Local maximality

We will now prove our main theorem. This result will not only contain a new proof of Lemma 3.56, but also improves our results concerning theλ-invariant (i.e., Lemma 3.37 and Corollary 3.38). Furthermore, our method will even be fine enough to obtain information about ν-invariants. The key idea is to use, as in the study of theµ-invariant, modules of the form

A/(f(T)·A),

for some suitable distinguished polynomial f(T). We will in fact choose a sequence of polynomials and consider the corresponding ranks.

The following theorem is the most important result of Chapter 3.

Theorem 3.57. Let L/K be a Zp-extension. Let µ:=µ(L/K), λ:=λ(L/K).

Then the following holds.

(i) µ(L/K)is locally maximal, i.e., there exists an integer n∈Nsuch that for every M ∈U(L, n), we have

µ(M/K) ≤ µ(L/K).

(ii) If µ = 0, then λ(L/K) is locally maximal, i.e., there exists some n ∈ N such that for each M ∈ U(L, n), we have µ(M/K) = 0 and λ(M/K)≤λ(L/K).

(iii) More generally, if µ:=µ(L/K)∈N0 is arbitrary, then λ(L/K) is locally maximal on the setE^µ(K)ofZp-extensionsM/Ksatisfyingµ(M/K) =µ, i.e., there exists an integer n∈Nsuch that

λ(M/K) ≤ λ(L/K) for every M ∈U(L, n)∩ E^µ(K).

(iv) If E^µ,λ(K)denotes the set of Zp-extensionsM/K satisfying µ(M/K) =µ and λ(M/K) =λ, then there exists an integern∈Nsuch that

|M₁^(M)| = |M₁^(L)| and ν(M/K) = ν(L/K)

for every M ∈ U(L, n) ∩ E^µ,λ(K), i.e., the ν-invariant is locally con-stant in this set. Here M₁^(M) denotes the maximal finiteΛ-submodule of the projective limit A^(M) = lim←−A^(M)n , respectively.

Proof. (i) Let n > m≥e(L/K). We make use of the distinguished polyno-mials ν_(n,m)(T) introduced in Section 1.2. Sinceν_(n,m) = _(T^(T₊₁₎⁺¹⁾_pm^pn−1₋₁, the roots ofν_(n,m)in an algebraic closureQpofQp are of the formζ−1, where ζ^pn = 1,ζ^pm 6= 1, i.e.,ζ =ζ_pl is a primitivep^l-th root of unity,m < l≤n.

We note that

vp(ζ_p^l−1) = 1

p^l−1(p−1) < 1 p^m−1(p−1)

for every l > m, where vp denotes the extension of the usual p-adic val-uation to Qp(ζ_pl) (i.e., v_p(p) = 1). The degree of F_A(L)(T) is equal to λ = λ(L/K). We choose m large enough to ensure that _pm−1^λ(p−1) < 1.

Then, sinceF_A(L)(T) is a distinguished polynomial, we have v_p(F_A(L)(ζ_pl−1)) = λ

p^l−1(p−1) < 1 (3.1) for everyl > m. For everyl, there exist exactlyp^l−1(p−1) primitivep^l-th roots of unity. We may conclude that

|Λ/(F_A(L), ν_(n,m))| = Y

m<l≤n , ζ^pl= 1

p^{vp(FA(L)(ζ−1))}

= Y

m<l≤n

(p

λ

pl−1(p−1))^{pl−1(p−1)} = (p^λ)^n−m ,

3.3. LOCAL BOUNDEDNESS RESULTS 107 where the first product runs over the primitive p^l-th roots of unity, re-spectively. Indeed, Z := Λ/(ν_(n,m)) is isomorphic to a free Zp-module of rank deg(ν_(n,m)) =pⁿ−p^m, by the Division Lemma 1.10. Multiplication by T is aZp-linear map T :Z −→Z with eigenvaluesζ_p^l−1,m < l≤n.

Λ/(F_A(L), ν_(n,m)) is the cokernel of the linear map onZ given by multipli-cation by F_A(L)(T). This map has eigenvaluesF_A(L)(ζ_pl−1), m < l≤n, and the order of the cokernel equals the p-valuation of the determinant, which is the product of the eigenvalues. Note that ν_(n,m) is coprime to F_A(L)(T) for every n ≥m ≥e(L/K), by Proposition 1.44, and therefore F_A(L)(ζ_p^l−1)6= 0 for each m < l ≤n, i.e., |Λ/(F_A(L), ν_(n,m))| <∞ and rank_ν(n,m)(A^(L))<∞.

More generally, for every divisorf_j(T)^lj ofF_A(L)(T) arising in the decom-position ofE_A(L), we have|Λ/(ν_(n,m), fj(T)^lj)| = p^{(n−m)·lj·deg(fj(T))}. In particular,

rank_ν(n,m)(A^(L)) ≤ rank_ν(n,m)( ˜E_A(L)) +v_p(|M₁^(L)|)

| {z }

=:C

= rank_ν(n,m)(E_A(L)) + C (3.2)

= (pⁿ−p^m)·µ+ (n−m)·λ + C . Now we choose a neighbourhood U(L, w0) ofL such that

rankν_(n,m)(A^(M)) = rankν_(n,m)(A^(L)) < ∞ for every M ∈U(L, w0), using Lemma 3.44. Then

rankν_(n,m)(E_A(M)) = rankν_(n,m)( ˜E_A(M)) = rankν_(n,m)(A^(M)/M₁^(M))

≤ rankν_(n,m)(A^(M)) = rankν_(n,m)(A^(L)) (3.3) for these M.

LetM ∈U(L, w₀) be arbitrary, but fixed. We will develop a formula that will be useful to bound µ- and λ-invariants. The latter means bounding the degree λ^(M) := λ(M/K) of the characteristic polynomial F_A(M)(T).

For arbitraryl∈ {m+ 1, . . . , n}, it is not clear whether ^λ(M)

p^l−1(p−1) <1, i.e., whether (3.1) holds for F_A(M)(T).

We therefore letl1, . . . , lr∈ {m+ 1, . . . , n}denote the values oflfor which (3.1) fails. Then l_i =m+i, 1≤i≤r. Thus, v_p(F_A(M)(ζ_pl−1)) ≥1 for l ≤ m+r, and v_p(F_A(M)(ζ_pl −1)) = _pl−1^λ(M)(p−1) < 1 for l > m+r. Note that at the moment, we have not said anything aboutr (sor =n−m is possible) and therefore have not boundedλ^(M) yet.

However, we know that

rank_ν(n,m)(E_A(M)) = (pⁿ−p^m)·µ(M/K) +|Λ/(F_A(M)(T), ν_(n,m))|

(3.3)

≤ rankν_(n,m)(A^(L))

(3.2)

≤ (pⁿ−p^m)·µ+ (n−m)·λ+C ,

whereC =vp(|M₁^(L)|) has been defined above, and therefore

(pⁿ−p^m)µ(M/K) +p^m(p−1) +p^m+1(p−1) +. . .+p^m+r−1(p−1) + (n−m−r)·λ^(M)

= p^m(p^n−m−1)µ(M/K) +p^m(p^r−1) + (n−m−r)·λ^(M) (3.4)

≤ p^m(p^n−m−1)µ+ (n−m)·λ+C .

For every pair of integersn > m ≥e(L/K), we have found a neighbour-hood U(L, w₀), w₀ =w₀(n, m), such that for every M ∈U(L, w₀), (3.4) holds with a suitable integer

r=r(n, m, M)∈ {0, . . . , n−m}.

We will now choose special values fornandm, namely sequences (ni)i≥0, (m_i)i≥0 defined byn_i := 2iandm_i:=ifor everyi≥0. Ifi₁ ≥e(L/K) is large enough to ensure that _pi1−1^λ(p−1) <1 and

p^mi1(pⁱ¹−1) = pⁱ¹(pⁱ¹ −1) > i1·λ+C = (ni1−mi1)·λ+C , then (3.4) implies thatµ(M/K)≤µ=µ(L/K).

(ii) If µ(L/K) = 0, then Corollary 3.22, (ii) implies that there exists some

˜

w₀ ≥ w₀ such that µ(M/K) = 0 for each M ∈ U(L,w˜₀). In particular, for these M, (3.4) reduces to

p^m(p^r−1) + (n−m−r)·λ^(M) ≤ (n−m)·λ+C . Ifi2≥i1 is large enough to ensure that

p^mi2 = pⁱ² > i2·λ+C = (ni2 −mi2)·λ+C ,

then (3.4) implies that r(n_i, m_i, M) = 0 for every i ≥ i₂ and every M contained in the neighbourhood U(L, wi) ⊆ U(L,w˜0) corresponding to the pair (ni, mi).

Therefore, (3.4) yields

i·λ^(M) = (ni−mi)·λ^(M) ≤ i·λ+C (3.5) for everyi≥i2. Let now i≥max(i2, C+ 1). Ifλ^(M) > λ, then

(C+ 1)·(λ^(M)−λ) ≤ i·(λ^(M)−λ) ≤ C , contradiction. Thereforeλ^(M)≤λfor every M ∈U(L, w_i).

(iii) IfM satisfiesµ(M/K) =µ, then we may subtract µ·(pⁿ−p^m) on both sides of the inequality (3.4) and obtain the same inequality as in the proof of (ii); we then may proceed as above.

Note that (3.4) in general does not yield bounds for the λ-invariants of Zp-extensionsM/K withµ(M/K)≤µ−1, sincer≤n−mand therefore in this case, the inequality

p^m·(p^r−1) ≤ (µ−µ(M/K))

| {z }

≥1

·p^m·(p^n−m−1)

3.3. LOCAL BOUNDEDNESS RESULTS 109 is always true, for every choice of n and m. We will have to put further technical restrictions on the characteristic polynomialsF_A(M)(T) in order to obtain results concerning such Zp-extensions (compare Lemma 3.62 below).

(iv) In the proof of the preceding statements, we have used the inequality rank_ν(n,m)(A^(M)/M₁^(M))≤rank_ν(n,m)(A^(M)) (compare (3.3)). We can make this more precise, using the following

Proposition 3.58. Let λ ∈ Λ denote either a distinguished polynomial, or λ=p. Let M/K denote a Zp-extension, and assume thatλis coprime to the characteristic polynomial F_A(M)(T) of A^(M) (this means that we want λ6=p if µ(M/K)6= 0). Then

rank_λ(A^(M)) = rank_λ(A^(M)/M₁^(M)) + rank_λ(M₁^(M))

= rank_λ(A^(M)/M₁^(M)) + v_p(|M₁^(M)/(λ·M₁^(M))|). Proof. We will make use of Proposition 3.43. Recall that for every element λ∈Λ and every Λ-moduleN, we defined N[λ] :={x∈N |λ·x= 0}and Qλ(N) := _|N/(λ·N^|N[λ]|_)|, whenever both orders are finite (compare Definition 3.42).

In our situation, Proposition 3.43, (ii), applied to the exact sequence 0−→M₁^(M)−→A^(M)−→A^(M)/M₁^(M)−→0,

implies that Q_λ(A^(M)) = Q_λ(A^(M)/M₁^(M))·Q_λ(M₁^(M)). Since M₁^(M) is finite, we haveQ_λ(M₁^(M)) = 1, by Proposition 3.43, (i).

Furthermore, λ acts injectively on A^(M)/M₁^(M) ∼= E_A(M), using our as-sumption that λ is coprime to the characteristic polynomial of A^(M). ThereforeA^(M)[λ]⊆M₁^(M)[λ], andQλ(A^(M)/M₁^(M)) =p^{−rankλ(A(M)/M1(M))}. It follows that

p^{−rankλ(A(M)/M1(M))} = Q_λ(A^(M)) = |A^(M)[λ]|

|A^(M)/(λ·A^(M))|

= |M₁^(M)[λ]|

p^rankλ(A(M)) = p^{rankλ(M1(M))−rankλ(A(M))} , proving Proposition 3.58.

We therefore may replace inequality (3.3) by the equality

rank_ν(n,m)(A^(M)/M₁^(M)) + rank_ν(n,m)(M₁^(M)) = rank_ν(n,m)(A^(M)). Now

|M₁^(M)/(ν_(n,m)·M₁^(M))| =

n−1

Y

i=m

|(ν_(i,m)·M₁^(M))/(ν_(i+1,m)·M₁^(M))|.

Applying Nakayama’s Lemma 1.43 to the compact Λ-module M₁^(M), we see that

either M₁^(M) ={0} or ν_(m+1,m)·M₁^(M) 6=M₁^(M), i.e., |M₁^(M)/(ν_(m+1,m)·M₁^(M))| ≥p. Analogously,

either ν_(m+1,m)·M₁^(M)={0} or ν_(m+2,m)·M₁^(M)6=ν_(m+1,m)·M₁^(M), i.e., |(ν_(m+1,m)·M₁^(M))/(ν_(m+2,m)·M₁^(M))| ≥p and therefore

rankν_(n,m)(M₁^(M)) ≥ 2.

Inductively, we obtain that rank_ν(n,m)(M₁^(M))≥n−mas long as we don’t have ν_(n,m)·M₁^(M) ={0}. But in the latter case, M₁^(M)[ν_(n,m)] =M₁^(M), and therefore rankν_(n,m)(M₁^(M)) =vp(|M₁^(M)|). We have thus shown that

rank_ν(n,m)(M₁^(M)) ≥ n−m whenever |M₁^(M)| ≥ p^n−m. More generally, this argument shows that for every j≤n−m, we have

rankν_(n,m)(M₁^(M)) ≥ j whenever |M₁^(M)| ≥ p^j .

Choosingn_i= 2iandm_i=iwithi≥max(i₂, C+1), as in the proof of (ii), the inequality (3.5) from that proof yields that for every j≤i=n_i−m_i and everyM ∈U(L, wi) satisfying µ(M/K) =µ(L/K),

either |M₁^(M)|< p^j or i·λ^(M)+j≤i·λ+C . In particular, ifM ∈U(L, w_i) also satisfiesλ^(M) =λ, then

either |M₁^(M)|< p^j or j≤C for everyj≤i. Letting j=C+ 1, we may conclude that

|M₁^(M)| < p^C+1 = p· |M₁^(L)|, using the definition ofC, and therefore

|M₁^(M)| ≤ |M₁^(L)|.

Remark 3.59. Note that actually we have proved a bit more: If we apply Proposition 3.58 to bothA^(M) and A^(L), then we can turn the inequality (3.2) into an equality and replace the right-hand sidei·λ+C of (3.5) by the better upper boundi·λ+|M₁^(L)[ν_(ni,mi)]|. This means that

|M₁^(M)[ν_(ni,mi)]| = |M₁^(L)[ν_(ni,mi)]|

3.3. LOCAL BOUNDEDNESS RESULTS 111 for every M ∈ U(L, wi) satisfying λ^(M) = λ, provided that i ≥ i2. In particular, ifi≥vp(|M₁^(L)|)≥vp(|M₁^(M)|), then

M₁^(L)[ν_(ni,mi)] = M₁^(L) and M₁^(M)[ν_(ni,mi)] = M₁^(M)

for every M ∈ U(L, wi) satisfying µ(M/K) = µ and λ(M/K) = λ, and thus

|M₁^(M)| = |M₁^(L)|

for these M. We will give another proof of this fact in Corollary 3.75, under the assumption that µ(L/K) = 0.

Now fix i as in Remark 3.59. For M ∈ U(L, w_i), we let Y_i^(M) denote the kernel of the i-th projection mapA^(M) −→A^(M_i ⁾, respectively. Then A^(Mn ⁾ ∼= A^(M)/(ν_(n,i)·Y_i^(M)) for everyn≥i. Moreover,

|A^(M)_n | = |A^(M)/(ν_(n,i)·A^(M))| · |(ν_(n,i)·A^(M))/(ν_(n,i)·Y_i^(M))|

= p^{rankν(n,i)(A}

(M))

· |A^(M)_i | ·|Y_i^(M)∩M₁^(M)|

|M₁^(M)| for every n≥2i, because the map

φ_(n,i) : A^(M)/Y_i^{(M) //} (ν_(n,i)·A^(M))/(ν_(n,i)·Y_i^(M))

given by multiplication by ν_(n,i) is a surjective homomorphism having kernel (Y_i^(M) + M₁^(M))/Y_i^(M) (apply Proposition 1.44 and use the fact that ν_(n,i) annihilates M₁^(M) ifn−i≥i).

In particular, if U ⊆U(L, wi) is a sufficiently small neighbourhood, then

|Y_i^(M) ∩ M₁^(M)| = |Y_i^(L) ∩ M₁^(L)|

for every M ∈U. SinceY_i^(M) ∩M₁^(M) is the maximal finite Λ-submodule of Y_i^(M) ⊆A^(M), respectively, Proposition 3.58 implies that

rankν_(n,m)(Y_i^(M)) = rankν_(n,m)(Y_i^(L))

for every M ∈U and for all pairsn > m≥isatisfying n−m=m.

But then

|A^(M)_n | = |A^(M)_i | · |Y_i^(M)/(ν_(n,i)·Y_i^(M))| = |A^(L)_n |

for arbitrarily largen, proving thatν(M/K) = ν(L/K) for everyM ∈U. Corollary 3.60. Suppose that K/K is a Z^kp-extension, k ∈ N. Let E^⊆K(K) denote the set of Zp-extensions of K contained in K. If there exist an inte-ger 0 ≤ µ ∈ Z and a set P ⊆ I of prime ideals of K dividing p such that µ(M/K) =µ and P(M) =P for every M ∈ E^⊆K(K), then λ(M/K) is glob-ally boundedon the set E^⊆K(K).

Proof. Since µ(M/K) = µ for every M ∈ E^⊆K(K), λ is locally maximal, and in particular locally bounded, in appropriate neighbourhoods of every M ∈ E^⊆K(K). Moreover, since P(M) = P for every M, the set E^⊆K(K) is compact with respect to the Greenberg-R-topology (compare Remarks 3.26, (2)). This proves the corollary.

Remark 3.61. Note that this corollary generalises Greenberg’s Theorem 2.30, which is the caseµ= 0 andP =I ={p}.

We cannot say much in the case of a ‘jump’ of the µ-invariant, i.e., if µ(M/K) < µ(L/K) for some M ∈ U(L, n). In order to obtain boundedness results in this situation, we have to put technical assumptions on the involved characteristic polynomials, as in the following lemma. In fact, it seems likely that theλ-invariant can be unbounded in the neighbourhood of aZp-extension Lwhoseµ-invariant is ‘isolated’, i.e., ifµ(L/K)> µ(M/K) for infinitely many M contained in some small neighbourhood of L(compare Theorem 4.43).

Lemma 3.62. Let L/K be a Zp-extension. Write µ:= µ(L/K). Let further y ∈ N0. As in Theorem 3.57, we define the set E^y(K) to consist of those Zp-extensions M/K satisfying µ(M/K) = y. For every x ∈ N, we let E_x(K) denote the set of Zp-extensions M ∈ E(K) such that every coefficient of the characteristic polynomial F_A(M)(T), besides the leading coefficient, is divisible by p^x, respectively. Then there exists an integer w∈N such that the following holds:

For 0≤x≤µ,λ is bounded in each of the setsE^µ−x(K)∩ E_x+1(K)∩U(L, w), respectively.

Proof. We will use the notation introduced in the proof of Theorem 3.57. If F_A(M)(T) is a distinguished polynomial such that every coefficient, besides the leading one, is divisible byp^x+1, then either

v_p(F_A(M)(ζ_pl−1)) = λ^(M)

p^l−1(p−1) < x+ 1,

or v_p(F_A(M)(ζ_pl−1)) ≥ x+ 1. This means that the inequality (3.4) from the proof of Theorem 3.57 may be strengthened to

(pⁿ−p^m)·µ(M/K) + (x+ 1)·p^m(p^r−1) + (n−m−r)·λ^(M)

≤ (pⁿ−p^m)·µ + (n−m)·λ + C .

Choosing the sequences m_i :=i→ ∞,n_i :=m_i+i,i≥0, this implies that for ilarge enough to ensure that _pi−1^λ(p−1) <1, we have

p^mi(pⁱ−1)·µ(M/K) + (x+ 1)·p^mi(p^ri−1) + (i−ri)·λ^(M)

≤ p^mi(pⁱ−1)·µ + λ+ C .

Choosing i1 ∈Nsuch thatp^mi1(pⁱ¹ −1)> λ+C, we may conclude that ri < i for every pair (n_i, m_i) with i ≥ i₁ and every M contained in the neighbour-hood U(L, wi) ∩ E_x+1(K) ∩ E^µ−x(K) of L. In particular, by definition of ri,

3.3. LOCAL BOUNDEDNESS RESULTS 113

λ^(M)

p^l−1(p−1) < x+ 1 forl=mi1 +i1=ni1, and therefore λ^(M) < (x+ 1)·p²ⁱ¹⁻¹(p−1) is bounded in this (restricted) neighbourhood ofL.

Im Dokument A new approach to the investigation of Iwasawa invariants (Seite 117-125)