Documenta Math. 885 Change of Selmer Group for Big Galois Representations and Application to Normalization

The goal of this note is to prove, under some assumptions, a formula relating the Selmer groups of isogenous Galois representations. Local and global Euler-Poincaré characteristic formulas are key tools in the proof. With additional hypotheses, we use the isogeny formula to study how the formation of Selmer groups interacts with normalization of the coefficient ring and discuss how a main conjecture for a big Galois representation over a non-normal ring follows from a corresponding conjecture over the normalization. 2010 Mathematics Subject Classification: 11R23, 11R34

where R is a ring finite and free over the power series ring O[[X 1 , . . ., X n ]], with O the integer ring of a p-adic field, and T is a finitely-generated R-module.One can, under suitable hypotheses, attach a Selmer group Sel(ρ) to such ρ.This Selmer group is a finitely-generated R-module which is canonically defined in terms of the Galois cohomology of ρ.
The basic question we investigate below is the following.Given representations ρ 1 and ρ 2 as above on R-modules T 1 and T 2 which are isogenous, i.e., such that there is an R[G Q ]-linear homomorphism T 1 → T 2 with R-torsion cokernel, how are the Selmer groups Sel(ρ 1 ) and Sel(ρ 2 ) related?We prove the following formula relating the support divisors of Sel(ρ 1 ) and Sel(ρ 2 ) in terms of local and global invariants of the quotient Q = T 2 /φ(T 1 ) (see Theorem ?? for the precise statement).
Theorem.If T 1 and T 2 satisfy certain natural hypotheses (cf.??), then div Sel(ρ 1 ) − div Sel(ρ 1.2.Our main motivation (and a key example of this type of representation) comes from Hida theory.Let f be a p-ordinary cuspidal newform.By work of Hida [?], such f belongs to a p-adic family F of newforms, which can be viewed as a formal power series with coefficients in a ring R finite and free over where O is a suitable finite extension of Z p .The specializations of F at appropriate values of T are power series expansions of classical p-stabilized newforms of varying weight, level, and character.One can attach a Galois representation ρ F to F on a rank 2 module T over the ring R interpolating the p-adic Galois representation attached to the classical newforms arising as specializations of F .Many of the hypotheses imposed in ?? are automatically satisfied by these representations. 1.
3. An early investigation of how isogenies affect Iwasawa invariants was undertaken by Schneider [?], who gave a formula relating the µ-invariants for Selmer groups of isogenous abelian varieties over Z p -extensions of number fields.This formula was generalized by Perrin-Riou [?] to more general p-adic representations.More recently, Ochiai [?] has given a similar formula for invariants of big Galois representations with coefficients in a power series ring Z p [[T 1 , . . ., T n ]].Our isogeny formula is a generalization of Ochiai's and has a similar proof, which, in particular, depends on Euler-Poincaré characteristic formulas and Poitou-Tate duality.
1.4.In Theorem ??, we prove somewhat general Euler-Poincaré characteristic formulas for big Galois representations.For p > 2, the theorem can be deduced from the corresponding statements in Nekovář [?, 4.6.9 and 7.8.6](which exclude the case of p = 2).Our main result, the isogeny formula of Theorem ??, follows from a series of computations involving these.Fortunately, many of the needed computations are contained in Greenberg's series of papers [?, ?, ?].In a certain sense, therefore, this note may be viewed as an addition to that series.Some of the results contained here can also be found in the second author's thesis [?, Ch. 1].
1.5.Under an additional "p-criticality" assumption on the representation T (cf.??), we show in §?? that the corresponding normalized representation T obtained by extending scalars to the normalization R of R gives Selmer groups which, when considered as R-modules, have the same divisor on Spec R. Using this fact and some elementary commutative algebra, we discuss how a main conjecture for the representation T implies a corresponding main conjecture for T .Thus, under our admittedly somewhat strict hypotheses, main conjectures, roughly speaking, commute with normalization.This result should not be surprising to the experts; its study was suggested by Greenberg [?, §1].
1.6.We remark here on some of the hypotheses we impose, some of which could be considered rather strong.The conditions (??)-(??) and the p-criticality hypothesis imposed in §?? are somewhat standard and are known to hold for many of the representations arising "in nature" from the study of Hida families as discussed briefly above, with the possible exception of (??), which has nonetheless been extensively studied.There are two additional, less standard, hypotheses we employ.The first of these is that the Galois modules we consider are assumed to be free over the coefficient ring.There are two places where we make serious use of this hypothesis.The first is in the application of a result of Greenberg [?, Lemma 2.2.6] on vanishing of Galois invariants.We feel that this result is probably true for even torsion-free modules.The second is in the proof of Theorem ??, where we make use of the following property of free modules M over a ring R with module-finite normalization R: the divisor (on Spec R) associated to the torsion R-module (M ⊗ R R)/M is rank R M times the divisor associated to R/R.It is unclear to us whether there is a weaker hypothesis on R-modules which guarantees this to hold.
The second is the condition (??) on the rank of compact Selmer groups, which is necessary in order to conclude the surjectivity of a certain localization map.It is a difficult and interesting question whether this condition holds for representations arising from Hida theory and is not true in general (cf.[?, §4.9] or [?, §7(d)] for an example).
2 Notation 2.1.Fix a prime p.Let R be a complete Noetherian local domain with maximal ideal m and assume that R is finite and free over which are continuous in the sense that the representation space V = V ρ of ρ admits a G Σ -stable R-lattice T such that the induced representation, which by abuse of notation we still denote by ρ : G Σ → Aut R (T ), is continuous for the Krull topology on G Σ and the topology on Aut R (T ) induced by the topology on R.
In what follows, we shall be studying free lattices, i.e., R-submodules of (Frac R) ⊕n of rank n which are free R-modules.Without additional assumptions on R, it may not be the case that any continuous R-linear representation of Gal(K Σ /K) admits a Gal(K Σ /K)-stable lattice which is free as an R-module.

Ordinary data.
For notational convenience, we now define a notion of ordinary datum over R.Such a datum X consists of a pair (T, F), where T is a finitely-generated free R-module with continuous G Σ -action and F consists of We refer to the chain T ⊇ F + v T ⊇ 0 as the local filtration on T at v given by F. Given ordinary data X 1 = (T 1 , F 1 ) and X 2 = (T 2 , F 2 ), we define a homomorphism φ : X 1 → X 2 to be an R[G Σ ]-linear homomorphism φ : T 1 → T 2 which is compatible with the filtrations in the sense that φ( 2.4.We now define discrete modules associated to a datum X = (T, F).Denote by W * = W * X the discrete Galois module W * = Hom R T, R ∨ (1) dual to T .Thus, W * ∼ = T ∨ as an R-module.(Note that we do not define here a compact module T * with Galois action the Tate dual of that on T .)The filtrations F on T induce filtrations where where the homomorphism on the right is induced by the quotient W * → F − v W * and restriction to I v .Let X = (T, F) be an ordinary datum.Recall that Tate local duality gives a perfect pairing We define the Greenberg local conditions H 1 f (K v , T ) for T as the orthogonal complements of the Greenberg local conditions H 1 f (K v , W * ) for W * under this pairing.

Given a set of local conditions ∆ on an R[G
and define the Selmer group over K attached to M and ∆ by where the homomorphism on the right-hand side is induced by the obvious local-to-global map.If M = W * or T and ∆ is the set of Greenberg local conditions for M , then we omit the ∆ and denote the corresponding Selmer group by Sel(M ).For a G Σ -module M and i ≥ 0, we further define Shafarevich-Tate groups Thus, 1 (M ) = Sel ∆ (M ) for ∆ the set of local conditions defined by setting H 1 f (K v , M ) = 0 for all v ∈ Σ.The representations arising in the case of Hida families come equipped with additional structure that allows other natural definitions of local conditions (e.g., the so-called Bloch-Kato local conditions) which in general give rise to Selmer groups different from those discussed above.Ochiai has studied the relationship between these Selmer groups, cf.[?, §3].

2.7.
If M is a finitely-generated R-module and p ⊆ R is a prime ideal, then we denote the p-length of M by (2.7.1) A finitely-generated R-module M is said to be pseudo-null if lgth p M = 0 for every height 1 prime p ⊆ R. Equivalently, M is pseudo-null if the set Ass R (M ) of associated primes of M contains only primes of height 2 or greater.If M is cofinitely-generated, we say M is copseudo-null if M ∨ is pseudo-null.
If R has dimension 2 and finite residue field, then a finitely-generated, resp.cofinitely-generated, R-module is pseudo-null, resp.copseudo-null, if and only if it contains only finitely many elements.
2.8 Conditions on X. Fix an ordinary datum X = (T, F).Below, we often subject X to the following conditions. (2.8.1) (2.8.3) Sel(W * ) is a cotorsion R-module.
(2.8.4)No subquotient of W * [m] is isomorphic to µ p as a G K -module.
3 Duality formulas 3.1.This section is devoted to the proof of various duality results for Selmer groups.The first several subsections (up to ??) are devoted to the proof of the following theorem, the global (??) and local (??) Euler-Poincaré characteristic formulas.This theorem can be deduced from Nekovář [?, 4.6.9 and 7.8.6], at least in the case p > 2.
3.2 Theorem.Suppose K has r 1 real places and r 2 conjugate pairs of complex places.For any cofinitely-generated cotorsion R-module D and height and, for every non-archimedean prime v of K, Similarly, for v a non-archimedean prime of K, if v ∤ p, then set Thus, δ * (D) is the difference between the right-hand side and left-hand side of (??) (if * = Σ) or of (??) (if * is a prime of K), and we need to show that δ * (D) = 0.

3.4.
The proof of Theorem ?? proceeds by induction on lgth p D ∨ and dévissage.The base case is the following.
Proof.Consider the short exact sequence By hypothesis, we have lgth p (D/D[p]) ∨ = 0, so p / ∈ Supp(D/D[p]) ∨ .As Supp M ∨ ⊇ Supp H i (G, M ) ∨ for any i ≥ 0, any cofinitely-generated R-module M , and G = G Σ or G Kv , we see from the definition of δ * that δ * (D) = δ * (D[p]).We may therefore assume without loss of generality that D[p] = D.Under this assumption, D and all the H i (G, D) are cofinitely-generated R/p-modules, so lgth p D ∨ = corank R/p D and lgth p H i (G, D) ∨ = corank R/p H i (G, D) ∨ by (??).Rephrased in this way via coranks, the statement of the lemma becomes the same as [?, Prop.4.1].

Lemma (Dévissage). For any short exact sequence
Proof.As lgth p is additive in exact sequences, we may ignore the terms in the definition of δ * (D) which are multiples of lgth p D ∨ .The lemma is slightly more difficult when p = 2 and * = Σ, so let us first assume either p > 2 or * = Σ.If v is an archimedean prime of K, then, as p > 2, H i (K v , D) = 0 for i > 1 and D = A, B, or C. The result then follows from the long exact cohomology sequence and the fact that G Σ and G Kv have p-cohomological dimension 2 under our assumptions.Now suppose p = 2 and * = Σ.By the long exact G Σ -cohomology sequence, we have Recall ([?, Thm.4.10], e.g.) that for any discrete ind-finite given by the product of restrictions to decomposition groups at real places is an isomorphism for q ≥ 3, so the right-hand side of (??) is equal to As G Kv is cyclic of order 2 for v a real place, the cohomology groups H i (K v , D) are periodic of period 2 for i > 0, and all have equal p-length (cf.[?, Prop.4.18]).This implies that for real v, which, by the long exact G Kv -cohomology sequences, shows that the right hand side of (??) is equal to which proves the lemma for p = 2 and * = Σ.
3.6 Proof of Theorem ??.The statement is true when lgth p D ∨ = 0 by Lemma ??, so assume lgth p D ∨ > 0. Consider the short exact sequence Lemma ?? implies the result if lgth p pD ∨ = 0. Similarly, if lgth p D ∨ /pD ∨ = 0, then lgth p D ∨ = 0 by Nakayama's Lemma, so a fortiori lgth p pD ∨ = 0 and we are again done by Lemma ??.We may therefore assume that both lgth p pD ∨ and lgth p D ∨ /pD ∨ are positive and thus less than lgth p D ∨ .The theorem then follows from dévissage (Lemma ??) and induction.

Theorem (Poitou-Tate global duality).
There is a perfect pairing and a 9-term exact sequence , the theorem follows from the version for finite modules (see [?, Thm.I.4.10], for example) by taking limits.

3.8.
For a G Σ -module M with local filtrations at each v | p, e.g., for M arising from an ordinary datum, we define semi-local cohomology groups by Additionally, let loc i M : H i (G Σ , M ) → H i loc (M ) be the natural localization map.
3.10 Lemma.If X satisfies (??) and (??), then Sel(W * )/ ker loc 1 W * is a copseudo-null R-module.In particular, lgth p Sel(W * ) ∨ = lgth p ker(loc 1 W * ) ∨ for every height 1 prime p ⊆ R. If X further satisfies (??) and (??), then coker loc 1  W * is a copseudo-null R-module.Proof.The inflation-restriction sequence for where K ur v is the maximal unramified extension of K v .The lemma thus follows from the assumptions (??) and (??), which state that F − v W * (I v ) and W * (I v ) are copseudo-null R-modules.In case X also satisfies (??) and (??), then Lemma ?? gives that the homomorphism defining Sel(W * ) is surjective.The module (??) above is the kernel of the quotient map so the final statement in the lemma follows from the fact that (??) is copseudonull.

Isogenies
4.1.If T 1 and T 2 are finitely generated R-modules and φ : X 1 → X 2 is a homomorphism with torsion kernel and cokernel, then we say that φ is an isogeny or that T 1 and T 2 are isogenous if we do not wish to make the homomorphism explicit.Note that an isogeny of torsion-free R-modules is necessarily injective.Similarly, if W 1 and W 2 are cofinitely-generated R-modules, then we say that a homomorphism ψ : W 1 → W 2 is an isogeny (and that W 1 and W 2 are isogenous) if its Pontryagin dual ψ ∨ : are ordinary data, then we say that φ : X 1 → X 2 is an isogeny and hence that the X i are isogenous if φ : T 1 → T 2 is an isogeny.A homomorphism of ordinary data φ : X 1 → X 2 is an isogeny if and only if the induced homomorphism W * X2 → W * X1 is an isogeny.Isogeny is an equivalence relation on the categories of finitely-generated R-modules, cofinitely-generated R-modules, and ordinary data, cf.[?, §2].

4.2.
For the remainder of the section, fix ordinary data X 1 = (T 1 , F 1 ) and X 2 = (T 2 , F 2 ) and an isogeny φ : X 1 → X 2 .Our goal is to use Theorem ?? to prove a formula (Theorem ??) relating the p-lengths of Selmer groups for X 1 and X 2 in terms of various Galois invariants of the quotient module T 2 /φ(T 1 ), or, more precisely, its dual The key tools we need are the global Euler-Poincaré characteristic formulas above and Poitou-Tate duality, Theorem ??.The formula can be thought of as a reorganization of the information provided by Poitou-Tate duality under the assumptions (??)-(??).
4.3 Proposition.If X satisfies (??)-(??), then for all height 1 primes p ⊆ R, which implies the proposition by the first statement of Lemma ??.
5 Application to normalization 5.1.We now apply the main result of §?? to study how Selmer groups behave with respect to normalization.Assume R is reduced and let R be the integral closure of R in its total ring of fractions.A well-known result of Nagata [?, Thm.7] states that R is a finite R-module.If X is an ordinary datum over R, then set X = ( T , F), where T = T ⊗ R R and F + v T = (F + v T ) ⊗ R R. Since R is finite over R, we may view X as an ordinary datum over R or over R, and the natural inclusion T → T is an isogeny of ordinary data over R.

5.2.
Fix an ordinary datum X over R. For Φ = Frac R the fraction field of R, define V = T ⊗ R Φ, so V is a finite-dimensional Φ-vector space with a Φ-linear action of G Σ .The filtrations F induce filtrations We say that X is p-critical if α(X) = v|p ε v (x).For p-critical data, we have the following theorem regarding normalization.
5.3 Theorem.Let p ⊆ R be a height 1 prime.If p = 2, then assume that T (K v ) is a summand of T for each real place v of K.If X and X, both viewed as ordinary data over R, are p-critical and satisfy (??)-(??), then lgth p Sel(W * ) ∨ = lgth p Sel( W * ) ∨ .Documenta Mathematica 16 (2011) 885-899 5.5.In the below corollary to Theorem ??, we say two finitely-generated Rmodules M and N have the same divisor if lgth p M = lgth q N for all height 1 primes p ⊆ R. Similarly, we say a finitely-generated R-module has the same divisor as an element L ∈ R if M and R/L have the same divisor.
Corollary.Let 0 = L ∈ R and let L be the image of L in R. Using notation and assumptions as in Theorem ??, with the exception that we now view X as an ordinary datum over R, Sel(W * ) ∨ has the same divisor as L if and only if Sel( W * ) ∨ has the same divisor as L.
Proof.Viewing R as a rank 1 R-module, we use the formula [?, Lemma 11.7] to see that, for every height 1 prime q ⊆ R, lgth q R/( L) = lgth q R/(L), so the result follows by combining Theorem ?? and Lemma ??. 5.6.Corollary ?? states, roughly speaking, that, under some assumptions, the formation of the divisor of the Selmer group of an ordinary datum commutes with normalization.In a situation where there is a p-adic L-function belonging to R associated with the ordinary datum X, the corollary provides some flexibility in proving a main conjecture for X, in that such a conjecture can be proved equivalently before or after normalization.