Classical Iwasawa theory and infinite descent on a family of abelian varieties

Abstract

For primes $q \equiv 7 \ \mathrm {mod}\ 16$, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field $K = {\mathbb {Q}}(\sqrt{-q})$, which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.

Appendix A

Unlike the earlier part of this paper, p will now denote an arbitrary prime number throughout this Appendix. Also K will now denote either the rational field ${\mathbb {Q}}$, or an arbitrary imaginary quadratic field. Let ${\mathfrak {p}}$ be a prime of K such that the completion $K_{\mathfrak {p}}$ is ${\mathbb {Q}}_p$. We write F now for an arbitrary finite extension of K. However, if $K={\mathbb {Q}}$, we shall always assume in addition that F is totally real. By class field theory, K admits a unique ${\mathbb {Z}}_p$-extension which is unramified outside ${\mathfrak {p}}$, and we denote this ${\mathbb {Z}}_p$-extension by $K_\infty$. Let $K_n$ be its n-th layer. Write $\mathscr {S}$ for the set of primes of F lying above ${\mathfrak {p}}$. Let M be the maximal abelian p-extension of F, which is unramified outside $\mathscr {S}$. The following classical formula is due to the first author [3] when $K={\mathbb {Q}}$, and the first author and Wiles [5] when K is imaginary quadratic. Let $R_{\mathfrak {p}}$ be the ${\mathfrak {p}}$-adic regulator of F, whose definition is recalled below (A.2). Let $F_\infty = FK_\infty$.

Theorem A.1

With notation as above, the degree $[M:F_\infty ]$ is finite if and only if the ${\mathfrak {p}}$-adic regulator $R_{\mathfrak {p}}$ of F is non-zero. If $R_{\mathfrak {p}}\ne 0$, then

$$\begin{aligned} {[}M:F_\infty ]&= \frac{2p^{e-f+1}R_{\mathfrak {p}}h_F}{\omega _F \sqrt{\Delta _{\mathfrak {p}}}}\prod _{\mathfrak {g}\in \mathscr {S}}(1-(N\mathfrak {g})^{-1}),\quad \nonumber \\&\qquad \text {up to multiplication by a }p\text {-adic unit}. \end{aligned}$$

(A.1)

Here $h_F$ is the class number of F, $\omega _F$ is the number of roots of unity in F, $\Delta _{\mathfrak {p}}\in {\mathcal {O}}_{K_{\mathfrak {p}}}$ is a generator of the ${\mathfrak {p}}$-component ideal of the relative discriminant of F/K, and $N(\mathfrak {g})$ is the absolute norm of an ideal $\mathfrak {g}$ of F. Moreover, the integers e and f are defined by $F\cap K_\infty = K_e$ and $H\cap K_\infty = K_f$, respectively, where H denotes the Hilbert class field of K (thus $f=0$ when $K = {\mathbb {Q}}$).

This formula is stated and proved in [3] and [5] under the assumptions that the prime p is odd, and that p does not divide the class number of K. However, it is often precisely these cases which are needed for certain applications of the formula, as, for example, in the present paper. Thus the aim of this appendix is to show that, after some very slight modifications, the arguments given in the two original papers work in complete generality, and give a proof of Theorem A.1. All the notation we used in this appendix is compatible with [5].

Let $d=[F:K]$. We first recall the definition of the ${\mathfrak {p}}$-adic regulator $R_{\mathfrak {p}}$. Let $\varepsilon _1,\ldots , \varepsilon _{d-1}$ be a basis of the group $\mathcal {E}$ of global units of F modulo torsion, and put $\varepsilon _d=1+p$. Let $\overline{K}_{\mathfrak {p}}$ be a fixed algebraic closure of $K_{\mathfrak {p}}(={\mathbb {Q}}_p)$. Denote by $\phi _1,\ldots , \phi _d$ the distinct embeddings of F into $\overline{K}_{\mathfrak {p}}$. Let $\log$ denote the usual extension of the p-adic logarithmic function to $\overline{K}_{\mathfrak {p}}$. The ${\mathfrak {p}}$-adic regulator $R_{\mathfrak {p}}$ is then defined to be the $d\times d$ determinant

$$\begin{aligned} R_{\mathfrak {p}}= (d\log \varepsilon _d)^{-1} \det (\log (\phi _i(\varepsilon _j) ) )_{1\le i, j\le d}. \end{aligned}$$

(A.2)

In fact, since $\sum _{i=1}^{d} \log (\phi _i(\varepsilon _j) )= 0$ for $j=1, \ldots , d-1$, it is not difficult to see that $R_{\mathfrak {p}}= \det (\log (\phi _i(\varepsilon _j)))_{1\le i, j\le d-1}.$ Let $V=1+p{\mathbb {Z}}_p$ be the group of principal units of $K_{\mathfrak {p}}$, and define $\bar{V} =V\{ \pm 1 \}/\{ \pm 1 \}$. Note that the p-adic logarithm defines an isomorphism $\bar{V} \cong {\mathbb {Z}}_p$ for all primes p. Our arguments will hinge on the following elementary facts. Firstly, we have that $\log (\varepsilon _d)=\log (1+p)=2p$, up to a p-adic unit. Secondly, for an integer $n=p^u m$ with $p\not \mid m$, the image of the closure $\langle (1+p)^n \rangle \subset V$ under the map $V \rightarrow \bar{V}$ has index $p^u$ in $\bar{V}$.

Since K is either ${\mathbb {Q}}$ or an arbitrary imaginary quadratic field, we can now only assert that there will be only finitely many primes of $K_\infty$ lying above ${\mathfrak {p}}$. We fix such one prime $\tilde{{\mathfrak {p}}}$ of $K_\infty$ lying above ${\mathfrak {p}}$. Let $\varPsi _n$ be the completion of $K_n$ at the unique prime below $\tilde{{\mathfrak {p}}}$. Let $\varPsi _\infty = \cup _{n} \varPsi _n$, so that ${\mathrm {Gal}}(\Psi _\infty / \Psi _0) \cong {\mathbb {Z}}_p$. Now our hypothesis $H\cap K_\infty = K_f$ clearly implies that $\varPsi _\infty /\varPsi _f$ is totally ramified. Let $V_n$ denote the group of principal units of $\varPsi _n$ which are $\equiv 1$ modulo the maximal ideal. Thus $V_0 = V$. Let $N_n$ denote the norm map from $V_n$ to V, and define $\bar{N}_n$ to be the composite map

$$\begin{aligned} \bar{N}_n: V_n \xrightarrow {N_n} V \rightarrow \bar{V}. \end{aligned}$$

Of course, $\bar{N}_n=N_n$ for $p>2$.

Lemma A.2

For each $n\ge f$, we have $\bar{N}_n(V_n)= \bar{V}^{p^{n-f}}$.

Proof

Recall that the inertial subgroup of ${\mathfrak {p}}$ in the extension $K_\infty /K$ is ${\mathrm {Gal}}(K_\infty /K_f)$. It follows then from local class field theory that, assuming $n \ge f$, the Artin map induces two isomorphisms

$$\begin{aligned} V/\bigcap _{n\ge f} N_n(V_n) \cong G(\varPsi _\infty /\varPsi _f) \quad \text { and }\quad V/{N_n(V_n)}\cong G(\varPsi _n/\varPsi _f). \end{aligned}$$

Since $G(\varPsi _\infty /\varPsi _f)\cong {\mathbb {Z}}_p$, it follows that we must have $-1 \in \bigcap _{n\ge f} N_n(V_n)$, whence

$$\begin{aligned} \bar{V}/{\bar{N}_n(V_n)} \cong V/N_n(V_n) \cong {\mathbb {Z}}/{p^{n-f}{\mathbb {Z}}}, \, (n \ge f). \end{aligned}$$

But the group $\bar{V}$ is isomorphic to ${\mathbb {Z}}_p$, and so $\bar{N}_n(V_n)$, being a closed subgroup of index $p^{n-f}$, must be equal to $\bar{V}^{p^{n-f}},$ and the proof of the lemma is complete. $\square$

Define $U_1=\prod _{\mathfrak {g}\in \mathscr {S}}{ U_{\mathfrak {g},1}}$, where $U_{\mathfrak {g},1}$ denotes the local units $\equiv 1 \ \mathrm {mod}\ {\mathfrak {g}}$ in the completion of F at ${\mathfrak {g}}$, and let $N_{F/K}$ be the norm map from $U_1$ to V. Define $\bar{N}_{F/K}$ to be the composite map

$$\begin{aligned} \bar{N}_{F/K}: U_1 \xrightarrow {N_{F/K}} V \rightarrow \bar{V}. \end{aligned}$$

The following two lemmas describe the kernel and image of $\bar{N}_{F/K}$. For each $n \ge 0$, let $F_n$ be the n-th layer of the ${\mathbb {Z}}_p$-extension $F_\infty /F$, and let $C_n$ be the idele class group of $F_n$. Put

$$\begin{aligned} Y = \cap _{n \ge 0}N_{F_n/F}C_n. \end{aligned}$$

(A.3)

Lemma A.3

$Y\cap U_1$ is the kernel of $\bar{N}_{F/K}$.

Proof

This is equivalent to showing that $Y\cap U_1$ is the inverse image of $\pm 1$ under the map $N_{F/K}$. By Lemma A.2, $\bigcap _n N_n(V_n)=\{\pm 1 \}$ for $p=2$ whence $\bigcap \bar{N}_n(V_n)=1$. Granted this, the rest of the proof is exactly the same as that given in [5, Lemma 5], and we omit the details. $\square$

Lemma A.4

Let L be the p-Hilbert class field of F and let the integer k be defined by $L\cap F_\infty =F_k$. Then $\bar{N}_{F/K}(U_1)=\bar{V}^{p^{e+k-f}}$.

Proof

The proof is essentially the same as the one of [5, Lemma 6]. Firstly, one needs to replace $N_{F/K}$ and $N_n$ there by $\bar{N}_{F/K}$ and $\bar{N}_n$, respectively. Clearly, $k+e-f\ge 0$. By Lemma A.3, $\bar{N}_{k+e}(V_{k+e})=\bar{V}^{p^{k+e-f}}$. Thus one also needs to replace the integer e in [5] by $e-f$, and t by $t+f$ consequently. The rest of the argument is the same as in [5, Lemma 6]. $\square$

Let $\mathcal {E}_1$ be the group of global units of F, which are $\equiv 1\bmod \mathfrak {g}$ for each $\mathfrak {g}\in \mathscr {S}$. Let $j: F\rightarrow \prod _{\mathfrak {g}\in \mathscr {S}}F_\mathfrak {g}$ be the canonical embedding. Define D to be the ${\mathbb {Z}}_p$-submodule of $U_1$ which is generated by $j(\mathcal {E}_1)$ and $j(\varepsilon _d)$.

Lemma A.5

The index of D in $U_1$ is finite if and only if $R_{\mathfrak {p}}\ne 0$. If $R_{\mathfrak {p}}\ne 0$, then

$$\begin{aligned} {[}U_1:D] = \frac{2pd}{\omega _F \sqrt{\Delta _{\mathfrak {p}}}} \prod _{\mathfrak {g}\in \mathscr {S}}(1-(N\mathfrak {g})^{-1}) \quad \text {up to a }p\text {-adic unit}. \end{aligned}$$

Proof

This is proven in [5, Lemma 7,8 and 9]. The proof there applies to our case without change, but we remind the reader that, when $p=2$, $\log (\varepsilon _d)=\log (3)=2p$, up to a 2-adic unit. This gives the extra factor 2 in Lemma A.5 when comparing with [5, Lemma 9]. $\square$

We now come to the crucial lemma. If $p>2$, we clearly have $N_{F/K}(j(\mathcal {E}_1))=\{1\}$, since $N_{F/K}(j(\mathcal {E}_1))$ is contained in $1+p{\mathbb {Z}}_p$. If $p=2$, then the unit group of K is $\{\pm 1 \}$, since 2 does not split in ${\mathbb {Q}}(i)$ or ${\mathbb {Q}}(\sqrt{-3})$. Thus $N_{F/K}(\overline{j(\mathcal {E}_1)})\subset \{\pm 1\}$. In other words, $\overline{j(\mathcal {E}_1)}$ is contained in the kernel of the map $\bar{N}_{F/K}$. Thus Lemma A.3 shows that $\overline{j(\mathcal {E}_1)}$ is contained in $Y\cap U_1$.

Lemma A.6

Let the integer k be as in Lemma A.4. The index of $\overline{j(\mathcal {E}_1)}$ in $Y\cap U_1$ is finite if and only if $R_{\mathfrak {p}}\ne 0$. If $R_{\mathfrak {p}}\ne 0$, then

$$\begin{aligned} {[}Y\cap U_1:\overline{j(\mathcal {E}_1)}]= \frac{2p^{e+k-f+1}R_{\mathfrak {p}}}{\omega _F \sqrt{\Delta _{\mathfrak {p}}}}\prod _{\mathfrak {g}\in \mathscr {S}}(1-(N\mathfrak {g})^{-1})\quad \text {up to a }p\text {-adic unit}. \end{aligned}$$

Proof

We have the commutative diagram with exact rows

The exactness of the first row follows from Lemmas A.3 and A.4. Note that $\bar{N}_{F/K}(D)$ is the closure of the image of $\langle (1+p)^d \rangle$ in $\bar{V}$, which coincides with $\bar{V}^d$. Since $\bar{V}\cong {\mathbb {Z}}_p$, we have that $[\bar{V}^{p^{e+k-f}}: \bar{V}^d]=d/p^{e+k-f}$, up to a p-adic unit. Thus Lemma A.6 follows from Lemma A.5. $\square$

Proof of Theorem A.1

As in [5, Theorem 11], noting that we have already shown that $\overline{j(\mathcal {E}_1)}$ is contained in $Y\cap U_1$, it is a standard consequence of global class field theory that

$$\begin{aligned} (Y\cap U_1)/\overline{j(\mathcal {E}_1)} \cong G(M/LF_\infty ), \end{aligned}$$

where, as in Lemma A.4, L is the p-Hilbert class field of F. It follows that

$$\begin{aligned} {[}M:F_\infty ]=[M:LF_\infty ][LF_\infty :F_\infty ] =[Y\cap U_1:\overline{j(\mathcal {E}_1)} ]h_F/p^k \quad \text {up to a }p\text {-adic unit}. \end{aligned}$$

The last equality follows from $[LF_\infty :F_\infty ]=[L:L\cap F_\infty ]$ and the definition of k given in Lemma A.4. Now, Theorem A.1 follows from Lemma A.6. $\square$

We end with the following remark. Let F be a totally real number field of degree d. The reason for proving Theorem A.1 in 1974 was that it provided the first general evidence that Iwasawa’s then revolutionary discovery of his p-adic “main conjecture” for totally real F lying inside the field generated by all p-power roots of unity might hold in complete generality for the cyclotomic ${\mathbb {Z}}_p$-extension of all totally real number fields F. The deep subsequent work of Mazur-Wiles and Wiles has happily shown this to be true for all totally real F and all odd primes p. However, the situation for the prime $p=2$ still has not been completely settled, and we want to just point out that, even in this special case, Theorem A.1 is in perfect accord with the “main conjecture” which we believe to be true. Let $\mathfrak {M}$ be the maximal abelian p-extension of F, which is unramified outside p and the infinite primes of F. Let $F_\infty$ be the cyclotomic ${\mathbb {Z}}_p$-extension of F. Let $I_F$ be the idèle group of F. It follows from class field theory that we have the following commutative diagram; all the maps are clearly surjective and the kernels of the maps on the rows are finite groups of order prime to p:

Lemma A.7

The kernel of the vertical map on the left is isomorphic to $({\mathbb {R}}^\times /{\mathbb {R}}_{>0})^d$.

Proof

Note that $\prod _{v\mid \infty }{\mathbb {R}}^\times$ naturally maps onto this kernel. Thus, the assertion will directly follow from the following identity in $I_F$:

$$\begin{aligned} \left( \prod _{v\mid \infty }{\mathbb {R}}^\times \right) \bigcap \left( \overline{ F^\times \prod _{v\not \mid p\infty }U_v \prod _{v\mid \infty }{\mathbb {R}}_{>0}}\right) = \prod \limits _{v\mid \infty } {\mathbb {R}}_{>0}. \end{aligned}$$

(A.4)

To see this identity, note that by definition an element $(a_v)_v$ of $\prod _{v\mid \infty }{{\mathbb {R}}^\times }\subset I_K$ has component 1 at every finite place. Let y be an element in the second group of (A.4). Then y is a limit of $x^{(n)} \cdot b^{(n)}$ where $x^{(n)}\in F^\times$ and $b^{(n)} =(b^{(n)}_v)_v\in \prod _{v\not \mid p\infty } \prod _{v\mid \infty }{\mathbb {R}}_{>0}$ for each n. But note that $b^{(n)}_v=1$ if $v\mid p$. This forces $x^{(n)}=1$ for each n whence $b^{(n)}_v>0$ for $v\mid \infty$ and n is large. This proves (A.4), completing the proof of the lemma. $\square$

Thus, thanks to this lemma, we obtain the following result from Theorem A.1.

Theorem A.8

Let F be a totally real number field of degree d. Let $F_\infty$ be the cyclotomic ${\mathbb {Z}}_p$-extension of F, and let $\mathfrak {M}$ be the maximal abelian p-extension of F which is unramified outside p and the infinite primes. Let e and $h_F$ as defined in Theorem A.1. Let $\Delta$ be the discriminant of F. Assuming the p-adic regulator $R_p$ of F is nonzero, we have

$$\begin{aligned} {[}\mathfrak {M}:F_\infty ]= \frac{2^{d}p^{e+1}R_p h_F}{ \sqrt{\Delta }}\prod _{v\mid p}(1-(Nv)^{-1})\quad , \text {up to a }p\text {-adic unit}. \end{aligned}$$

Let $\zeta _{F,p}(s)$ be the p-adic zeta function of F, constructed by P. Cassoun-Nogues and P. Deligne and K. Ribet. Assuming that $R_p \ne 0$, Colmez [6] proved that the residue of $\zeta _{F,p}$ at $s=1$ is

$$\begin{aligned} \frac{2^{d-1}R_p h_F}{\sqrt{\Delta }} \prod _{v\mid p}(1-(Nv)^{-1}). \end{aligned}$$

We end by pointing out that Colmez’s formula and Theorem A.8 are in perfect accord for all primes p, including $p=2$, via the following “main conjecture” of Iwasawa theory. Let $\mathfrak {M}_\infty$ be the maximal abelian p-extension of $F_\infty$ which is unramified outside ${\mathfrak {p}}$ and the infinite primes, and put ${\mathfrak {X}}_\infty = {\mathrm {Gal}}(\mathfrak {M}_\infty /F_\infty )$. Let $\Gamma = {\mathrm {Gal}}(F_\infty /F)$, and let $\Lambda (\Gamma )$ be the Iwasawa algebra of $\Gamma$. Now, fixing a topological generator $\gamma$ of $\Gamma$, we can identify $\Lambda (\Gamma )$ with the ring of formal power series ${\mathbb {Z}}_p[[T]]$ by mapping $\gamma$ to $1+T$. Let $\kappa : \Gamma \rightarrow {\mathbb {Z}}_p^\times$ be the cyclotomic character of $\Gamma$, and put $u = \kappa (\gamma )$. Thus, by the very definition of the cyclotomic character, we have $u - 1$ is equal to $2p^{e+1}$ up to a p-adic unit. Now Iwasawa [15] has shown that ${\mathfrak {X}}_\infty$ is a finitely generated torsion $\Lambda (\Gamma )$-module with no non-zero finite $\Lambda (\Gamma )$-submodule. Thus, by the structure theory of such modules, we can associate to ${\mathfrak {X}}_\infty$ a characteristic power series $f_{{\mathfrak {X}}_\infty }(T)$ in ${\mathbb {Z}}_p[[T]]$. Moreover, $({\mathfrak {X}}_\infty )_\Gamma = {\mathrm {Gal}}(\mathfrak {M}/F_\infty )$. It then follows from the Euler characteristic formula that, assuming $R_p \ne 0$, we have

$$\begin{aligned} {[}\mathfrak {M}:F_\infty ] = f_{{\mathfrak {X}}_\infty }(0), \, \, \text {up to a }p\text {-adic unit}. \end{aligned}$$

(A.5)

Now the “main conjecture” for ${\mathfrak {X}}_\infty$ asserts that, for a suitable choice of the characteristic power series $f_{{\mathfrak {X}}_\infty }(T)$, we have

$$\begin{aligned} \zeta _{F,p}(s) = f_{{\mathfrak {X}}_\infty }(u^{s-1} - 1)/(u^s - u), \end{aligned}$$

(A.6)

where, as above, $\zeta _{F,p}(s)$ is the p-adic zeta function of F. Also

$$\begin{aligned} u^{s-1} - 1 = (s-1)\log (u)\,\, + \, \, \text { higher powers of }(s-1), \end{aligned}$$

(A.7)

and $\log (u) = 2p^{e+1}$, up to a p-adic unit. If we now combine Theorem A.8 with (A.5), (A.6), (A.7), we see that, when $R_p \ne 0$, the “main conjecture” does indeed predict Colmez’s residue formula up to a p-adic unit for all primes p, as claimed above.