (S(Fd ))) ⊆ ChΛ(Fd−1 ) (S(Fd−1 )) (2.8) and this relation would be enough to be able to define the projective limit of the ChΛ(Fd ) (S(Fd )) in Λ(F). 3. Descent for totally ramified extensions pramSel The main examples we have in mind are extensions satisfying the following AssRam Assumption 3.1. There exists an e > 0 such that all places which ramify in F/F are totally ramified in F/Fe . An example is the p-cyclotomic extension of Fq (T ) generated by the p-torsion of the Carlitz module (see e.g. [11, Chapter 12]). As customary in Iwasawa theory over number fields, to simplify notations and proofs, we shall assume that e = 0 (i.e., all the ramified places are totally ramified), but all the statements will hold for the general case as well (simply considering Fn for n ≫ 0). CharIdCoker Theorem 3.2. Assume F/F verifies Assumption 3.1 with e = 0, then, for d ≫ 0, there exists a ν > 0 such that ChΛ(Fd−1 ) ((Coker add−1 )∨ ) = (pν ) . Moreover if no unramified place v ∈ SA splits completely in F, then, for d ≫ 0, we have ChΛ(Fd−1 ) ((Coker add−1 )∨ ) = (1) . Proof. The proof of Proposition 2.6 shows that the Λ(Fd−1 )-modules (Coker add−1 )∨ and (Ker cdd−1 )∨ are pseudo-isomorphic for d > 3. Moreover, by the proof of Lemma 2.4, we Y Hv (Fd )∨ (since Hv (Fd ) = 0 for all v 6∈ SA ). By know that (Ker cdd−1 )∨ is a quotient of v∈SA equation (2.4), we have TanChId (3.1) ChΛ(Fd−1 ) (Hv (Fd )∨ ) = Λ(Fd−1 )ChΛ(Fd−1,v ) (hv (Fd )∨ ) . We also saw that, for a ramified prime v, Hv (Fd )∨ (which is the same as hv (Fd )∨ , because v is totally ramified) is finitely generated over Zp , hence pseudo-null over Λ(Fd−1,v ) = Λ(Fd−1 ) for d > 3. Now we check the unramified primes in SA . If v splits completely in F/F , then Λ(Fd−1,v ) ≃ Zp for any d and, since (by Lemma 2.4) hv (Fd )∨ is finite, we get ChΛ(Fd−1,v ) (hv (Fd )∨ ) = |hv (Fd )∨ | = (pνv ) . If v is inert in some extension Fd /Fd−1 , then Λ(Fd−1,v ) ≃ Zp and Λ(Fr,v ) ≃ Zp [[td ]] for any r > d . Hence ChΛ(Fr−1,v ) (hv (Fr )∨ ) = (1) for any r > d + 1 . These local informations and (3.1) yield the theorem.  CHARACTERISTIC IDEALS AND SELMER GROUPS 6 d Now we deal with the other extra term of equation (2.7), i.e., ChΛ(Fd−1 ) (S(Fd )Γd−1 ). Note first that, taking duals d (S(Fd )Γd−1 )∨ ≃ S(Fd )∨ /(γd − 1)S(Fd )∨ = Sel(Fd )/(γd − 1)Sel(Fd ) , so we work on the last module. From now on we put γ := γd and we shall need the following (see also [13, Proposition 4.3.2]) Surjgamma-1 Lemma 3.3. We have Hf1l (Xd , A[p∞ ]) = (γ − 1)Hf1l (Xd , A[p∞ ]) . Proof. Let α ∈ Hf1l (Xd , A[p∞ ]) and take K ⊂ Fd (with [K : F ] < ∞) and m > 0 such that α ∈ Hf1l (XK , A[pm ]): their existence is guaranteed by the fact that lim Hf1l (XK , A[pm ]) Hf1l (Xd , A[p∞ ]) = lim −→ −→ K m (where, as usual, XK denotes Spec(K) and we use the X for infinite extensions of F ). Now let K∞ /K be a Zp -extension of K whose Galois group is topologically generated by a power s(K) of γ (i.e., let γ p be the largest power of γ which acts trivially on K, then Gal(K∞ /K) = s(K) p <γ >) and whose layers we denote by Kn . Take t > m, consider the restrictions Hf1l (XK , A[pm ]) → Hf1l (XKt , A[pm ]) → Hf1l (XK∞ , A[pm ]) and denote by xt the image of x. Since xt is fixed by Gal(Kt /K) and annihilated by Kt pm 6 pt = |Gal(Kt /K)|, one has that xt is in the kernel of the norm NK , i.e., in the 1 1 m (Galois) cohomology group H (Kt /K, Hf l (XK∞ , A[p ]). In particular, via the inflation map, 2 be the kernel of the restriction map from xt ∈ H 1 (K∞ /K, Hf1l (XK∞ , A[pm ]). Let Kerm Hf2l (XK , A[pm ]) to Hf2l (XK∞ , A[pm ]), then, from the Hochschild-Serre spectral sequence, we get a sequence HochSerKer 2 → H 1 (K∞ /K, Hf1l (XK∞ , A[pm ]) → H 3 (K∞ /K, A(K∞ )[pm ]) . Kerm (3.2) The rightmost term is 0 because the p-cohomological dimension of Zp is 1. Regarding the 2 note that, by [5, Lemma 3.3], H 2 (X , A) = 0. Hence, from the cohomology sequence Kerm K fl arising from pm A[pm ] ֒→ A−−−։A , one has an isomorphism Hf2l (XK , A[pm ]) ≃ Hf1l (XK , A)/pm . Consider the commutative diagram (with m2 > m1 ) Hf1l (XK , A)/pm1 ∼ // H 2 (XK , A[pm1 ]) fl pm2 −m1  Hf1l (XK , A)/pm2 ∼  // H 2 (XK , A[pm2 ]) . fl An element of Hf1l (XK , A)/pm1 of order pr goes to zero via the vertical map on the left as soon as m2 > m1 + r, hence the direct limit provides lim Hf1l (XK , A)/pm = 0 and, eventually, −→ 2 = 0 as well. By (3.2) lim Kerm −→ m m 0 = lim H 1 (K∞ /K, Hf1l (XK∞ , A[pm ]) = H 1 (K∞ /K, Hf1l (XK∞ , A[p∞ ]) , −→ m which yields Hf1l (XK∞ , A[p∞ ]) = (γ p s(K) − 1)Hf1l (XK∞ , A[p∞ ]) = (γ − 1)Hf1l (XK∞ , A[p∞ ]) . CHARACTERISTIC IDEALS AND SELMER GROUPS 7 We get the claim by taking the direct limit on the finite subextensions K.  d PNullThm Theorem 3.4. For any d > 3 we have ChΛ(Fd−1 ) (S(Fd )Γd−1 ) = (1). Proof. Consider the following diagram DiagTan (3.3) Sel(Fd )  // H 1 (Xd , A[p∞ ]) φ fl  γ−1 γ−1 Sel(Fd )  // H 1 (Xd , A[p∞ ]) φ fl  Y // // Coker(φ) γ−1   (where Hi (Xd , A) := // H1 (Xd , A) γ−1   // H1 (Xd , A) // // Coker(φ) Hfi l (Xd,w , A)[p∞ ] and the surjectivity of the second vertical arrow w comes from the previous lemma). Inserting G(Fd ) := Im(φ), we get two diagrams DiagTan1 (3.4) Sel(Fd )  φ // H 1 (Xd , A[p∞ ]) fl  γ−1 γ−1 γ−1   Sel(Fd )  // // G(Fd ) φ // H 1 (Xd , A[p∞ ]) fl   // // G(Fd ) G(Fd )  // H1 (Xd , A)  γ−1  // // Coker(φ) γ−1 γ−1  G(Fd )   // H1 (Xd , A)  // // Coker(φ) . From the snake lemma sequence of the first one, we obtain the isomorphism QuotSelIso d d G(Fd )Γd−1 /Im(φΓd−1 ) ≃ Sel(Fd )/(γ − 1)Sel(Fd ) (3.5) d d (where φΓd−1 is the restriction of φ to Hf1l (Xd , A[p∞ ])Γd−1 ). The snake lemma sequence of the second diagram (its “upper” row) yields an isomorphism CokerphiIso d d d H1 (Xd , A)Γd−1 /G(Fd )Γd−1 ≃ Coker(φ)Γd−1 . (3.6) d d The injection G(Fd )Γd−1 ֒→ H1 (Xd , A)Γd−1 induces an exact sequence d d d d d d G(Fd )Γd−1 /Im(φΓd−1 ) ֒→ H1 (Xd , A)Γd−1 /Im(φΓd−1 ) ։ H1 (Xd , A)Γd−1 /G(Fd )Γd−1 d d (with a little abuse of notation we are considering Im(φΓd−1 ) as a submodule of H1 (Xd , A)Γd−1 via the natural injection above) which, by (3.5) and (3.6), yields the sequence Tan42 d d Sel(Fd )/(γ − 1)Sel(Fd ) ֒→ Coker(φΓd−1 ) ։ Coker(φ)Γd−1 . (3.7) Now consider the following diagram H 1 (Γdd−1 , A[p∞ ])  φdd−1 // H 1 (Xd−1 , A[p∞ ]) fl // H 1 (Xd , A[p∞ ])Γdd−1 fl φd−1  H1 (Γdd−1 , A)     // H1 (Xd−1 , A)  φ // 0 Γd d−1 // H1 (Xd , A)Γdd−1  // // H2 (Γd , A) d−1 where: • the vertical maps are all induced by the product of restrictions φ; • the horizontal lines are just the Hochschild-Serre sequences for global and local cohomology; • the 0 in the upper right corner comes from H 2 (Γdd−1 , A[p∞ ]) = 0; • the surjectivity on the lower right corner comes from H2 (Xd−1 , A) = 0, which is a direct consequence of [8, Theorem III.7.8]. CHARACTERISTIC IDEALS AND SELMER GROUPS This yields a sequence (from the snake lemma) EqCoker (3.8) d Coker(φd−1 ) → Coker(φΓd−1 ) → H2 (Γdd−1 , A) = Y 8 H 2 (Γdd−1,w , A(Fd,w ))[p∞ ] . w The module Coker(φd−1 ). The Kummer map induces a surjection H 1 (Xd−1 , A[p∞ ]) ։ H 1 (Xd−1 , A)[p∞ ] which fits in the diagram H 1 (Xd−1 , A[p∞ ]) φd−1  λd−1 H 1 (X d−1 // H1 (Xd−1 , A) 66 , A)[p∞ ] . This induces natural surjective maps Im(φd−1 ) ։ Im(λd−1 ) and, eventually, Coker(λd−1 ) ։ Coker(φd−1 ). For any finite extension K/F we have a similar map λK : H 1 (XK , A)[p∞ ] → H1 (XK , A) whose cokernel verifies Coker(λK )∨ ≃ Tp (SelAt (K)p ) (by [4, Main Theorem]), where At is the dual abelian variety of A. Moreover there is an embedding of Tp (SelAt (K)p ) into the p-adic completion of H0 (XK , At ) (recall that, by Tate local duality, At (Kv ) = H 0 (Kv , At ) is the Pontrjagin dual of H 1 (Kv , A), see [8, Theorem III.7.8]). Taking limits on all the finite subextensions of Fd−1 we find similar relations Coker(λd−1 )∨ ≃ Tp (SelAt (Fd−1 )p ) ֒→ lim lim H0 (XK , At )/pn = lim lim At (K)[p∞ ]/pn . ←− ←− ←− ←− K n K n Hence Coker(λd−1 )∨ embeds into a finitely generated Zp -module, i.e., it is Λ(Fd−1 ) pseudonull for any d > 3. The modules H 2 (Γdd−1,w , A(Fd,w ))[p∞ ]. If the prime splits completely in Fd /Fd−1 , then obviously H 2 (Γdd−1,w , A(Fd,w ))[p∞ ] = 0. If the place is ramified or inert, then Γdd−1,w ≃ Zp . Consider the exact sequence A(Fd,w )[p]   // A(Fd,w ) p // // pA(Fd,w ) , which yields a surjection H 2 (Γdd−1,w , A(Fd,w )[p]) // // H 2 (Γd d−1,w , A(Fd,w ))[p] . The module on the left is trivial because cdp (Zp ) = 1, hence H 2 (Γdd−1,w , A(Fd,w ))[p] = 0 and this yields H 2 (Γdd−1,w , A(Fd,w ))[p∞ ] = 0 as well. d The sequence (3.8) implies that Coker(φΓd−1 ) is Λ(Fd−1 ) pseudo-null for d > 3 and, by (3.7), we get Sel(Fd )/(γ − 1)Sel(Fd ) is pseudo-null as well. Therefore d ChΛ(Fd−1 ) (S(Fd )Γd−1 ) = ChΛ(Fd−1 ) ((Sel(Fd )/(γ − 1)Sel(Fd ))∨ ) = (1) .  A direct consequence of equation (2.7) and Theorems 3.2 and 3.4 is CorTan CharIdSel5 Corollary 3.5. Assume F/F verifies Assumption 3.1 with e = 0, then for any d ≫ 0 (3.9) d πd−1 (ChΛ(Fd ) (S(Fd ))) = (pν ) · ChΛ(Fd−1 ) (S(Fd−1 )) . Moreover if no unramified places v ∈ SA splits completely in F, then, for d ≫ 0 we have CharIdSel4 (3.10) d πd−1 (ChΛ(Fd ) (S(Fd ))) = ChΛ(Fd−1 ) (S(Fd−1 )) . CHARACTERISTIC IDEALS AND SELMER GROUPS 9 Whenever equation (3.10) or (3.9) holds one can define CharIdSel Definition 3.6. The pro-characteristic ideal of S(F) is f Λ(F ) (S(F)) := lim(πF )−1 (ChΛ(F ) (S(Fd ))) = Ch d d ←− Fd \ −1 πF (ChΛ(Fd ) (S(Fd ))) d d>r (for any large enough integer r) where πFd : Λ(F) → Λ(Fd ) is the natural projection map. We remark that Definition 3.6 only depends on the extension F/F and not on the filtration of Zdp -extension we choose inside it. Indeed with two different filtrations {Fd } and {Fd′ } we can define a third one by putting F0′′ := F and Fn′′ = Fn Fn′ ∀n > 1 . By Corollary 3.5, the limits of the characteristic ideals of the filtrations we started with coincide with the limit on the filtration {Fn′′ } (see [3, Remark 3.11] for an analogous statement for characteristic ideals of class groups). This pro-characteristic ideal could play a role in the Main Conjecture of Iwasawa theory for a “cyclotomic” extension of F as the algebraic counterpart of a p-adic L-function associated to A and F (see [1, Section 5] or [2, Section 3] for similar statements but with Fitting ideals). Anyway, at present, the problem of formulating a (conjectural) description of this ideal in terms of a natural p-adic L-functions (i.e., a general non-noetherian Iwasawa Main Conjecture) is still wide open. However, we can say something if A is already defined over the constant field of F . FinalThm e:lltt Theorem 3.7. [Non-noetherian IMC for constant abelian varieties] Assume A/F is a constant abelian variety and let F/F be a totally ramified extension as above. Then there exists an element θA,F interpolating the classical L-function L(A, χ, 1) (where χ varies among characters of Gal(F/F )) such that one has an equality of ideals in Λ(F) (3.11) f Λ(F ) (S(F)) = (θA,F ) . Ch Proof. This is a simple consequence of [7, Theorem 1.3]. Namely, the element θA,L is defined in [7, §7.2.1] for any abelian extension L/F unramified outside a finite set of places. It satisfies d (θ πd−1 A,Fd ) = θA,Fd−1 by construction and the interpolation formula (too complicated to report it here) is proved in [7, Theorem 7.3.1]. Since A has good reduction everywhere, (3.10) always holds, so both sides of (3.11) are defined. Finally [7, Theorem 1.3] proves that ChΛ(Fd ) (S(Fd )) = (θA,Fd ) for all d and (3.11) follows by just taking a limit.  References BL BBL BBL2 GAT07 GAT12 Gr1 LLTT Mi NSW [1] A. Bandini - I. Longhi, Control theorems for elliptic curves over function fields, Int. J. Number Theory 5 (2009), no. 2, 229–256. [2] A. Bandini - F. Bars - I. 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C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia E-mail address: francesc@mat.uab.cat Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren Ai Road, Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou, Jiangsu, 215123, China E-mail address: Ignazio.Longhi@xjtlu.edu.cn