Iwasawa Theory

Chapter 6
The Iwasawa main conjecture

6.1. Semi-local units modulo cyclotomic units

Let $F = ℚ (μ_{𝑚𝑝})$ , where $p ∤ m$ and $m ≢ 2 mod 4$ . Let us first observe that we have a map $Col : 𝒰_{\infty}^{(χ)} \to Λ_{χ}$ . Note that each place of $ℚ (μ_{m})$ over $p$ is totally ramified in $F_{\infty}$ . Let $V_{p}$ denote the set of places over $p$ in any intermediate field. Recall that

𝒰_{\infty} ≅ ⨁_{v \in V_{p}} 𝒰_{v, \infty} .

Let $𝒪$ be the valuation ring of a place $w$ of $ℚ (μ_{m})$ over $p$ . (Note that such places are totally ramified in $F = ℚ (μ_{𝑚𝑝})$ , and even in $F_{\infty}$ .) Then $𝒪$ is free of rank one over $ℤ_{p} [Δ_{p}]$ , for $Δ_{p}$ the decomposition group at $p$ in $Δ = Gal (F ∕ ℚ)$ . Giving

⨁_{v \in V_{p}} 𝒪

the structure of a $ℤ_{p} [Gal (ℚ (μ_{m}) ∕ ℚ)]$ -module by allowing $Δ$ to permute the factors, it is then is free of rank one as a $ℤ_{p} [Gal (ℚ (μ_{m}) ∕ ℚ)]$ -module, and we can think of $1$ in the term for $w$ as a generator. Similarly, then, we obtain

\tilde{Λ} = ℤ_{p} ⟦ ℤ_{p, m}^{\times} ⟧ ≅ \prod_{v \in V_{p}} 𝒪 ⟦ ℤ_{p}^{\times} ⟧,

and we can use this to define a Coleman map $Col : 𝒰_{\infty} \to \tilde{Λ}$ as the direct sum of the Coleman maps at the places over $p$ .

Proposition 6.1.1. §

There is a homomorphism $Col : 𝒰_{\infty} \to \tilde{Λ}$ induced by the Coleman maps at the places of $F_{\infty}$ above $p$ , fitting in an exact sequence

0 \to ℤ_{p} [Δ ∕ Δ_{p}] (1) \to 𝒰_{\infty} \to_{}^{Col} \tilde{Λ} \to ℤ_{p} [Δ ∕ Δ_{p}] (1) \to 0

of $\tilde{Λ}$ -modules.

Corollary 6.1.2. §

Let $χ$ be a nontrivial Dirichlet $p$ -adic character of conductor dividing $𝑚𝑝$ . Then the Coleman map induces a map $𝒰_{\infty}^{(χ)} \to Λ_{χ}$ which is an isomorphism if and only if $χ ω^{- 1} (p) - 1 \in 𝒪_{χ}^{\times}$ .

We aim to prove the following theorem that, at least in the case of $F = ℚ (μ_{p})$ , was proven by Iwasawa.

Theorem 6.1.3. §

Let $χ$ be a nontrivial, even primitive Dirichlet character of conductor $m$ or $𝑚𝑝$ , where $p$ is an odd prime and $m$ is a positive integer prime to $p$ . Let $F = ℚ (μ_{𝑚𝑝})$ , and let $F_{\infty}$ be its cyclotomic $ℤ_{p}$ -extension. Then there is an exact sequence

0 \to A_{χ} (1) \to 𝒰_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)} \to Λ_{χ} ∕ (g_{χ}^{'}) \to 0,

where $A_{χ} = 𝒪_{χ} ∕ (χ ω^{- 1} (p) - 1)$ , and where $g_{χ}^{'} = T^{- δ} g_{χ}$ for $δ = 1$ if $χ ω^{- 1} (p) = 1$ and $δ = 0$ otherwise, where $g_{χ} \in Λ_{χ}$ satisfies

g_{χ} (u^{s} - 1) = L_{p} (χ, 1 - s)

for all $s \in ℤ_{p}$ .

Corollary 6.1.4. §

With the notation of Theorem 6.1.3, suppose that $χ ω^{- 1} (p) - 1 \in 𝒪_{χ}^{\times}$ . Then

𝒰_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)} ≅ Λ_{χ} ∕ (g_{χ}) .

Proposition 6.1.5. §

Suppose that $p ∤ φ (m)$ . Let $χ$ be a nontrivial, even Dirichlet $p$ -adic character of conductor dividing $𝑚𝑝$ , and let $f$ be the prime-to- $p$ part of its conductor. The $Λ_{χ}$ -module $𝒞_{\infty}^{(χ)}$ is generated by the image of the norm compatible sequence ${(1 - ζ_{f p^{n}})}_{n}$ .

Proof.

Since only primes over $p$ ramify in $F_{\infty} ∕ F$ , the norm compatible sequences of elements of $F_{n}^{\times}$ are all norm compatible sequences of $p$ -units. The group of norm compatible sequences of cyclotomic $p$ -units in $F_{\infty} ∕ F$ is generated as a $\tilde{Λ}$ -module by ${(1 - ζ_{d p^{n}})}_{n}$ for $d$ dividing $m$ . Such a sequence is of true units if $f \neq 1$ . From this, it is easy to see that $𝒞_{\infty}$ is similarly generated by the ${(1 - ζ_{d p^{n}})}_{n}$ and ${({(1 - ζ_{p^{n}})}^{σ_{c} - 1})}_{n}$ , where $c \in ℤ$ is a primitive root modulo $p$ . The elements $e_{χ} {(1 - ζ_{d p^{n}})}_{n}$ vanish unless $f$ is a multiple of $d$ . On the other hand, $e_{χ} (1 - ζ_{d p^{n}})$ for $d$ a multiple of $f$ equals $e_{χ}$ times the norm for $ℚ (ζ_{d p^{n}}) ∕ ℚ (ζ_{f p^{n}})$ of the elements $(1 - ζ_{d p^{n}})$ . That is, $e_{χ} {(1 - ζ_{d p^{n}})}_{n}$ is a $\tilde{Λ}$ -multiple of $e_{χ} {(1 - ζ_{f p^{n}})}_{n}$ for $f$ dividing $d$ . □

Let us now focus on the case that $F = ℚ (μ_{p})$ , for which we suppose that $E = ℚ_{p}$ . Note that $U_{\infty} = 𝒰_{\infty} \times μ_{p - 1}$ . Consider the cyclotomic unit

u_{n, c} = \frac{ζ_{p^{n}}^{- c ∕ 2} - ζ_{p^{n}}^{c ∕ 2}}{ζ_{p^{n}}^{- 1 ∕ 2} - ζ_{p^{n}}^{1 ∕ 2}}

for $c$ prime to $p$ , and let $u_{c} = {(u_{n, c})}_{n} \in U_{\infty}$ . Let ${\tilde{ζ}}_{p, c} = Col (u_{n, c})$ .

Proposition 6.1.6. §

For $k \geq 1$ , we have

\int_{ℤ_{p}^{\times}} x^{k} d {\tilde{ζ}}_{p, c} (x) = (1 - c^{k}) (1 - p^{k - 1}) ζ (1 - k) = (1 - c^{k}) L_{p} (ω^{k}, 1 - k) .

Proof.

We employ the Coleman power series

f (T) = \frac{{(1 + T)}^{- c ∕ 2} - {(1 + T)}^{c ∕ 2}}{{(1 + T)}^{- 1 ∕ 2} - {(1 + T)}^{1 ∕ 2}}

and the change of variables $T = e^{t} - 1$ . By Lemma 5.4.32, we have that

δ_{k} (u_{c}) = \frac{d^{k}}{d t^{k}} \log f (e^{t} - 1) ∣_{t = 0} .

We have

\begin{array}{l} \frac{d}{𝑑𝑡} \log f (e^{t} - 1) & = \frac{1}{2} (\frac{1}{e^{- t} - 1} - \frac{1}{e^{t} - 1}) - \frac{c}{2} (\frac{1}{e^{- 𝑐𝑡} - 1} - \frac{1}{e^{𝑐𝑡} - 1}) \\ = \sum_{k = 0}^{\infty} \frac{B_{k}}{2 \cdot k!} ({(- t)}^{k - 1} - t^{k - 1} + c ({(𝑐𝑡)}^{k - 1} - {(- 𝑐𝑡)}^{k - 1})) = \sum_{k = 0}^{\infty} \frac{B_{k}}{k!} (c^{k} - 1) t^{k - 1}, \end{array}

δ_{k} (u_{c}) = (c^{k} - 1) \frac{B_{k}}{k} = (1 - c^{k}) ζ (1 - k) .

The result then follows from Proposition 5.4.35. □

We then have the following.

Corollary 6.1.7. §

The $ℤ_{p}$ -valued measure $E_{c}^{(0)}$ on $ℤ_{p}^{\times}$ satisfying

\int_{ℤ_{p}^{\times}} h (x) d E_{c}^{(0)} (x) = \int_{ℤ_{p}^{\times}} x^{- 1} h (x) d E_{c}^{(1)} (x)

for all $h \in C (ℤ_{p}, ℂ_{p})$ is equal to ${\tilde{ζ}}_{p, c}$ .

Proof

Sketch of proof of Theorem 6.1.3 for $ℚ (μ_{p})$ . Note that $𝒞_{\infty}$ is topologically generated by the elements $u_{c}$ , and in particular it is generated as a $ℤ_{p} ⟦ ℤ_{p}^{\times} ⟧ ≅ Λ [Δ]$ -module, where $Δ = Gal (F_{\infty} ∕ ℚ_{\infty})$ , by $u_{c}$ for any integer $c$ that is a primitive root modulo $p$ . Proposition 6.1.6 tells us that

Col (𝒞_{\infty}) = ℤ_{p} ⟦ ℤ_{p}^{\times} ⟧ {\tilde{ζ}}_{p, c} .

Note that ${\tilde{ζ}}_{p} = {(1 - σ_{c})}^{- 1} {\tilde{ζ}}_{p, c}$ is independent of $c$ , though it is not quite integral, though it becomes integral up application of any element in the augmentation ideal $I$ of $Λ [Δ]$ . If $k ≢ 0 mod p - 1$ , then $1 - c^{k} \in ℤ_{p}^{\times}$ . The “equivariant” version of Iwasawa’s theorem is then proven: it reads

𝒰_{\infty} ∕ 𝒞_{\infty} ≅ Λ [Δ] ∕ I {\tilde{ζ}}_{p} .

Recall that a character $ω^{k}$ of $Δ$ defines an in this case surjective homomorphism $\tilde{ω^{k}} : Λ [Δ] \to Λ$ of $Λ$ -algebras. For even $k$ , the image $\tilde{ω^{k}} ({\tilde{ζ}}_{p})$ is nonzero, and it is integral if and only if $k ≢ 0 mod p - 1$ . A simple check yields using Corollary 6.1.7 yields that the power series corresponding to $\tilde{ω^{k}} ({\tilde{ζ}}_{p})$ (or $T$ times it if $k \equiv 0 mod p - 1$ ) is $g_{ω^{k}}$ , so we have Iwasawa’s theorem. □

6.2. The Ferrero-Washington theorem

Theorem 6.2.1 (Ferrero-Washington). §

Let $F$ be a finite abelian extension of $ℚ$ , and let $F_{\infty}$ be its cyclotomic $ℤ_{p}$ -extension for a prime $p$ . Then $μ (X_{\infty}) = 0$ .

The following is immediate from the theorem and Proposition 3.4.2.

Corollary 6.2.2. §

Let $F$ be an abelian extension of $ℚ$ , and let $F_{\infty}$ be its cyclotomic $ℤ_{p}$ -extension for an odd prime $p$ . Then the $p$ -torsion subgroup of $X_{\infty}^{-}$ is zero.

In this section, we prove the Ferrero-Washington theorem in the case of $F = ℚ (μ_{p})$ for an odd prime $p$ . We follow their original proof in this case.

Notation 6.2.3. §

For $a \in ℤ_{p}$ and a nonnegative integer $m$ , let ${[a]}_{m} \in ℤ$ denote the unique integer with $0 \leq a < p^{m + 1}$ to which $a$ is congruent modulo $p^{m + 1}$ . Let $δ_{0} (a) = {[a]}_{0}$ and $δ_{m} (a) = p^{- m} ({[a]}_{m} - {[a]}_{m - 1})$ if $m \geq 1$ .

We may think of $δ_{m} (a)$ as the coefficient of $p^{m}$ in the usual $p$ -adic expansion of $a$ .

Proposition 6.2.4. §

The $μ$ -invariant of $X_{\infty}$ is nonzero if and only if there exists an even integer $k ≢ 0 mod p - 1$ such that

\sum_{ξ \in μ_{p - 1} (ℤ_{p})} δ_{m} (𝑎𝜉) ξ^{k - 1} \equiv 0 mod p

for all $m \geq 0$ and all $a \in ℤ_{p}$ .

Proof.

Since $X_{\infty}^{(ω)}$ is trivial, we need only show that the $μ$ -invariant $μ_{k}$ of $X_{\infty}^{(ω^{1 - k})}$ is zero for every even $k$ with $2 \leq k \leq p - 3$ . Since $f_{ω^{k}}$ annihilates $X_{\infty}^{(ω^{1 - k})}$ , it suffices to show that $f_{ω^{k}}$ is not in $p ℤ_{p} ⟦ T ⟧$ . For $b \in ℤ_{p}$ , let $1 \leq i_{m} (b) \leq p^{m}$ be such that ${⟨ b ⟩}_{p} \equiv {(1 + p)}^{i_{m} (b)} mod p^{m + 1}$ . The expression for $f_{ω^{k}}$ given by Remark 5.3.10 reduces to

f_{ω^{k}} \equiv - \frac{1}{p^{m}} \sum_{\begin{array}{c} b = 1 \\ p ∤ b \end{array}}^{p^{m + 1}} b ω^{k - 1} (b) {(T + 1)}^{p^{m} - i_{m} (b)} mod ω_{m} .

Since $ω_{m} \equiv T^{p^{m}} mod p$ , the congruence holds modulo $(p, T^{p^{m}})$ as well. To say that $μ_{k}$ is nonzero is then equivalent to saying that every coefficient of a power of $T + 1$ in each such expansion as we vary $m$ is zero. Let $T_{m} (a)$ denote the set of positive integers $b < p^{m + 1}$ that are prime to $p$ and satisfy $i_{m} (b) \equiv i_{m} (a) mod p^{m}$ . By what we have just said, we have $μ_{k} > 0$ if and only if

\sum_{b \in T_{m} (a)} b ω^{k - 1} (b) \equiv 0 mod p^{m + 1}

for all $a \in ℤ_{p}$ and $m \geq 0$ . Note that $i_{m} (b) \equiv i_{m} (a) mod p^{m + 1}$ if and only if there exists $ξ \in μ_{p - 1} (ℤ_{p})$ such that ${[b]}_{m} = {[𝜉𝑎]}_{m}$ . For given $a \equiv 1 mod p$ and $ξ$ there is exactly one $0 < b < p^{m}$ with $p ∤ b$ having this property. Since $ω (b) = ω (ξ)$ , we then have $μ_{k} > 0$ if and only if

\sum_{ξ \in μ_{p - 1} (ℤ_{p})} {[𝑎𝜉]}_{m} ξ^{k - 1} \equiv 0 mod p^{m + 1} .

As $δ_{0} (𝑎𝜉) = {[𝑎𝜉]}_{0}$ and $δ_{m} (𝑎𝜉) = p^{- m} ({[𝑎𝜉]}_{m} - {[𝑎𝜉]}_{m - 1})$ for all $m \geq 1$ , the result follows from the equivalence of $μ_{k} > 0$ with the latter congruences. □

Definition 6.2.5. §

A sequence ${(b_{i})}_{i \geq 1}$ of tuples in $[0, 1)^{r}$ is uniformly distributed if for every product $U \subseteq {(0, 1)}^{r}$ of open intervals in $(0, 1)$ , the volume of $U$ is proportional by a positive real number, independent of $U$ , to the density of the $b_{i}$ in $U$ , which is to say the limit of $\frac{1}{N} | {i \leq N ∣ b_{i} \in U} |$ as $N \to \infty$ .

Definition 6.2.6. §

For $r \geq 1$ , we say that $(a_{1}, \dots, a_{r}) \in ℤ_{p}^{r}$ is normal if the sequence of tuples

(p^{- m} {[a_{1}]}_{m - 1}, \dots, p^{- m} {[a_{r}]}_{m - 1})

with $m \geq 1$ is uniformly distributed in $[0, 1)^{r}$ .

We omit the proof of the following.

Theorem 6.2.7 (Weyl). §

A sequence ${(b_{i, 1}, \dots, b_{i, r})}_{i \geq 1}$ of tuples in $[0, 1)^{r}$ is uniformly distributed if and only if for every tuple $(t_{1}, \dots, t_{r}) \in ℤ^{r} - {0}$ , we have

\lim_{N \to \infty} \frac{1}{N} \sum_{i = 1}^{N} e^{2 𝜋𝑖 \sum_{j = 1}^{r} b_{i, j} t_{j}} = 0 .

Proposition 6.2.8. §

For $r \geq 1$ , let $b_{1}, \dots, b_{r} \in ℤ_{p}$ be such that $b_{1}, \dots, b_{r}$ are $ℚ$ -linearly independent. Then the complement of the set of $a \in ℤ_{p}$ with $(a b_{1}, \dots, a b_{r})$ normal has Haar measure zero.

Proof.

Let $t = (t_{1}, \dots t_{r}) \in ℤ^{r} - {0}$ , and let $c = \sum_{j = 1}^{r} b_{j} t_{j}$ , which is nonzero by the linear independence of the $b_{j}$ . For $a \in ℤ_{p}$ , we have

{[𝑎𝑐]}_{m - 1} \equiv 𝑎𝑐 \equiv \sum_{j = 1}^{r} a b_{j} t_{j} \equiv \sum_{j = 1}^{r} {[a b_{j}]}_{m - 1} t_{j} mod p^{m} .

Therefore, that $(a b_{1}, \dots, a b_{r})$ is normal is equivalent by the criterion of Weyl to the statement that

\lim_{N \to \infty} \frac{1}{N} \sum_{m = 1}^{N} e^{2 𝜋𝑖 p^{- m} {[𝑎𝑐]}_{m - 1}} = 0

for all $t$ . We claim that this holds outside a set of $a$ of measure zero for each $t$ . Since there are only countably many $t$ , this implies the result. We also suppose that $t \notin p ℤ^{r}$ , as the convergence to zero of limit in question is unaffected by dividing by a power of $p$ .

Set

p_{N} (a) = \frac{1}{N} \sum_{m = 1}^{N} e^{2 𝜋𝑖 p^{- m} {[𝑎𝑐]}_{m - 1}},

and note that

\int_{ℤ_{p}} | p_{N} (a) |^{2} 𝑑𝑎 = \frac{1}{N} + \frac{1}{N^{2}} \sum_{\begin{array}{c} m, n = 1 \\ m \neq n \end{array}}^{N} \int_{ℤ_{p}} e^{2 𝜋𝑖 (p^{- n} {[𝑎𝑐]}_{n - 1} - p^{- m} {[𝑎𝑐]}_{m - 1})} 𝑑𝑎 = \frac{1}{N},

each integral in the latter sum being zero, being a multiple of a sum over all $p^{n}$ th (resp., $p^{m}$ th) roots of unity if $n > m$ (resp., $m > n$ ). It follows that

\sum_{M = 1}^{\infty} \int_{ℤ_{p}} | p_{M^{2}} (a) |^{2} 𝑑𝑎 = \sum_{M = 1}^{\infty} \frac{1}{M^{2}} = \frac{π^{2}}{6}

is finite, which forces $\lim_{M \to \infty} p_{M^{2}} (a) = 0$ outside of a set of measure zero. Note also that for any $N \in ℤ$ with $M^{2} \leq N < {(M + 1)}^{2}$ , we have

| p_{N} (a) | < | p_{M^{2}} (a) | + \frac{2 M}{N} \leq | p_{M^{2}} (a) | + \frac{2}{M},

as $p_{N} (a) - p_{M^{2}} (a)$ is a sum of $N - M^{2}$ roots of unity. Thus ${(p_{N} (a))}_{N}$ has limit $0$ outside of the same measure zero set. □

Proposition 6.2.9. §

Set $s = \frac{p - 1}{2}$ and $r = φ (p - 1)$ . Let $b_{1}, \dots, b_{s} \in ℤ_{p}$ be such that $(b_{1}, \dots, b_{r})$ is normal, $b_{i} b_{1}^{- 1} \notin ℤ$ for all $2 \leq i \leq s$ , and

b_{i} = \sum_{j = 1}^{r} c_{i, j} b_{j}

for some $c_{i, j} \in ℤ$ for all $r < i \leq s$ . Then there exist nonnegative integers $m$ and $n$ such that $δ_{n} (b_{j}) = δ_{m} (b_{j})$ for all $2 \leq j \leq s$ , while $δ_{n} (b_{1}) = 1$ and $δ_{m} (b_{1}) = 0$ .

Proof.

Take $x_{1} = \frac{1}{p}$ , and let $x_{2}, \dots, x_{r} \in (0, 1)$ be such that the $x_{1}, \dots, x_{r}$ are $ℚ$ -linearly independent. For $r < i \leq s$ , set

x_{i} = ⟨ \sum_{j = 1}^{r} c_{i, j} x_{j} ⟩ .

If $x_{i} \in ℚ$ for such an $i$ , then $c_{i, j} = 0$ for $2 \leq j \leq r$ by the assumed linear indepedence, so $x_{i} = c_{1, j} x_{1}$ . But this would imply that $b_{i} b_{1}^{- 1} \in ℤ$ , contradicting our hypotheses. Thus, the $x_{i}$ for $2 \leq i \leq s$ are all irrational.

Let $y_{1} \in (0, \frac{1}{p})$ , set $y_{i} = x_{i}$ for $2 \leq i \leq r$ , and set $y_{i} = ⟨ \sum_{j = 1}^{r} c_{i, j} y_{j} ⟩$ for $r < i \leq s$ . Suppose that $x_{1} - y_{1}$ is sufficiently small so that for each $2 \leq i \leq s$ , we have an $0 \leq a < p$ such that $x_{i}, y_{i} \in (\frac{a}{p}, \frac{a + 1}{p})$ . Since $(b_{1}, \dots, b_{r})$ is normal, there exists an $m \geq 0$ such that

| p^{- m - 1} {[b_{i}]}_{m} - y_{i} | < 𝜖

for a given choice of $𝜖 > 0$ . for all $1 \leq i \leq r$ . For $r < i \leq s$ , we have

p^{- m - 1} ({[b_{i}]}_{m} - \sum_{j = 1}^{r} c_{i, j} {[b_{j}]}_{m}) \in ℤ,

| p^{- m - 1} {[b_{i}]}_{m} - y_{i} | \leq \sum_{j = 1}^{r} | c_{i, j} | | p^{- m - 1} {[b_{j}]}_{m} - y_{i} | < \sum_{i = 1}^{r} | c_{i, j} | \cdot 𝜖,

noting that both terms are in $(0, 1)$ in the middle step. We may take $𝜖$ small enough that small enough $p^{- m - 1} {[b_{i}]}_{m}$ lies in the same open interval $(\frac{a}{p}, \frac{a + 1}{p})$ as $y_{i}$ for all $1 \leq i \leq s$ . We then have

p^{- 1} δ_{m} (b_{i}) < p^{- m} {[b_{i}]}_{m} < p^{- 1} (δ_{m} (b_{i}) + 1),

so $δ_{m} (b_{i}) = ⌊ p y_{i} ⌋$ , and we note that $⌊ p y_{i} ⌋ = ⌊ p x_{i} ⌋$ for $i \geq 2$ . Since $y_{1} < \frac{1}{p}$ , we have $δ_{m} (b_{1}) = 0$ .

Now repeat the argument, but this time replace $y_{1}$ with $z_{1}$ where $\frac{1}{p} < z_{1} < \frac{2}{p}$ and $z_{1} - x_{1}$ is small enough. We then again obtain an $n \geq 0$ such that $δ_{n} (b_{i}) = ⌊ p x_{i} ⌋$ , this time for all $i$ , noting that $⌊ p x_{1} ⌋ = 1$ . Thus $δ_{n} (b_{i}) = δ_{m} (b_{i})$ for all $i \geq 2$ , while $δ_{n} (b_{1}) = 1 > 0 = δ_{m} (b_{1})$ . □

Proof

Proof of Theorem 6.2.1 for $F = ℚ (μ_{p})$ with $p$ odd. Set $s = \frac{p - 1}{2}$ and $r = φ (p - 1)$ . Let $ξ$ be a primitive $(p - 1)$ th root of unity. Note that $ξ^{s + 1} = - ξ$ , so for $a \in ℤ_{p}$ we have $δ_{m} (- 𝑎𝜉) = p - 1 - δ_{m} (𝑎𝜉)$ so long as $m \geq 1 + v_{p} (a)$ . It follows that

\sum_{i = 1}^{p - 1} δ_{m} (𝑎𝜉) ξ^{i (k - 1)} = 2 \sum_{i = 1}^{s} δ_{m} (𝑎𝜉) ξ^{i (k - 1)} - (p - 1) \sum_{i = 1}^{s} ξ^{i (k - 1)}

(6.2.1)

for all $a \in ℤ_{p}$ and even integers $k$ . The $ξ^{i}$ with $1 \leq i \leq r$ are linearly independent: in fact, they form a $ℤ$ -basis of $ℤ [μ_{p - 1}] \subset ℤ_{p}$ . Let $a \in ℤ_{p}$ be such that $(𝑎𝜉, a ξ^{2}, \dots, a ξ^{r})$ is normal, and set $b_{i} = a ξ^{i}$ for each $1 \leq i \leq s$ . Then the conditions of Proposition 6.2.9 are satisfied for the $b_{i}$ , so we can find nonnegative integers $m$ and $n$ as in its statement.

Suppose that $μ (X_{\infty}^{-}) > 0$ . By Proposition 6.2.4, there exists an even $2 \leq k \leq p - 1$ such that

\sum_{i = 1}^{p - 1} δ_{l} (𝑎𝜉) ξ^{i (k - 1)} \equiv 0 mod p

for all $l \geq 0$ . Applying (6.2.1), we then have that

\begin{array}{l} 2 ξ^{k - 1} & = 2 (\sum_{i = 1}^{s} δ_{n} (𝑎𝜉) ξ^{i (k - 1)} - \sum_{i = 1}^{s} δ_{m} (𝑎𝜉) ξ^{i (k - 1)}) \\ = \sum_{i = 1}^{p - 1} δ_{n} (𝑎𝜉) ξ^{i (k - 1)} - \sum_{i = 1}^{p - 1} δ_{m} (𝑎𝜉) ξ^{i (k - 1)} \equiv 0 mod p, \end{array}

providing the desired contradiction. □

6.3. The main conjecture over $ℚ$

In its most classical form, the main conjecture of Iwasawa theory, or Iwasawa main conjecture, states that the characteristic ideals of odd eigenspaces of $X_{\infty}$ are generated by the power series interpolating corresponding $p$ -adic $L$ -functions in the case that $F$ is an abelian field and $F_{\infty}$ is its cyclotomic $ℤ_{p}$ -extension. We refer to this as the main conjecture over the rationals, since it deals with fields cut out by abelian characters of the absolute Galois group over $ℚ$ . Its formulation in print is due to Greenberg. While the main conjecture was actually proven by Mazur and Wiles in 1984, we shall label it as a conjecture here in order to discuss its equivalent forms. We discuss its proof in later sections.

Conjecture 6.3.1 (The Iwasawa Main Conjecture). §

Let $p$ be an odd prime. Let $χ$ be a nontrivial, even finite order $p$ -adic character of $G_{ℚ}$ of conductor not divisible by $p^{2}$ , and let $F$ be the the fixed field of the kernel of $χ$ . For the cyclotomic $ℤ_{p}$ -extension $F_{\infty}$ of $F$ , we have

{char}_{Λ_{χ}} X_{\infty}^{(ω χ^{- 1})} = (f_{χ}),

where $f_{χ} \in Λ_{χ}$ satisfies

f_{χ} ({(1 + p)}^{s} - 1) = L_{p} (χ, s)

for all $s \in ℤ_{p}$ .

We can reformulate the main conjecture in terms of the $p$ -ramified Iwasawa module.

Proposition 6.3.2. §

The Iwasawa main conjecture is equivalent to the statement that

{char}_{Λ_{χ}} 𝔛_{\infty}^{(χ)} = (g_{χ}),

where $g_{χ} \in Λ_{χ}$ satisfies

g_{χ} ({(1 + p)}^{1 - s} - 1) = L_{p} (χ, s)

for all $s \in ℤ_{p}$ .

Proof.

By Corollary 3.4.9, we have a pseudo-isomorphism

𝔛_{\infty}^{(χ)} ≃ {(X_{\infty}^{(ω χ^{- 1})})}^{ι} (1),

and pseudo-isomorphic modules have the same characteristic ideal. We then have

g_{χ} (T) = f_{χ} (u {(1 + T)}^{- 1} - 1),

and the result follows. □

We can also reformulate the main conjecture as a comparison between global units modulo cyclotomic units and the plus part of the Iwasawa module. This formulation eschews the use of $L$ -functions.

Theorem 6.3.3. §

The Iwasawa main conjecture is equivalent to the statement that

{char}_{Λ_{χ}} (E_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)}) = {char}_{Λ_{χ}} (X_{\infty}^{(χ)}) .

Proof.

From the first exact sequence of Proposition 3.3.6, we obtain an exact sequence

0 \to E_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)} \to 𝒰_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)} \to 𝔛_{\infty}^{(χ)} \to X_{\infty}^{(χ)} \to 0 .

Iwasawa’s theorem tells us that the characteristic ideal of the second term has characteristic ideal $(g_{χ})$ . Since the alternating product of characteristic ideals of Iwasawa modules in an exact sequence of finite length is $1$ , we have that

{char}_{Λ_{χ}} (E_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)}) = {char}_{Λ_{χ}} (X_{\infty}^{(χ)})

if and only if ${char}_{Λ_{χ}} 𝔛_{\infty}^{(χ)} = (g_{χ})$ . The latter statement is an equivalent form of the main conjecture by Proposition 6.3.2. □

Mazur and Wiles proved the following interesting consequence of the main conjecture.

Theorem 6.3.4 (Mazur-Wiles). §

Let $p$ , $F$ , $χ$ , and $𝒪_{χ}$ be as in the Iwasawa main conjecture, and suppose that $χ$ has prime-to- $p$ order. We then have

| A_{F}^{(ω χ^{- 1})} | = | B_{1, χ ω^{- 1}} |_{χ}^{- 1},

where $| \cdot |_{χ}$ denotes the normalized multiplicative valuation on the unramified extension $𝒪_{χ}$ of $ℤ_{p}$ .

In particular, the converse to Herbrand’s theorem (due to Ribet) holds.

We also note that any one divisibility of characteristic ideals in the main conjecture for all $χ$ of the Galois group of a given totally real abelian field implies the other. This is a consequence of the following result, which can be derived using the analytic class number formula (for instance, using Sinnott’s work).

Proposition 6.3.5. §

Let $F$ be an abelian, CM extension of $ℚ$ of conductor not divisible by $p^{2}$ , and let $G = Gal (F^{+} ∕ ℚ)$ . Let $f = \prod_{χ \in \hat{G}} f_{χ} \in ℤ_{p} ⟦ T ⟧$ , and let $μ (f) = μ (Λ ∕ (f))$ and $λ (f) = λ (Λ ∕ (f))$ . Then

\begin{array}{rcl} μ (X_{\infty}^{-}) = μ (f) & and & λ (X_{\infty}^{-}) = λ (f) . \end{array}

As a final note, we treat the powers of the variable $T$ itself that appear in the ideals of the main conjecture.

Proposition 6.3.6. §

We have that $T ∣ {char}_{Λ} X_{\infty}^{(ω χ^{- 1})}$ if and only if $χ ω^{- 1} (p) = 1$ .

Proof.

Let $N_{F_{\infty} ∕ F} : E_{\infty} \to E_{F}$ be projection to the first term of a norm compatible sequence. Consider the exact sequence

E_{F} ∕ N_{F_{\infty} ∕ F} E_{\infty} \to \ker (⨁_{v \in V_{p} (F)} Γ_{v} \to Γ) \to {(X_{\infty})}_{Γ} \to A_{F}

that exists by Theorem 1.3.15. We take $ω χ^{- 1}$ -eigenspaces. Since $A_{F}$ is finite, $E_{F}^{(ω χ^{- 1})} = 0$ , and $Γ^{(ω χ^{- 1})} = 0$ , we have that

{(X_{\infty}^{(ω χ^{- 1})})}_{Γ} ≃ {(⨁_{v \in V_{p} (F)} Γ_{v})}^{(ω χ^{- 1})},

and the latter isomorphic to $ℤ_{p}$ or $0$ depending on whether $χ ω^{- 1} (p) = 1$ or not. □

Theorem 6.3.7 (Ferrero-Greenberg). §

We have $T^{2} ∤ f_{χ}$ , and $T ∣ f_{χ}$ if and only if $χ ω^{- 1} (p) = 1$ .

We can see from this (and Sinnott’s work, for instance) that $T^{2} ∤ {char}_{Λ} X_{\infty}^{(ω χ^{- 1})}$ for all $χ$ as well, so the same power of $T$ divides both $f_{χ}$ and ${char}_{Λ} X_{\infty}^{(ω χ^{- 1})}$ .

6.4. The Euler system of cyclotomic units

Let $m > 1$ be a positive integer, and let $F = ℚ {(μ_{m})}^{+}$ . Let $Δ = Gal (F ∕ ℚ)$ . Consider the set $𝒫$ of nontrivial products of distinct prime numbers that split completely in $F$ , which is to say are congruent to $\pm 1$ modulo $m$ . For any $r \in 𝒫$ , we set $F_{r} = F (μ_{r})$ for brevity, and we let $G_{r} = Gal (F_{r} ∕ F)$ , which is isomorphic to $Gal (ℚ (μ_{r}) ∕ ℚ)$ by restriction. For $ℓ ∣ r$ , we view $G_{ℓ}$ as the subgroup $Gal (F_{r} ∕ F_{r ∕ ℓ})$ of $G_{r}$ . With this identification, if we let $N_{r} \in ℤ [G_{r}]$ be the norm element, we then have

N_{r} = \prod_{ℓ ∣ r} N_{ℓ},

the product being (implicitly) taken over primes. Fix a generator $σ_{ℓ}$ of $G_{ℓ}$ for each prime $ℓ \in 𝒫$ , and let $φ_{ℓ}$ denote the Frobenius in $G_{r}$ for any $r \in 𝒫$ not divisible by $ℓ$ .

Definition 6.4.1. §

For $r \in 𝒫$ , the $r$ th derivative element is

D_{r} = \prod_{ℓ ∣ r} D_{ℓ} \in ℤ [G_{r}]

where for a prime $ℓ \in 𝒫$ , we set

D_{ℓ} = \sum_{i = 1}^{ℓ - 2} i σ_{ℓ}^{i} .

The $ℓ$ th derivative element has the following key property.

Lemma 6.4.2. §

For $ℓ \in 𝒫$ , we have

(σ_{ℓ} - 1) D_{ℓ} = ℓ - 1 - N_{ℓ} .

Proof.

We have

\begin{array}{l} σ_{ℓ} D_{ℓ} = \sum_{i = 1}^{ℓ - 2} i σ_{ℓ}^{i + 1} = \sum_{i = 1}^{ℓ - 1} (i - 1) σ_{ℓ}^{i} = \sum_{i = 1}^{ℓ - 1} i σ_{ℓ}^{i} - \sum_{i = 1}^{ℓ - 1} σ_{ℓ}^{i} = (D_{ℓ} + ℓ - 1) - N_{ℓ} . \end{array}

□

Fix a primitive $m$ th root a unity $ζ_{m}$ and a primitive $ℓ$ th root of unity $ζ_{ℓ}$ for each $ℓ \in 𝒫$ . For $r \in 𝒫$ , set $ζ_{r} = \prod_{ℓ ∣ r} ζ_{ℓ}$ . Let

α_{r} = (ζ_{m} ζ_{r} - 1) (ζ_{m}^{- 1} ζ_{r} - 1) \in F_{r},

which is a cyclotomic unit if $r \neq 1$ or $m$ is composite. It has two key properties: the first is that

α_{r} \equiv α_{r ∕ ℓ} mod 𝔏

for every prime $𝔏$ of $F_{r}$ over $ℓ$ . The second is the so-called Euler system relation found in the following lemma. Note that we use additive notation for the multiplicative action of the group ring.

Lemma 6.4.3. §

We have $N_{ℓ} α_{r} = (φ_{ℓ} - 1) α_{r ∕ ℓ}$ .

Proof.

Set $s = \frac{r}{ℓ}$ . We have

N_{ℓ} (ζ_{m} ζ_{r} - 1) = \prod_{i = 1}^{ℓ - 1} (ζ_{m} ζ_{ℓ}^{i} ζ_{s} - 1) = \frac{ζ_{m}^{ℓ} ζ_{s}^{ℓ} - 1}{ζ_{m} ζ_{s} - 1} = (φ_{ℓ} - 1) (ζ_{m} ζ_{s} - 1),

and replacing $ζ_{m}$ with $ζ_{m}^{- 1}$ , we have the lemma. □

Fix an odd positive integer $n$ , and let $𝒫_{n}$ denote the subset of elements of $𝒫$ that are products of primes that are $1$ modulo $n$ .

Lemma 6.4.4. §

If $r \in 𝒫_{n}$ , then $D_{r} α_{r} \in {(F_{r}^{\times} ∕ F_{r}^{\times n})}^{G_{r}}$ .

Proof.

We prove this by induction on the number of primes dividing $r$ , the case that the number is zero, i.e., $r = 1$ , being clear. If $r = ℓ𝑠$ for some prime $ℓ$ and $s$ in $𝒫_{n}$ , then

(σ_{ℓ} - 1) D_{r} α_{r} = (ℓ - 1 - N_{ℓ}) D_{s} α_{r} = (ℓ - 1) D_{s} α_{r} + (1 - φ_{ℓ}) D_{s} α_{s}

by the Euler system relation. The latter of course agrees with $(1 - φ_{ℓ}) D_{s} α_{s}$ modulo ${(F_{r}^{\times})}^{ℓ - 1}$ . Now, by induction we have $D_{s} α_{s} \in F_{s}^{\times n}$ , and since $ℓ \in 𝒫_{n}$ , this tells us that $(σ_{ℓ} - 1) D_{r} α_{r} \in F_{r}^{\times n}$ . Since this holds for all $ℓ$ , we have proven the lemma. □

Note that $μ_{n} \cap F = {1}$ since $F$ is totally real and $n$ is odd, and this and the fact that $n$ and $r$ are relatively prime tell us that $μ_{n} \cap F (μ_{r}) = {1}$ . We therefore have that $μ_{n}$ has trivial $G_{F_{r}}$ -invariants, so the sequence of base terms in the Hochschild-Serre spectral sequence yields an isomorphism

{(F_{r}^{\times} ∕ F_{r}^{\times n})}^{G_{r}} \to_{}^{\sim} F^{\times} ∕ F^{\times n}

inverse to the inflation map $H^{1} (G_{F}, μ_{n}) \to H^{1} {(G_{F_{r}}, μ_{n})}^{G_{r}}$ . Let $κ_{r} \in F^{\times} ∕ F^{\times n}$ denote the image of $D_{r} α_{r}$ under this map.

Terminology 6.4.5. §

The element $κ_{r}$ is called the Kolyvagin derivative of $α_{r}$ .

Remark 6.4.6. §

Note that for any $y \in F_{ℓ}^{\times}$ , the element $(1 - σ_{ℓ}) y = \frac{y}{σ_{ℓ} y}$ is necessarily a unit at primes over $ℓ$ . As $ℓ$ splits completely in $F$ and all primes over it are totally ramified in $F_{ℓ} ∕ F$ , it makes sense to take the image of $(σ_{ℓ} - 1) y$ in

{(𝒪_{F} ∕ ℓ 𝒪_{F})}^{\times} ≅ \prod_{𝔩 ∣ ℓ} {(𝒪_{F} ∕ 𝔩 𝒪_{F})}^{\times} ≅ \prod_{𝔏 ∣ ℓ} {(𝒪_{F_{ℓ}} ∕ 𝔏 𝒪_{F_{ℓ}})}^{\times} .

Let ${\tilde{κ}}_{r}$ denote a lift of $κ_{r}$ to $F^{\times}$ . Write

D_{r} α_{r} = {\tilde{κ}}_{r} β_{r}^{n}

for some $β_{r} \in F_{r}^{\times}$ .

Lemma 6.4.7. §

The fractional ideal $β_{r} 𝒪_{F_{r}}$ is invariant under $G_{r}$ .

Proof.

For $σ \in G_{r}$ , the element $(σ - 1) β_{r}$ is an $n$ th root of $(σ - 1) D_{r} α_{r}$ , since ${\tilde{κ}}_{r} \in F$ . In particular, $(σ - 1) β_{r}$ is a unit for all $σ \in G_{r}$ , and the result follows from this. □

Let $I_{ℓ}$ denote the subgroup of the ideal group $I_{F}$ of $F$ generated by the prime ideals $𝔩$ in $𝒪_{F}$ dividing a rational prime $ℓ$ . Then $I_{F} = \oplus_{ℓ} I_{ℓ}$ , where the direct sum is taken over all primes.

Lemma 6.4.8. §

If $r \in 𝒫_{n}$ and $ℓ$ is prime with $ℓ ∤ r$ , then we may choose ${\tilde{κ}}_{r}$ so that $β_{r} \in F_{r}^{\times}$ is a unit at all primes over $ℓ$ .

Proof.

Note that the choice of ${\tilde{κ}}_{r}$ is canonical up to an element of $F^{\times n}$ , so $β_{r}$ is similarly-well determined exactly up to an element of $F^{\times}$ . Since no prime over $ℓ$ ramifies in $F_{r} ∕ F$ , we have that the $G_{r}$ -fixed part of the summand of $I_{F_{r}}$ generated by primes over $ℓ$ is $I_{ℓ}$ . By Lemma 6.4.7, we can find $a \in F^{\times}$ such that $a β_{r}$ is a unit at all primes over $ℓ$ , as required. □

For $a \in F^{\times} ∕ F^{\times n}$ , we let ${[a]}_{ℓ}$ to denote the image of $a 𝒪_{F}$ in $I_{ℓ} ∕ n I_{ℓ}$ under the canonical projection.

Lemma 6.4.9. §

Let $ℓ \in 𝒫_{n}$ . Then there exists a unique $Δ$ -equivariant surjection

Π_{ℓ} : {(𝒪_{F} ∕ ℓ 𝒪_{F})}^{\times} \to I_{ℓ} ∕ n I_{ℓ}

such that

Π_{ℓ} ((1 - σ_{ℓ}) x) = {[N_{ℓ} x]}_{ℓ}

for all $x \in F_{ℓ}^{\times} ∕ F_{ℓ}^{\times n}$ .

Proof.

Since $F_{ℓ} ∕ F$ is tamely ramified at each prime dividing $ℓ$ , the $Δ$ -equivariant map

p_{ℓ} : F_{ℓ}^{\times} ∕ F_{ℓ}^{\times n} \to_{}^{1 - σ_{ℓ}} {(𝒪_{F} ∕ ℓ 𝒪_{F})}^{\times}

that exists by Remark 6.4.6 is surjective. Similarly, the $Δ$ -equivariant map $q_{ℓ} : F_{ℓ}^{\times} ∕ F_{ℓ}^{\times n} \to I_{ℓ} ∕ n I_{ℓ}$ given by $q_{ℓ} (x) = {[N_{ℓ} x]}_{ℓ}$ is surjective as all primes dividing $ℓ$ in $F$ are totally ramified in $F_{ℓ}$ .

For $x \in F_{ℓ}^{\times} ∕ F_{ℓ}^{\times n}$ , we have $p_{ℓ} (x) = 0$ if and only if the order $ℓ - 1$ of the residue field of each prime $𝔏$ over $ℓ$ in $F_{ℓ}$ divides the valuation $v_{𝔏} (x)$ , which of course implies that $ℓ - 1$ divides $v_{𝔩} (N_{ℓ} (x))$ for each prime $𝔩$ of $F$ over $L$ . Since $ℓ \in 𝒫_{n}$ , we then have ${[N_{ℓ} (x)]}_{ℓ} = 0$ . Consequently, the map $q_{ℓ}$ factors through the map $p_{ℓ}$ , producing the unique map $Π_{ℓ}$ . □

Let

π_{ℓ} : {a \in F^{\times} ∕ F^{\times n} ∣ {[a]}_{ℓ} = 0} \to I_{ℓ} ∕ n I_{ℓ}

be the map that takes an element $a$ to the value of $Π_{ℓ}$ on the image of $a$ in ${(𝒪_{F} ∕ ℓ 𝒪_{F})}^{\times}$ .

Remark 6.4.10. §

From the proof of Lemma 6.4.9, we have that $x \in \ker π_{ℓ}$ if and only if $x$ is an $n$ th power modulo $𝔩$ for all prime $𝔩$ dividing $ℓ$ .

Proposition 6.4.11. §

For any $r \in 𝒫_{n}$ and prime $ℓ$ , we have

{[κ_{r}]}_{ℓ} = {\begin{matrix} π_{ℓ} (κ_{r ∕ ℓ}) & if ℓ ∣ r \\ 0 & if ℓ ∤ r . \end{matrix}

Proof.

If $ℓ ∤ r$ , then we saw in Lemma 6.4.8 that $β_{r}$ may be chosen to be a unit at all primes over $ℓ$ , in which case ${\tilde{κ}}_{r}$ will also be a unit at $ℓ$ , and therefore ${[κ_{r}]}_{ℓ} = 0$ .

If $ℓ ∣ r$ , then write $r = ℓ𝑠$ . We choose $β_{s}$ to be a unit at primes over $ℓ$ . Since $β_{r}^{n}$ is a unit times an element of $F^{\times}$ , we have that $v_{𝔏} (β_{r}^{n})$ is a multiple of the ramification index $ℓ - 1$ for each prime $𝔏$ of $F_{r}$ over $ℓ$ . Since such primes are unramified over $F_{ℓ}$ , we can find $ν \in F_{ℓ}^{\times}$ such that $β_{r} ν^{(ℓ - 1) ∕ n}$ is a unit at all primes over $ℓ$ . Since $N_{ℓ} ν$ and $ν^{ℓ - 1}$ have the valuation at each $𝔏$ over $ℓ$ and $β_{r}^{- n} 𝒪_{F_{r}} = {\tilde{κ}}_{r} 𝒪_{F_{r}}$ , we therefore have ${[N_{ℓ} ν]}_{ℓ} = {[κ_{r}]}_{ℓ}$ .

Fix a prime $𝔏$ over $ℓ$ in $F_{r}$ . Since $𝔏$ is ramified over $F$ , we have

(1 - σ_{ℓ}) ν^{(ℓ - 1) ∕ n} \equiv (σ_{ℓ} - 1) β_{r} mod 𝔏 .

Since ${\tilde{κ}}_{r}, {\tilde{κ}}_{s} \in F$ , we have

\begin{matrix} (σ_{ℓ} - 1) β_{r}^{n} = (σ_{ℓ} - 1) D_{r} α_{r} = (ℓ - 1 - N_{ℓ}) D_{s} α_{r} \\ = (ℓ - 1) D_{s} α_{r} - (φ_{ℓ} - 1) D_{s} α_{s} = (ℓ - 1) D_{s} α_{r} - (φ_{ℓ} - 1) β_{s}^{n}, \end{matrix}

the third equality by the Euler system relation. Since $α_{r} \equiv α_{s} mod 𝔏$ , and

(φ_{ℓ} - 1) β_{s} \equiv (ℓ - 1) β_{s} mod 𝔏

by definition of the Frobenius, we have

(σ_{ℓ} - 1) β_{r} \equiv \frac{D_{s} α_{r}^{(ℓ - 1) ∕ n}}{(φ_{ℓ} - 1) β_{s}} \equiv {(\frac{D_{s} α_{s}}{β_{s}^{n}})}^{(ℓ - 1) ∕ n} \equiv {\tilde{κ}}_{s}^{(ℓ - 1) ∕ n} mod 𝔏 .

In other words, the elements $(1 - σ_{ℓ}) ν$ and ${\tilde{κ}}_{s}$ differ by an $\frac{ℓ - 1}{n}$ th root of unity modulo primes over $ℓ$ in $F_{ℓ}$ . We then have that $Π_{ℓ} ((1 - σ_{ℓ}) ν) = π_{ℓ} (κ_{s})$ . By Lemma 6.4.9, we have the result. □

Now suppose that $p$ is an odd prime, and let $n = p^{k}$ for some $k \geq 1$ . The following theorem guarantees the existence of enough primes for our application.

Proposition 6.4.12. §

Given an ideal class $𝔠 \in A_{F}$ , a finite $ℤ [Δ]$ -submodule $M$ of $F^{\times} ∕ F^{\times n}$ , and a Galois-equivariant map $𝜃 : M \to ℤ ∕ 𝑛ℤ [Δ]$ , there exist infinitely many primes $𝔩 \in 𝔠$ that lie over some prime $ℓ \in 𝒫_{n}$ such that $M$ has trivial image in $I_{ℓ} ∕ n I_{ℓ}$ and there exists a unit $u \in {(ℤ ∕ 𝑛ℤ)}^{\times}$ such that

π_{ℓ} (x) = 𝑢𝜃 (x) 𝔩 mod n I_{ℓ}

for all $x \in M$ .

Proof.

Let $E = F (μ_{n})$ and $H$ be the $p$ -Hilbert class field of $F$ . The inertia group at any prime over $p$ in $Gal (E ∕ F)$ has index at most $2$ , so $H \cap E = F$ as $p$ is odd. Note that $Gal (E (\sqrt[n]{M}) ∕ E)$ injects into $Hom (M, μ_{n})$ by Kummer theory. The element $ρ \in Gal (E ∕ F)$ corresponding to complex conjugation acts as $1$ on $M$ and as $- 1$ on $μ_{m}$ , so $ρ$ acts as $- 1$ on $Hom (M, μ_{n})$ . It also acts as $1$ on $Gal (𝐻𝐸 ∕ E)$ , so $E (\sqrt[n]{M}) \cap 𝐻𝐸 = E$ . Since $H \cap E = F$ , it follows that $E (\sqrt[n]{M}) \cap H = F$ .

Since $μ_{n} \cap F = {1}$ , we have ${\hat{H}}^{0} (Gal (E ∕ F), μ_{n}) = 0$ and therefore $H^{1} (Gal (E ∕ F), μ_{n}) = 0$ as $Gal (E ∕ F)$ is cyclic. The natural map $F^{\times} ∕ F^{\times n} \to E^{\times} ∕ E^{\times n}$ is therefore an injection, and we see that the injection

Gal (E (\sqrt[n]{M}) ∕ E) \to Hom (M, μ_{n})

is in fact an isomorphism.

Fix a primitive $n$ th root of unity $ζ_{n}$ , and define a homomorphism $ι : (ℤ ∕ 𝑛ℤ) [Δ] \to μ_{n}$ on group elements by $ι (1) = ζ_{n}$ and $ι (δ) = 1$ for $δ \neq 1$ . The homomorphism $ι \circ 𝜃 : M \to μ_{n}$ corresponds to an element $τ \in Gal (E (\sqrt[n]{M}) ∕ E)$ satisfying

\frac{τ (\sqrt[n]{x})}{\sqrt[n]{x}} = ι \circ 𝜃 (x)

for all $x \in M$ .

By what we have shown, restriction maps define an isomorphism

Gal (𝐻𝐸 (\sqrt[n]{M}) ∕ F) ≅ Gal (H ∕ F) \times Gal (E ∕ F) \times Gal (E (\sqrt[n]{M}) ∕ E) .

So, we may choose $σ \in Gal (𝐻𝐸 (\sqrt[n]{M}) ∕ F)$ such that $σ |_{E (\sqrt[n]{M})} = τ$ and $σ |_{H}$ corresponds to $𝔠 \in A_{F}$ via the Artin isomorphism. By the Čebotarev density theorem, there exist infinitely many primes $ℓ$ that are unramified in $E (\sqrt[n]{M})$ and for which the Frobenius $φ_{ℓ}$ at $ℓ$ has the same conjugacy class as $σ$ in $Gal (𝐻𝐸 (\sqrt[n]{M}) ∕ ℚ)$ .

Now fix such a prime $ℓ$ , and let $𝔩$ be a prime lying over it. Here then are its most easily derived properties. Since $σ |_{F} = 1$ , the prime $𝔩$ has degree $1$ , or in other words $ℓ \in 𝒫$ . Since $σ |_{H}$ corresponds to $𝔠$ , we have $𝔩 \in 𝔠$ . Since $σ |_{E} = 1$ , the prime $𝔩$ splits in $E ∕ F$ , so $ℓ \in 𝒫_{n}$ . Since $ℓ$ is unramified in the Galois extension $E (\sqrt[n]{M})$ of $ℚ$ , we have ${[x]}_{ℓ} = 0$ for all $x \in M$ .

The component of $π_{ℓ} (x) \in I_{ℓ} ∕ n I_{ℓ}$ as $𝔩$ is trivial if and only if $x$ is an $n$ th power modulo $𝔩$ , as in Remark 6.4.10. On the other hand, $𝜃 (x) 𝔩 \in I_{ℓ} ∕ n I_{ℓ}$ is trivial if and only if $ι \circ 𝜃 (x) = 1$ , so if and only if $τ |$ fixes $\sqrt[n]{x}$ , and then if and only if $φ_{𝔩}$ fixes $\sqrt[n]{x}$ , and then finally if and only if $x$ is an $n$ th power modulo $𝔩$ . Thus, there exists a $u \in {(ℤ ∕ 𝑛ℤ)}^{\times}$ such that the $𝔩$ -component of $π_{ℓ} (x)$ and $𝑢𝜃 (x) 𝔩$ agree for all $x \in M$ . The map

π_{ℓ} (x) - 𝑢𝜃 (x) 𝔩 : M \to ⨁_{\begin{array}{c} 𝔩^{'} ∣ ℓ \\ 𝔩 \neq 𝔩^{'} \end{array}} (ℤ ∕ 𝑛ℤ) 𝔩 \subset I_{ℓ} ∕ n I_{ℓ}

is $Δ$ -equivariant as the difference of $Δ$ -equivariant maps, so its image is $ℤ ∕ 𝑛ℤ [Δ]$ -stable, but its image also lies in a subgroup of $I_{ℓ} ∕ n I_{ℓ}$ containing no nontrivial $ℤ ∕ 𝑛ℤ [Δ]$ -submodule, as $Δ$ acts transitively on the primes of $F$ over $ℓ$ . It follows that $π_{ℓ} (x) = 𝑢𝜃 (x) 𝔩$ for all $x \in M$ . □

Recall that $E_{F} = E_{F} \otimes_{ℤ} ℤ_{p}$ , and set $𝒞_{F} = C_{F} \otimes_{ℤ} ℤ_{p}$ .

Lemma 6.4.13. §

Suppose that $χ : Δ \to 𝒪_{χ}^{\times}$ is a $p$ -adic character of $Δ$ . Extending $χ$ to a primitive Dirichlet character, if $1 - χ (ℓ) \in 𝒪_{χ}^{\times}$ for all $ℓ ∣ m$ , then $e_{χ} (1 - ζ_{m})$ generates $𝒞_{F}^{(χ)}$ as an $𝒪_{χ}$ -module.

Proof.

Let $ζ_{d} = ζ_{m}^{m ∕ d}$ for $d$ dividing $m$ . The group $𝒞_{F}$ is the intersection with $E_{F}$ of the $ℤ_{p} [Δ]$ -module generated by the elements $1 - ζ_{d}$ for $d$ dividing $m$ . Since the norm from $ℚ (ζ_{d})$ to $ℚ (ζ_{e})$ for $e$ dividing $d$ of the element $1 - ζ_{d}$ is $1 - ζ_{e}$ so long as every prime dividing $d$ also divides $e$ , we can reduce this generating set to the set of $1 - ζ_{d}$ with $(d, \frac{m}{d}) = 1$ . In general, if $ℓ_{1}, \dots, ℓ_{k}$ are the primes dividing $d$ but not $e$ , then the norm of $1 - ζ_{d}$ is the application of $(1 - φ_{ℓ_{1}}^{- 1}) \dots (1 - φ_{ℓ_{k}}^{- 1})$ to $1 - ζ_{e}$ . Projecting to the $χ$ -isotypical quotient, we have that it becomes the multiple of the image of $1 - ζ_{e}$ by $(1 - χ {(ℓ_{1})}^{- 1}) \dots (1 - χ {(ℓ_{k})}^{- 1})$ , which is a unit by assumption. □

We may now bound the orders of eigenspaces of even eigenspaces of $p$ -parts of class groups. The proof of the following result using Euler systems is due to Kolyvagin. We suppose that $m$ is divisible by $4$ if it is even.

Theorem 6.4.14. §

Suppose that $p ∤ | Δ |$ , and let $χ$ be a primitive finite order $p$ -adic character of $Δ$ . Then the order of $A_{F}^{(χ)}$ divides the order of ${(E_{F} ∕ 𝒞_{F})}^{(χ)}$ .

Proof.

Let $𝒪$ be the $ℤ_{p}$ -algebra generated by the image of $χ$ , and let $f$ be its residue degree. Let $a_{χ} = | A_{F}^{(χ)} |^{1 ∕ f}$ and $q_{χ} = | {(E_{F} ∕ 𝒞_{F})}^{(χ)} |^{1 ∕ f}$ , and set $n = a_{χ} q_{χ}$ . Let $𝔠_{1}, \dots, 𝔠_{q}$ be ideal classes generating $A_{F}^{(χ)}$ as an $𝒪$ -module.

Set $δ_{r} = e_{χ} κ_{r}$ for $r \in 𝒫_{n}$ . Primitivity and the fact that $p ∤ | Δ |$ imply, by Lemma 6.4.13, that $𝒞_{F}^{(χ)}$ is free of rank $1$ over $𝒪_{χ}$ , generated by $δ_{1} = e_{χ} α_{1} = e_{χ} {(ζ_{m} - 1)}^{2}$ . Then $q_{χ}$ is the maximal integer $t_{0}$ such that $δ_{1} \in {(F^{\times t_{0}} ∕ F^{\times n})}^{(χ)}$ .

Let $1 \leq i \leq g$ , and suppose that for each $1 \leq j < i$ , we have found primes $𝔩_{j} \in 𝔠_{j}$ lying over primes $ℓ_{j} \in 𝒫_{n}$ such that for $r_{j} = \prod_{h = 1}^{j} ℓ_{h}$ and $t_{j} \leq n$ the largest power of $p$ such that

δ_{r_{j}} \in {(F^{\times t_{j}} ∕ F^{\times n})}^{(χ)},

one has $t_{j} ∣ t_{j - 1}$ and

\frac{t_{j - 1}}{t_{j}} 𝔠_{j} \in 𝒪 (𝔠_{1}, \dots, 𝔠_{j - 1}) .

(6.4.1)

We look for $𝔩_{i}$ with the same properties.

Let $M_{i}$ be the $𝒪$ -submodule of $F^{\times} ∕ F^{\times n}$ generated by $δ_{r_{i - 1}}$ . Define

𝜃_{i} : M_{i} \to {((ℤ ∕ 𝑛ℤ) [Δ])}^{(χ)}, 𝜃_{i} (δ_{r_{i - 1}}) = t_{i - 1} e_{χ} .

By Proposition 6.4.12, there exists a prime $𝔩_{i} \in 𝔠_{i}$ over some $ℓ_{i} \in 𝒫_{n}$ and satisfying ${[δ_{r_{i - 1}}]}_{ℓ_{i}} = 0$ and

π_{ℓ_{i}} (δ_{r_{i - 1}}) = u_{i} t_{i - 1} e_{χ} 𝔩_{i}

for some $u_{i} \in {(ℤ ∕ 𝑛ℤ)}^{\times}$ . Now let $r_{i} = \prod_{j = 1}^{i} ℓ_{j}$ and $t_{i} \leq n$ be the largest power of $p$ such that $δ_{r_{i}} \in F^{\times t_{i}} ∕ F^{\times n}$ .

We have by Proposition 6.4.11 that

{[δ_{r_{i}}]}_{ℓ_{i}} = π_{ℓ_{i}} (δ_{r_{i - 1}}) = u_{i} t_{i - 1} e_{χ} 𝔩_{i}

(6.4.2)

in $I_{ℓ_{i}} ∕ n I_{ℓ_{i}}$ . Since $δ_{r_{i}} \in F^{\times t_{i}} ∕ F^{\times n}$ , this forces $t_{i} ∣ t_{i - 1}$ . In particular, $t_{i}$ divides $t_{0} = q_{χ}$ , and therefore $a_{χ} = \frac{n}{q_{χ}}$ divides $\frac{n}{t_{i}}$ .

Proposition 6.4.11 also tells us that ${[δ_{r_{i}}]}_{ℓ} = 0$ unless $ℓ ∣ r_{i}$ . Thus $δ_{ℓ_{i}}$ has a $t_{i}$ th root in $F^{\times}$ and nonzero valuation modulo $n$ only at primes dividing $ℓ_{1}, \dots, ℓ_{i}$ . It follows that $\frac{1}{t_{i}} {[δ_{r_{i}}]}_{ℓ_{i}}$ has trivial image in the quotient of $A_{F}^{(χ)}$ by the $𝒪$ -span of the classes $𝔠_{1}, \dots, 𝔠_{i - 1}$ of $𝔩_{1}, \dots, 𝔩_{i - 1}$ . Moreover, we have by (6.4.2) that

\frac{1}{t_{i}} {[δ_{r_{i}}]}_{ℓ_{i}} \equiv u_{i} \frac{t_{i - 1}}{t_{i}} e_{χ} 𝔩_{i} mod \frac{n}{t_{i}} I_{ℓ_{i}},

This implies that $\frac{t_{i - 1}}{t_{i}} 𝔠_{i} \in 𝒪 (𝔠_{1}, \dots, 𝔠_{i - 1})$ , completing the recursion.

Multiplying together (6.4.1) for $1 \leq j \leq g$ gives that $a_{χ}$ divides

\prod_{i = 1}^{g} \frac{t_{i - 1}}{t_{i}} = \frac{q_{χ}}{t_{g}},

which clearly divides $q_{χ}$ . □

6.5. The main conjecture via Euler systems

Let $p$ be an odd prime. Let $m$ be a positive integer not divisible by $p$ and divisible by $4$ if $m$ is even. Set $F = ℚ (μ_{𝑚𝑝})$ , and let $F_{n} = ℚ (μ_{m p^{n}})$ for $n \geq 1$ and $F_{\infty} = ⋃_{n = 1}^{\infty} F_{n}$ . Let $Γ (n) = Γ^{p^{n - 1}} = Gal (F_{\infty} ∕ F_{n})$ and $Γ_{n} = Gal (F_{n} ∕ F) ≅ ℤ ∕ p^{n - 1} ℤ$ . Let $χ : {(ℤ ∕ 𝑚𝑝ℤ)}^{\times} \to 𝒪^{\times}$ be an even character of order prime to $p$ , where $𝒪 = 𝒪_{χ}$ , which we also view as a primitive $𝒪$ -valued Dirichlet character. Set $Λ = 𝒪 ⟦ Γ ⟧$ and $Λ_{n} = 𝒪 [Γ_{n}]$ . We make a usual choice of identification of $Λ$ with $𝒪 ⟦ T ⟧$ .

Lemma 6.5.1. §

a.: The restriction map ${(𝔛_{\infty})}_{Γ (n)}^{(χ)} \to 𝔛_{n}^{(χ)}$ is an isomorphism.
b.: The restriction map ${(X_{\infty})}_{Γ (n)}^{(χ)} \to X_{n}^{(χ)}$ has trivial kernel unless $χ (p) = 1$ and $χ \neq 1$ , in which case it is isomorphic to $𝒪$ . It has trivial cokernel unless $χ = 1$ , in which case it is a finite quotient of $Γ_{n}$ that is zero for sufficiently large $n$ .
c.: The inverse limit of norm maps ${(𝒰_{\infty})}_{Γ (n)}^{(χ)} \to 𝒰_{n}^{(χ)}$ is an injection unless $χ ω^{- 1} (p) = 1$ and a surjection unless $χ (p) = 1$ .
d.: The inverse limit of norm maps ${(𝒞_{\infty})}_{Γ (n)}^{(χ)} \to 𝒞_{n}^{(χ)}$ is an injection which is an isomorphism if $χ (p) \neq 1$ .

Note that if $M$ is a finitely generated $Λ$ -module such that $M_{Γ}$ is finite, then it is torsion and $T$ does not divide its characteristic ideal, so $M^{Γ}$ is finite as well, and in particular $M^{Γ}$ is contained in the maximal finite submodule $M_{fin}$ of $M$ .

Proposition 6.5.2. §

Suppose that $χ (p) \neq 1$ and $χ ω^{- 1} (p) \neq 1$ . Then there exists an open ideal $𝔞$ of $Λ$ that annihilates both the kernel and cokernel of the inverse limit of norm maps $N_{n} : {(E_{\infty})}_{Γ_{n}}^{(χ)} \to E_{n}^{(χ)}$ .

Proof.

Consider the following two commutative diagrams with top rows arising from taking the $Γ (n)$ -homology of Proposition 3.3.6 (noting Theorem 3.3.4) and the bottom rows coming from Theorem 1.5.4 (noting Theorem 1.5.21):

Coinvariants and finite-level reciprocity sequences. A full diagram description follows.

Diagram description: Coinvariants and finite-level reciprocity sequences

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: (X subscript (infinity) superscript (( chi ))) superscript (capital Gamma (n)); column 2: (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)); column 3: (fraktur X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)); column 4: (X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )); column 3: fraktur X subscript (n) superscript (( chi )); column 4: A subscript (n) superscript (( chi )); column 5: 0.

Arrows and lines:

An arrow from (X subscript (infinity) superscript (( chi ))) superscript (capital Gamma (n)) to (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)), without a label.
An arrow from (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to (fraktur X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)), without a label.
An arrow from (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )), labelled pi subscript (n).
An arrow from (fraktur X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to (X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)), without a label.
An arrow from (fraktur X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to fraktur X subscript (n) superscript (( chi )), labelled isomorphism symbol.
An arrow from (X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to 0 (row 1, column 5), without a label.
An arrow from (X subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to A subscript (n) superscript (( chi )), labelled isomorphism symbol.
An arrow from 0 (row 2, column 1) to script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )), without a label.
An arrow from script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )) to fraktur X subscript (n) superscript (( chi )), without a label.
An arrow from fraktur X subscript (n) superscript (( chi )) to A subscript (n) superscript (( chi )), without a label.
An arrow from A subscript (n) superscript (( chi )) to 0 (row 2, column 5), without a label.

and

Coinvariants and finite-level unit sequences. A full diagram description follows.

Diagram description: Coinvariants and finite-level unit sequences

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) superscript (capital Gamma (n)); column 2: (script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)); column 3: (script U subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)); column 4: (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: script E subscript (n) superscript (( chi )); column 3: script U subscript (n) superscript (( chi )); column 4: script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )); column 5: 0.

Arrows and lines:

An arrow from (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) superscript (capital Gamma (n)) to (script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)), without a label.
An arrow from (script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to (script U subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)), without a label.
An arrow from (script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to script E subscript (n) superscript (( chi )), labelled N subscript (n).
An arrow from (script U subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)), without a label.
An arrow from (script U subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to script U subscript (n) superscript (( chi )), labelled isomorphism symbol.
An arrow from (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to 0 (row 1, column 5), without a label.
An arrow from (script U subscript (infinity) superscript (( chi )) / script E subscript (infinity) superscript (( chi ))) subscript (capital Gamma (n)) to script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )), labelled pi subscript (n).
An arrow from 0 (row 2, column 1) to script E subscript (n) superscript (( chi )), without a label.
An arrow from script E subscript (n) superscript (( chi )) to script U subscript (n) superscript (( chi )), without a label.
An arrow from script U subscript (n) superscript (( chi )) to script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )), without a label.
An arrow from script U subscript (n) superscript (( chi )) / script E subscript (n) superscript (( chi )) to 0 (row 2, column 5), without a label.

By Lemma 6.5.1 and our assumption that $χ (p) \neq 1$ , we have that the rightmost two vertical arrows in the first diagram and the middle vertical arrow in the second diagram are isomorphisms. In particular, we have exact sequences

{(X_{\infty}^{(χ)})}^{Γ (n)} \to {(𝒰_{\infty}^{(χ)} ∕ E_{\infty}^{(χ)})}_{Γ (n)} \to_{}^{π_{n}} 𝒰_{n}^{(χ)} ∕ E_{n}^{(χ)} \to 0 .

(6.5.1)

and

{(𝒰_{\infty}^{(χ)} ∕ E_{\infty}^{(χ)})}^{Γ (n)} \to {(E_{\infty}^{(χ)})}_{Γ (n)} \to_{}^{N_{n}} E_{n}^{(χ)} \to \ker π_{n} \to 0 .

(6.5.2)

Since ${(X_{\infty}^{(χ)})}_{Γ (n)} ≅ A_{n}^{(χ)}$ is finite, so is ${(X_{\infty}^{(χ)})}^{Γ (n)}$ , and being contained in the maximal finite $Λ$ -submodule of $X_{n}^{(χ)}$ , it has order bounded independent of $n$ . By (6.5.1) and (6.5.2), we then have that $coker N_{n} ≅ \ker π_{n}$ also has bounded order. The group $𝔛_{n}^{(χ)} ≅ {(𝔛_{\infty}^{(χ)})}_{Γ (n)}$ is finite by our assumption on $χ$ and the theorem of Ferrero-Greenberg, so its subgroup $𝒰_{n}^{(χ)} ∕ E_{n}^{(χ)}$ is finite as well. Equation (6.5.1) then also yields that ${(𝒰_{\infty}^{(χ)} ∕ E_{\infty}^{(χ)})}_{Γ (n)}$ is finite, so $\ker N_{n} ≅ {(𝒰_{\infty}^{(χ)} ∕ E_{\infty}^{(χ)})}^{Γ (n)}$ is finite of order bounded in $n$ by the order of the maximal finite submodule of $𝒰_{\infty}^{(χ)} ∕ E_{\infty}^{(χ)}$ (which is in fact trivial). Letting $𝔞$ be the annihilator of ${(X_{\infty}^{(χ)})}_{fin} \oplus {(𝒰_{\infty}^{(χ)} ∕ E_{\infty}^{(χ)})}_{fin}$ , we are done. □

Let $h_{χ}$ denote a characteristic power series of $𝔛_{\infty}^{(χ)}$ , and let $j_{χ}$ denote a characteristic power series of $E_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)}$ .

Proposition 6.5.3. §

Suppose that $χ (p) \neq 1$ . Then there exists an open ideal $𝔞$ of $Λ$ such that for all $λ \in 𝔞$ and $n \geq 1$ , there exists a $Λ_{n}$ -module homomorphism $𝜃_{n, λ} : E_{n}^{(χ)} \to Λ_{n}$ such that

𝜃_{n, λ} (𝒞_{\infty}^{(χ)}) = λ j_{χ} Λ_{n} .

Proof.

Under our assumption that $χ (p) \neq 1$ , we have $𝒰_{\infty}^{(χ)} ≅ Λ$ via the product of Coleman maps by Corollary 6.1.2. Consequently, its submodule $E_{\infty}^{(χ)}$ is torsion-free of rank one. In particular, there exists an injective pseudo-isomorphism $𝜃 : E_{\infty}^{(χ)} \to Λ$ . By Proposition 6.1.5, the $Λ$ -module $𝒞_{\infty}^{(χ)}$ is cyclic, so

𝜃 (𝒞_{\infty}^{(χ)}) = {char}_{Λ_{χ}} (Λ_{χ} ∕ 𝜃 (𝒞_{\infty}^{(χ)})) = {char}_{Λ_{χ}} (E_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)}) = (j_{χ}) .

Now let $𝔞$ be as in Proposition 6.5.2. Let $𝜃_{n} : {(E_{\infty}^{(χ)})}_{Γ (n)} \to Λ_{n}$ be the map induced by $𝜃$ . For any $λ \in 𝔞$ and $u \in E_{n}^{(χ)}$ , we let $𝜃_{n, λ} (u)$ be $𝜃_{n} (v)$ for any $v$ with $N_{n} (v) = 𝜆𝑢$ , which exists since $λ$ annihilates $coker N_{n}$ and is unique since $𝜃_{n}$ is necessarily trivial on the finite kernel of $N_{n}$ . The result then follows by definition of $𝜃_{n, λ}$ . □

Lemma 6.5.4. §

Suppose that $χ (p) \neq 1$ . Let $f_{i}$ with $1 \leq i \leq g$ be such that $X_{\infty}^{(χ)} ≃ \prod_{i = 1}^{g} Λ ∕ (f_{i})$ . Then there exists an open ideal $𝔟$ of $Λ$ such that, for each $n \geq 1$ , there exist elements $𝔠_{1}, \dots, 𝔠_{g}$ of $A_{n}^{(χ)}$ such that the annihilator $Ann (𝔠_{𝔦})$ of each $𝔠_{i}$ as an element of the $Λ_{n}$ -module $A_{n}^{(χ)} ∕ Λ_{n} (𝔠_{1}, \dots, 𝔠_{i - 1})$ satisfies $𝔟 Ann (𝔠_{i}) \subseteq f_{i} Λ_{n}$ .

Proof.

By the given pseudo-isomorphism, there exists an exact sequence

0 \to ⨁_{i = 1}^{g} Λ ∕ (f_{i}) \to X_{\infty}^{(χ)} \to Q \to 0

with $Q$ finite. Taking $Γ_{n}$ -homology and noting that ${(X_{\infty}^{(χ)})}_{Γ (n)} ≅ A_{n}^{(χ)}$ by assumption, we obtain an exact sequence

Q^{Γ (n)} \to ⨁_{i = 1}^{g} Λ_{n} ∕ f_{i} Λ_{n} \to A_{n}^{(χ)} \to Q_{Γ (n)} \to 0 .

(6.5.3)

Let $𝔟$ be the $Λ$ -annihilator of $Q$ . Let $𝔠_{i}$ be the image of the generator $e_{i}$ of the $i$ th summand $Λ_{n} ∕ f_{i} Λ_{n}$ in $A_{n}^{(χ)}$ .

If $x \in Λ_{n}$ is such that $x \cdot 𝔠_{i} \in Λ_{n} (𝔠_{1}, \dots, 𝔠_{n})$ , then we have $x e_{i}$ is in the sum of the image of $Q^{Γ (n)}$ and $\oplus_{j \neq i} Λ_{n} ∕ f_{j} Λ_{n}$ in $\oplus_{j = 1}^{g} Λ_{n} ∕ f_{j} Λ_{n}$ by (6.5.3). Since $𝔟$ annhilates $Q$ , any $λ \in 𝔟$ satisfies $𝜆𝑥 e_{i} \in \oplus_{j \neq i} Λ_{n} ∕ f_{j} Λ_{n}$ , but $𝜆𝑥 e_{i}$ is clearly in the $i$ th summand, so $𝜆𝑥 e_{i} = 0$ . In other words, $𝔟 Ann (𝔠_{i}) \subseteq f_{i} Λ_{n}$ . □

Let us now work over $F_{n}^{+}$ .

Lemma 6.5.5. §

Let $r \in 𝒫_{m}$ for a power $m$ of $p$ , let $ℓ$ be a prime divisor of $r$ , and let $𝔮$ be a prime of $F_{n}^{+}$ over $ℓ$ . Let $B$ be the subgroup of $A_{n}^{+}$ generated by the primes dividing $\frac{r}{ℓ}$ . Let $𝔠 = e_{χ} [𝔮] \in A_{n}^{(χ)}$ . Let $δ_{r} = e_{χ} κ_{r}$ , and let $M = Λ_{n} δ_{r} \subseteq {(F^{\times} ∕ F^{\times m})}^{(χ)}$ . Suppose that we can choose

m \geq | A_{n}^{(χ)} | \cdot | {(I_{ℓ} ∕ m I_{ℓ})}^{(χ)} ∕ Λ_{n} {[δ_{r}]}_{ℓ} | .

Let $I \subseteq Λ_{n}$ denote the annihilator of the image of $𝔠$ in $A_{n}^{+} ∕ B$ . Suppose also that $λ, f \in Λ_{n}$ are such that

\begin{array}{rcl} 𝜆𝐼 \subset f Λ_{n}, & and & Λ_{n} ∕ f Λ_{n} is finite . \end{array}

Then there exists a $Λ_{n}$ -module homomorphism

𝜃 : M \to {(Λ_{n} ∕ m Λ_{n})}^{(χ)}

such that for

η : F^{\times} ∕ F^{\times n} \to Λ_{n} ∕ m Λ_{n}, η (x) 𝔮 = π_{ℓ} (x),

we have

𝜃 (δ_{r}) = 𝜆𝜂 (δ_{r}) .

Proof.

By assumption, we have $m \cdot A_{n}^{(χ)} = 0$ . Define $\tilde{η} : F_{n}^{\times} \to Λ_{n}$ by $\tilde{η} (x) 𝔮 = {(x)}_{ℓ}$ , where ${(x)}_{ℓ}$ denotes the image of $(x)$ in $I_{ℓ}$ so that $\tilde{η}$ lifts $η$ . Let ${\tilde{δ}}_{r} \in F_{n}^{\times}$ be a lift of $δ_{r}$ . Then $({\tilde{δ}}_{r})$ is a multiple of $m$ at primes not dividing $r$ , so its image in $A_{n}^{+} ∕ B$ is $η (δ_{r}) 𝔮$ , but also zero as the image of a principal ideal. Thus $η (δ_{r}) \in I$ , and so $λ \tilde{η} ({\tilde{δ}}_{r}) \in f Λ_{n}$ by assumption. Since $Λ_{n} ∕ f Λ_{n}$ is finite, we may set

α = \frac{λ \tilde{η} ({\tilde{δ}}_{r})}{f} \in Λ_{n} .

We define $𝜃$ as in the lemma as the unique $Λ_{n}$ -module homomorphism with $𝜃 (δ_{r}) = α$ , if it exists. If $a \in Λ_{n}$ is such that $a δ_{r} = 0$ , then ${[a δ_{r}]}_{ℓ} = 0$ . For $h = | A_{n}^{(χ)} |$ , we have

\frac{m}{h} \cdot {(I_{ℓ} ∕ m I_{ℓ})}^{(χ)} \subseteq Λ_{n} {[δ_{r}]}_{ℓ},

so we must have $a \in h Λ_{n}$ . Writing $a {\tilde{δ}}_{r} = x^{m}$ for some $x \in F_{n}^{\times}$ , we have $e_{χ} {[x]}_{ℓ} = {[\frac{1}{m} a {\tilde{δ}}_{r}]}_{ℓ}$ , and $(x)$ has valuation a multiple of $h$ at primes not dividing $r$ . Since $h \cdot A_{n}^{(χ)} = 0$ , the element $\frac{1}{m} {(a {\tilde{δ}}_{r})}_{ℓ}$ has trivial image in $A_{n}^{(χ)} ∕ B$ . In other words, $\frac{1}{m} \tilde{η} (a {\tilde{δ}}_{r}) \cdot 𝔠 \in B$ . Then

𝑎𝛼𝑓 = 𝑎𝜆 \tilde{η} ({\tilde{δ}}_{r}) \in 𝑚𝑓 Λ_{n},

so $𝜃$ is $0$ on $a δ_{r}$ . Therefore, $𝜃$ is well-defined. □

We now come to our proof of a divisibility in the main conjecture. For now, the proof is omitted.

Theorem 6.5.6. §

If $χ (p) \neq 1$ and $χ ω^{- 1} (p) \neq 1$ , then $char (X_{\infty}^{(χ)})$ divides $char (E_{\infty}^{(χ)} ∕ 𝒞_{\infty}^{(χ)})$ .

6.6. Geometry of modular curves

The original approach of Mazur and Wiles to the main conjecture was a heavily involved study of Galois actions on the cohomology of modular curves, inspired by the work of Ribet in his proof of the converse to Herbrand’s theorem, which looked at the Galois representations attached to a newform satisfying a mod $p$ congruence with an Eisenstein series. The work of Wiles was a significant refinement, and in some sense simplification, of the work of Mazur-Wiles that employed Hida theory and Galois representations constructed out of pseudo-representations to complete the proof of the more general main conjecture over totally real extensions of $ℚ$ . Back in the setting of the main conjecture over $ℚ$ , a further simplification of Wiles’ work can be found in the work of Masami Ohta (for primes $p \geq 5$ ). In this setting, the Galois representations that Wiles constructs are quotients of inverse limits of cohomology groups of modular curves, so one can study cohomology directly. It is this approach that we will attempt to roughly sketch here. For this, we will have to assume substantially more background than earlier in these notes, so we will try to focus on ideas to compensate for this.

For a given level $N \geq 4$ , the modular curve $X_{1} (N)$ may be defined as a scheme over $ℤ$ . Over, $ℤ [\frac{1}{N}]$ , it is a compactification of the fine moduli scheme $Y_{1} (N)$ that represents the functor that to a $ℤ [\frac{1}{N}]$ -scheme $S$ associates the set of pairs $(E, P)$ , where $E$ is an elliptic curve over $S$ and $P$ is a point of order $N$ generating a subgroup scheme of $E_{∕ S}$ isomorphic to ${(ℤ ∕ 𝑁ℤ)}_{∕ S}$ . If we consider the base change $\bar{X_{1} (N)}$ of $X_{1} (N)$ to $\bar{ℚ}$ , then its $p$ -adic étale cohomology group $𝒯_{N} = H_{ét}^{1} (\bar{X_{1} (N)}, ℚ_{p} (1))$ has a continuous action of $G_{ℚ}$ that is unramified outside of the primes over $N$ and $\infty$ .

There is also an action on cohomology of Hecke operators given by correspondences. To describe this, we remark that a choice of embedding $\bar{ℚ} ↪ ℂ$ gives rise to an isomorphism

𝒯_{N} \to_{}^{\sim} H^{1} (X_{1} (N) (ℂ), ℤ_{p})

of $ℚ_{p}$ -vector spaces, where the right-hand side is singular cohomology. This isomorphism commutes with the actions of Hecke operators, so we can describe them on the right side. Recall that $X_{1} (N) (ℂ)$ is a quotient of the union $ℍ^{*}$ of the upper-half place $ℍ$ and $ℚ \cup {\infty}$ by the congruence subgroup

Γ_{1} (N) = {(\begin{matrix} a & b \\ c & d \end{matrix}) \in {SL}_{2} (ℤ) ∣ (c, d) \equiv (0, 1) mod N} .

For a prime $ℓ$ , set

Γ_{1} (N, ℓ) = {(\begin{matrix} a & b \\ c & d \end{matrix}) \in Γ_{1} (N) ∣ c \equiv 0 mod ℓ} .

Consider the diagram

Two maps defining a modular-curve correspondence. A full diagram description follows.

Diagram description: Two maps defining a modular-curve correspondence

This is a correspondence, with two maps from the upper modular-curve quotient to the two copies of X subscript 1 of N evaluated at the complex numbers. Psi subscript ell is induced by multiplication by ell on the extended upper half-plane; pi subscript ell is induced by the identity.

Objects, listed by row and column:

Row 1, from left to right: column 2: capital Gamma subscript (1)(N, ell ) backslash blackboard H superscript (star).
Row 2, from left to right: column 1: X subscript (1)(N)( blackboard C ); column 3: X subscript (1)(N)( blackboard C ).

Arrows and lines:

An arrow from capital Gamma subscript (1)(N, ell ) backslash blackboard H superscript (star) to X subscript (1)(N)( blackboard C ) (row 2, column 1), labelled psi subscript (ell).
An arrow from capital Gamma subscript (1)(N, ell ) backslash blackboard H superscript (star) to X subscript (1)(N)( blackboard C ) (row 2, column 3), labelled pi subscript (ell).

where $ψ_{ℓ}$ is induced by multiplication by $ℓ$ on $ℍ^{*}$ and $π_{ℓ}$ is induced by the identity. This gives rise to two correspondences on $H^{1} (X_{1} (N) (ℂ), ℤ_{p})$ which are in a sense dual: we take the dual correspondence $T^{*} (ℓ)$ given by pullback by $ψ_{ℓ}$ followed by pushforward by $π_{ℓ}$ . (The usual Hecke correspondence $T (ℓ)$ is given instead by ${(ψ_{ℓ})}_{*} π_{ℓ}^{*}$ .) We also have dual diamond operators ${⟨ j ⟩}^{*}$ for $j \in {(ℤ ∕ 𝑁ℤ)}^{\times}$ (inverse to the usual ones) that are the automorphisms induced by the maps on $Y_{1} (N)$ given by $(\begin{matrix} a & b \\ c & d \end{matrix}) \in {SL}_{2} (ℤ)$ with $d \equiv j^{- 1} mod N$ . We let $𝔥 (N)$ denote the Hecke algebra of endomorphisms of $H^{1} (X_{1} (N) (ℂ), ℤ_{p})$ generated by these dual correspondences and diamond operators. Back on étale cohomology, the Galois and Hecke actions commute.

If $N ∣ M$ , then we have trace maps $Tr : 𝒯_{M} \to 𝒯_{N}$ given on singular cohomology by summing over $Γ_{1} (N) ∕ Γ_{1} (M)$ -conjugates (upon pullback to $𝒯_{M}$ via the injective map induced by the identity on $ℍ$ ). One key reason for our use of dual Hecke operators is that the trace map commutes with their actions. In particular, if we consider a tower of modular curves $X_{1} (N p^{n})$ for a fixed $N \geq 1$ not divisible by $p$ and $n \geq 1$ , then we have an inverse limit of cohomology groups ${lim_{\leftarrow}}_{n} 𝒯_{m p^{n}}$ under trace maps. Of particular interest to us is the $T^{*} (p)$ -ordinary part $𝒯 = {lim_{\leftarrow}}_{n} 𝒯_{m p^{n}}^{ord}$ of $H$ : it is the maximal direct summand of $H$ on which $T^{*} (p)$ acts invertibly. The ordinary part $𝔥^{*} = {lim_{\leftarrow}}_{n} 𝔥 {(m p^{n})}^{ord}$ inverse limit of Hecke algebras acting on $𝒯$ . This Hecke algebra $𝔥^{*}$ is known as Hida’s ordinary (dual, cuspidal) $ℤ_{p}$ -Hecke algebra of tame level $m$ .

One of the key properties of Hida’s ordinary Hecke algebra $𝔥$ is it nicely encapsulates the structure of ordinary parts of cuspidal Hecke algebras of all weights and levels. The Hecke algebra is free of finite rank over the Iwasawa algebra $Λ = ℤ_{p} ⟦ T ⟧$ , where $T = {⟨ 1 + p ⟩}^{*} - 1$ . If for $k \geq 2$ and $n \geq 1$ , the ordinary part of the weight $k$ , level $N p^{n}$ Hecke algebra that acts on $H^{1} {(X_{1} (m p^{n}) (ℂ), {Sym}^{k - 1} (ℤ_{p}^{2}))}^{ord}$ , is isomorphic to $𝔥 ∕ ({(1 + T)}^{p^{n}} - {(1 + p)}^{p^{n} (k - 2)})$ . Moreover, the latter cohomology group is isomorphic to the quotient of the free of finite rank $Λ$ -module $𝒯$ by the action of ${(1 + T)}^{p^{n}} - {(1 + p)}^{p^{n} (k - 2)}$ .

It is perhaps more typical to speak of Hida’s Hecke algebra as acting on the space of ordinary $Λ$ -adic cusp forms via the usual (not dual) action of Hecke operators. (The algebras of usual and dual Hecke algebras are isomorphic via the map that takes a Hecke operator to the corresponding dual operator.) For this, one has the theory of $Λ$ -adic modular forms, which are $q$ -expansions with coefficients in $Λ$ that specialize upon plugging in ${(1 + p)}^{k - 2} - 1$ for $T$ to weight $k$ cusp forms for each (or, equivalently, all but finitely many) $k \geq 2$ . For an eigenform to be $T (p)$ -ordinary means that its $p$ th Fourier coefficient is a unit. Again, we have the same sort of good control when we specialize at various weights and levels. Let us denote the $𝔥$ -module of $Λ$ -adic cusp forms by $𝒮$ . Hida proved that the pairing $𝔥 \times 𝒮 \to Λ$ of $Λ$ -modules that takes $(T, f)$ to the $q$ -coefficient of $T f$ is perfect, so $𝔥 ≅ {Hom}_{Λ} (𝒮, Λ)$ and $𝒮 ≅ {Hom}_{Λ} (𝔥, Λ)$ . Moreover, $𝒮 \otimes_{Λ} 𝒬$ , where $𝒬$ is the quotient field of $Λ$ , is free of rank one over $𝔥 \otimes_{Λ} 𝒬$ .

One sees that $𝒯$ fits in an exact sequence of $ℤ_{p} ⟦ G_{ℚ_{p}} ⟧$ -modules of the form

0 \to 𝒯_{sub} \to 𝒯 \to 𝒯_{quo} \to 0,

where $𝒯_{quo}$ has unramified action and is noncanonically isomorphic to the space of ordinary $Λ$ -adic cusp forms via an isomorphism that switches dual and usual Hecke actions. The key point here is that for the Galois representation $𝒯$ to be ordinary for $T^{*} (p)$ means also to be ordinary in the sense of $p$ -adic Hodge theory, which insures that it has a filtration of the above form. The Hecke operator $T^{*} (p)$ acts as the Frobenius $φ_{p}$ on $𝒯_{quo}$ . The characteristic polynomial of the Frobenius $φ_{ℓ}$ for $ℓ ∤ 𝑚𝑝$ acting on the rank two module $𝒯 \otimes_{Λ} 𝒬$ is an $𝔥 \otimes_{Λ} 𝒬$ -representation with $T^{*} (p)$ -action given by $x^{2} - T^{*} (ℓ) x + ℓ {⟨ ℓ ⟩}^{*}$ . One might roughly think of $𝒯$ as encapsulating all of the $p$ -adic Galois representations attached to ordinary cusp forms of tame level (dividing) $m$ at once.

A version of Poincaré duality, modified to be compatible with the inverse limit, sets up a perfect pairing of $Λ$ -modules $(,) : 𝒯 \times 𝒯 \to Λ$ such that $(T x, y) = (x, 𝑇𝑦)$ for $x, y \in 𝒯$ and $T \in 𝔥$ , and this induces a perfect pairing $𝒯_{sub} \times 𝒯_{quo} \to Λ$ . From this and the duality between Hida’s Hecke algebra and ordinary $Λ$ -adic cusp forms, we see that $𝒯_{sub} ≅ 𝔥$ . We remark that we may lift $𝒯_{quo} \otimes_{Λ} 𝒬$ to a subspace of $𝒯 \otimes_{Λ} 𝒬$ complementary to $𝒯_{sub} \otimes_{Λ} 𝒬$ . We would preferably lift $𝒯_{quo}$ itself, but it is not clear one can do this if $𝜃 ω^{- 1} (p) = 1$ . However, we can get away with something close in all eigenspaces using the action of a chosen element $ν$ of the inertia group $I_{p}$ at $p$ with $ν (ζ_{p^{n}}) = ζ_{p^{n}}^{1 + p}$ for all $n$ . Set $u = (1 + T) (1 + p)$ and $Λ^{'} = Λ [{(u - 1)}^{- 1}]$ . We declare $𝒯^{+}$ to be the $𝔥 \otimes_{Λ} Λ^{'}$ -submodule fixed by $ν$ . This clearly works, as the determinant in $𝒬^{\times}$ of the action of $ν$ is $u$ , but $ν$ acts trivially on the quotient $𝒯_{quo} \otimes_{Λ} Λ^{'}$ . We set $𝒯^{-} = 𝒯_{sub} \otimes_{Λ} Λ^{'}$ .

By picking an ordered basis of $𝒯 \otimes_{Λ} 𝒬$ from $𝒯^{-}$ and $𝒯^{+}$ , respectively, we see that the Galois representation

ρ : G_{ℚ} \to {GL}_{2} (𝔥 \otimes_{Λ} 𝒬), ρ = (\begin{matrix} a & b \\ c & d \end{matrix})

is upper-triangular on $G_{ℚ_{p}}$ and has the form

ρ |_{I_{p}} = (\begin{matrix} \det ρ & b \\ 0 & 1 \end{matrix})

on the inertia subgroup $I_{p}$ . We are particularly interested in the map $c$ .

Let $I$ denote the ideal of $𝔥$ generated by all $T^{*} (ℓ) - 1 - ℓ {⟨ ℓ ⟩}^{*}$ for primes $ℓ ∤ 𝑚𝑝$ and $T^{*} (ℓ) - 1$ for primes $ℓ ∣ 𝑚𝑝$ , and fix an even $p$ -adic character $𝜃$ of ${(ℤ ∕ 𝑚𝑝ℤ)}^{\times}$ of conductor $m$ or $𝑚𝑝$ . Set $χ = 𝜃 ω^{2}$ . The image $I_{𝜃}$ of $I$ in $𝔥^{(𝜃)}$ (which corresponds to the $𝜃^{- 1}$ -eigenspace of the usual non-cuspidal Hecke algebra acting on the space $Λ$ -adic cuspidal modular forms) is the image of the ideal of the Hecke algebra acting on $Λ$ -adic modular forms that is the annihilator of the $Λ$ -adic Eisenstein series

G_{𝜃^{- 1}} = \frac{1}{2} g_{χ}^{0} + \sum_{n = 1}^{\infty} \sum_{\begin{array}{c} d ∣ n \\ (d, 𝑚𝑝) = 1 \end{array}} d 𝜃^{- 1} (d) ⟨ κ (d) ⟩ q^{n},

where $κ (d)$ is the projection of $d \in ℤ_{p, m}^{\times}$ into $1 + p ℤ_{p}$ . Here $g_{χ}^{0} = g_{χ}$ if $𝜃𝜔 (p) \neq 1$ and $g_{χ}^{0} = {(T - p)}^{- 1} g_{χ}$ otherwise.

The quotient ${(𝔥 ∕ I)}^{(𝜃)}$ measures, in a sense, the failure of the above Eisenstein series $G_{𝜃^{- 1}}$ to be a cusp form. This Eisenstein series induces map from Hida’s full modular Hecke algebra $ℌ$ acting on the space of $Λ$ -adic modular forms to $Λ_{𝜃^{- 1}}$ , taking $T (ℓ)$ to the corresponding Fourier coefficient, and its kernel is the Eisenstein ideal in the $𝜃^{- 1}$ -eigenspace of this Hecke algebra. On the dual cuspidal Hecke algebra $𝔥^{(𝜃)}$ , this yields a surjection ${(𝔥 ∕ I)}^{(𝜃)} \to Λ_{𝜃} ∕ ι (g_{χ}^{0})$ since $G_{𝜃^{- 1}}$ becomes a cusp form when reduced modulo its constant term. In fact, this surjection is an isomorphism for $𝜃 \neq ω^{2}$ , though we shall not require it in our proof.

Now suppose that $f_{χ} \notin {(Λ_{χ} [T^{- 1}])}^{\times}$ . Note that $T$ divides $f_{χ}$ if and only if $χ ω^{- 1} (p) = 𝜃𝜔 (p) = 1$ . (Recall that $f_{χ} ({(1 + p)}^{s} - 1) = L_{p} (χ, s)$ for all $s \in ℤ_{p}$ .). By the result of Ferrero and Greenberg, $T$ exactly divides $f_{χ}$ in the “exceptional” case that $χ ω^{- 1} (p) = 1$ , and $T ∤ f_{χ}$ for non-exceptional $χ$ .

We shall be interested in the $𝜃$ -eigenspaces (under the action of diamond operators) of our Galois representation $ρ$ that is defined by $𝒯^{(𝜃)} \otimes_{Λ} 𝒬$ , so we view $ρ$ as taking values in ${GL}_{2} (𝔥^{(𝜃)} \otimes_{Λ} 𝒬)$ by projection.

Lemma 6.6.1. §

For $σ, τ \in G_{ℚ}$ , the elements $a (σ) - \det ρ (σ)$ , $d (σ) - 1$ , and $b (σ) c (τ)$ of $𝔥^{(𝜃)} \otimes_{Λ} 𝒬$ are all contained in $I_{𝜃} \subset 𝔥^{(𝜃)}$ .

Proof.

Note that $𝔥 = {End}_{𝔥} (𝔥) = {End}_{𝔥} (𝒮)$ , so $a$ and $d$ take values in $𝔥$ , and moreover $b (σ) c (τ) \in 𝔥$ for all $σ, τ \in G_{ℚ}$ since compositions of elements in ${Hom}_{𝔥} (𝒮, 𝔥)$ and ${Hom}_{𝔥} (𝔥, 𝒮)$ lie in one of the aforementioned endomorphism groups.

It suffices to show the containments in question on Frobenius elements $φ_{ℓ}$ (or their “geometric” inverses) at $ℓ ∤ 𝑁𝑝$ by the Čebotarev density theorem. One has that

a (φ_{ℓ}^{- 1}) + d (φ_{ℓ}^{- 1}) = ℓ^{- 1} T (ℓ) = ℓ^{- 1} ⟨ ℓ ⟩ T^{*} (ℓ) \equiv 1 + ℓ^{- 1} ⟨ ℓ ⟩ mod I_{𝜃} .

Since $\det ρ (φ_{ℓ}^{- 1}) = ℓ^{- 1} ⟨ ℓ ⟩$ for all $ℓ$ , we therefore have

a (σ) + d (σ) \equiv 1 + \det ρ (σ) mod I_{𝜃}

for all $σ \in G_{ℚ}$ . The element $ν$ used to lift $𝒯_{quo}$ satisfies

ρ (ν) = (\begin{matrix} u & 0 \\ 0 & 1 \end{matrix}),

where $u = (1 + p) (T + 1)$ . Taking the trace of $ρ (𝜈𝜎)$ , we see that

𝑢𝑎 (σ) + d (σ) \equiv 1 + u \det ρ (σ) mod I_{𝜃},

again for all $σ$ . It follows that $a (σ) - \det ρ (σ) \in I_{𝜃}$ and $d (σ) - 1 \in I_{𝜃}$ .

Now consider $σ, τ \in G_{ℚ}$ and note that $a (𝜎𝜏) = a (σ) a (τ) + b (σ) c (τ)$ . Thus we have

b (σ) c (τ) = (a (𝜎𝜏) - \det ρ (𝜎𝜏)) - (a (σ) a (τ) - \det ρ (σ) \cdot \det ρ (τ)) \in I_{𝜃} .

□

Let $B$ (resp., $C$ ) denote the $𝔥 \otimes_{Λ} Λ^{'}$ -submodules of $𝔥^{(𝜃)} \otimes_{Λ} 𝒬$ generated by the elements $b (σ)$ (resp., $c (σ)$ ) with $σ \in G_{ℚ}$ . The $𝔥$ -module $𝐵𝐶$ of sums of products is an ideal of $𝔥^{(𝜃)}$ contained in $I_{𝜃}$ .

Lemma 6.6.2. §

The ideal $𝐵𝐶$ of $𝔥^{(𝜃)} \otimes_{Λ} Λ^{'}$ is a faithful $𝔥^{(𝜃)}$ -module.

Proof.

The map $δ : G_{ℚ} \to {(𝔥 ∕ 𝐵𝐶)}^{\times}$ induced by $σ \mapsto d (σ)$ is a homomorphism that is unramified outside of the primes over $m$ . It is then at most tamely ramified at these primes, so by class field theory the map factors through a quotient of $\prod_{ℓ ∣ m} ℤ_{ℓ}^{\times}$ . Since the pro-abelian group ${(𝔥 ∕ 𝐵𝐶)}^{\times}$ has finite prime-to- $p$ part and the group $\prod_{ℓ ∣ m} ℤ_{ℓ}^{\times}$ has finite $p$ -part, we see that the image of $δ$ is finite

For $ℓ ∤ 𝑚𝑝$ , we have

\begin{matrix} ℓ^{- 1} ⟨ ℓ ⟩ (T^{*} (ℓ) - 1 - ℓ {⟨ ℓ ⟩}^{*}) = a (φ_{ℓ}^{- 1}) + d (φ_{ℓ}^{- 1}) - \det ρ (φ_{ℓ}^{- 1}) - 1 \\ = - (a (φ_{ℓ}^{- 1}) - 1) (d (φ_{ℓ}^{- 1}) - 1) + b (φ_{ℓ}^{- 1}) c (φ_{ℓ}^{- 1}), \end{matrix}

By the Čebotarev density theorem, we can find infinitely many primes $ℓ ∤ 𝑚𝑝$ such that $d (φ_{ℓ}^{- 1}) - 1 \in 𝐵𝐶$ . For such an $ℓ$ , we have then $T^{*} (ℓ) - 1 - ℓ {⟨ ℓ ⟩}^{*} \in 𝐵𝐶$ . This element is not a zero divisor in $𝔥^{(𝜃)}$ (as it does not annihilate any ordinary $Λ$ -adic cuspidal eigenform with character $𝜃^{- 1}$ , which we do not verify here), so the annihilator of $𝐵𝐶$ in $𝔥^{(𝜃)}$ is trivial. □

We have the following corollary.

Corollary 6.6.3. §

The $𝔥^{(𝜃)} \otimes_{Λ} Λ^{'}$ -modules $B$ and $C$ are faithful.

Let $F = ℚ (μ_{𝑚𝑝})$ , and let $F_{\infty}$ be its cyclotomic $ℤ_{p}$ -extension.

Proposition 6.6.4. §

The map $\bar{c} : G_{ℚ} \to C ∕ I_{𝜃} C$ induced by $c$ restricts to a homomorphism on $G_{F_{\infty}}$ with the same image as $\bar{c}$ and which factors through $X_{\infty}^{(ω χ^{- 1})}$

Proof.

For $σ, τ \in G_{ℚ}$ , we have that

c (𝜎𝜏) = a (τ) c (σ) + c (τ) d (σ) \equiv \det ρ (τ) c (σ) + c (τ) mod I_{𝜃} .

Since $\det ρ$ factors through $Gal (F_{\infty} ∕ ℚ)$ , we see that $\bar{c}$ is a homomorphism, and it factors through $X_{\infty}$ since $c |_{I_{p}} = 0$ .

For $σ_{j} \in Gal (F_{\infty} ∕ ℚ)$ with $σ_{j} (ζ_{m p^{n}}) = ζ_{m p^{n}}^{j}$ for all $n$ , where $j \in ℤ_{p, m}^{\times}$ , we have

\det ρ (σ_{j}) = j_{p} {⟨ j ⟩}^{*} = 𝜔𝜃 (j) κ (j) {⟨ κ (j) ⟩}^{*} .

In particular, for $j \in {(ℤ ∕ 𝑚𝑝ℤ)}^{\times}$ and $τ \in G_{F_{\infty}}$ , we have

\bar{c} (σ_{j} τ σ_{j}^{- 1}) = \det ρ {(σ_{j})}^{- 1} \bar{c} (τ) = {(𝜔𝜃)}^{- 1} (j) \bar{c} (τ) = ω χ^{- 1} (j) \bar{c} (τ) .

Finally, letting $σ \in G_{ℚ}$ , the commutator $[ν, σ]$ lies in $G_{F_{\infty}}$ , and we have

\bar{c} ([ν, σ]) = (u^{- 1} - 1) \bar{c} (σ) .

Since $u^{- 1} - 1$ is a unit in $Λ^{'}$ , we are done. □

Using the fact that $C$ is a faithful $𝔥^{(𝜃)}$ -module and the theory of Fitting ideals, one can show that the characteristic ideal of $C ∕ I_{𝜃} C$ as a module over the algebra $Λ_{𝜃}$ of diamond operators is divisible by $g_{ω^{2} 𝜃^{- 1}}^{0}$ . Since $X_{\infty}^{(ω χ^{- 1})}$ maps surjectively to $C ∕ I_{𝜃} C$ via $\bar{c}$ , we obtain the following theorem (upon application of the theorem of Ferrero and Greenberg to deal with exceptional zeros).

Theorem 6.6.5. §

The ideal $(f_{χ})$ divides ${char}_{Λ_{χ}} X_{\infty}^{(ω χ^{- 1})}$ .

Chapter 6 The Iwasawa main conjecture

6.1. Semi-local units modulo cyclotomic units

Proposition 6.1.1. §

Corollary 6.1.2. §

Theorem 6.1.3. §

Corollary 6.1.4. §

Proposition 6.1.5. §

Proposition 6.1.6. §

Corollary 6.1.7. §

6.2. The Ferrero-Washington theorem

Theorem 6.2.1 (Ferrero-Washington). §

Corollary 6.2.2. §

Notation 6.2.3. §

Proposition 6.2.4. §

Definition 6.2.5. §

Definition 6.2.6. §

Theorem 6.2.7 (Weyl). §

Proposition 6.2.8. §

Proposition 6.2.9. §

6.3. The main conjecture over ℚ

Conjecture 6.3.1 (The Iwasawa Main Conjecture). §

Proposition 6.3.2. §

Theorem 6.3.3. §

Theorem 6.3.4 (Mazur-Wiles). §

Proposition 6.3.5. §

Proposition 6.3.6. §

Theorem 6.3.7 (Ferrero-Greenberg). §

6.4. The Euler system of cyclotomic units

Definition 6.4.1. §

Lemma 6.4.2. §

Lemma 6.4.3. §

Lemma 6.4.4. §

Terminology 6.4.5. §

Remark 6.4.6. §

Lemma 6.4.7. §

Lemma 6.4.8. §

Lemma 6.4.9. §

Remark 6.4.10. §

Proposition 6.4.11. §

Proposition 6.4.12. §

Lemma 6.4.13. §

Theorem 6.4.14. §

6.5. The main conjecture via Euler systems

Lemma 6.5.1. §

Proposition 6.5.2. §

Proposition 6.5.3. §

Lemma 6.5.4. §

Lemma 6.5.5. §

Theorem 6.5.6. §

6.6. Geometry of modular curves

Lemma 6.6.1. §

Lemma 6.6.2. §

Corollary 6.6.3. §

Proposition 6.6.4. §

Theorem 6.6.5. §

Chapter 6
The Iwasawa main conjecture

6.3. The main conjecture over $ℚ$