Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

Abstract

Let $E_{/\mathbb {Q}}\!$ be an elliptic curve, $p>3$ a good ordinary prime for E, and $K_\infty$ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions $K_n/k$ inside $K_\infty$; we obtain lower bounds on the former, and upper bounds on the latter’s size.

Appendix: Skew power series rings and characteristic ideals

Let G be a d-dimensional torsion-free p-adic Lie group, given by $H\rtimes \Gamma$, where $\Gamma \cong \mathbb Z_p$ and H is a uniform pro-p group. Let $\Lambda (\star )$ be the Iwasawa algebra $\mathbb Z_p[[\star ]]$ for $\star \in \{G, H,\Gamma \}$. Suppose that $\Gamma =\overline{\langle \gamma \rangle }$ and $H=\overline{\langle \sigma _1,\ldots ,\sigma _{d-1}\rangle }$. An element of $\Lambda (G)$ if a (possibly infinite) sum of $(\sigma _1-1)^{n_1}\cdots (\sigma _{d-1}-1)^{n_{d-1}}\cdot (\gamma -1)^n$ for some non-negative integers $n_1,\ldots ,n_{d-1},n$. We may identify $\Lambda (G)$ with the skew power series ring $R\llbracket X;\sigma ,\delta \rrbracket$, where $R=\Lambda (H)$, X is an indeterminant, which can be identified with $\gamma -1$, $\sigma :R\rightarrow R$ is a ring homomorphism and $\delta :R\rightarrow R$ is a $\sigma$-derivation. If $r\in R$,

$$\begin{aligned} Xr=\sigma (r)X+\delta (r). \end{aligned}$$

More generally, for $n\ge 1$, we have

$$\begin{aligned} X^nr=\sum _{i=0}^n(X^nr)_iX^i, \end{aligned}$$

(5.1)

where $(X^nr)_i$ are elements of R.

For $f=\sum _{i=0}^\infty r_iX^i$, we say that f has finite reduced order if $r_i\in \Lambda (H)^\times$ for some $i\ge 0$. Recall the Weierstrass Preparation Theorem of Venjakob in [30] states that such f admits the factorization

$$\begin{aligned} f=u\times \tilde{f}, \end{aligned}$$

where $u\in \Lambda (G)^\times$ and $\tilde{f}$ is a polynomial over $\Lambda (H)$ in X.

Lemma 5.1

Let $M=\Lambda (G)/I$ be a $\Lambda (G)$-module that is finitely generated over $\Lambda (H)$. Then, there exists $f\in I$ that has finite reduced order.

Proof

Let $\mathfrak {m}$ be the maximal ideal of $\Lambda (H)$. For each element $f\in I$, we may write $f=\sum _{i=0}^\infty r_iX^i$. Suppose that f does not have finite reduced order. Then, $r_i\in \mathfrak {m}$ for all i.

Note that we have the following isomorphism of $\Lambda (H)$-modules:

$$\begin{aligned} s_H:\Lambda (G)\rightarrow&\Lambda (H)^{\mathbb {N}}\\ \sum _{i=0}^\infty a_iX^i\mapsto&(a_i)_{i=0,1,\ldots }. \end{aligned}$$

The image of $\Lambda (G)f$ at the i-th component is

$$\begin{aligned} \sum _{k+\ell =i,k\le j}\Lambda (H)(X^jr_\ell )_k, \end{aligned}$$

as given by (5.1). Since we assume that $r_\ell \in \mathfrak {m}$ for all $\ell$, the $\Lambda (H)$-module above is contained in $\mathfrak {m}$ by [30, Lemma 2.1]. If this is the case for all $f\in I$, the image of M under $s_H$ in $\Lambda (H)^\mathbb {N}$ is non-trivial at all components, which contradicts the fact that M is finitely generated over $\Lambda (H)$. Therefore, we conclude that there must exist $f\in I$ with finite reduced order. $\square$

Lemma 5.2

Let $f=\sum _{i=0}^nr_iX^i$ be a polynomial in $\Lambda (G)$. There exists $r_0',\ldots ,r_n'\in \Lambda (H)$ such that

$$\begin{aligned} f=\sum _{i=0}^nX^ir_i'. \end{aligned}$$

Proof

Note that for all $m\ge 0$ and $h\in H$, there exists $h'\in H$ such that $\gamma ^mh=h'\gamma ^m$ by the fact that H is normal in G. The result follows by identifying X with $\gamma -1$. $\square$

For all $n\ge 1$, we write $H_n=H^{p^n}$, which is a subgroup of H and $H/H^{p^n}$ is a p-group of order $p^{(d-1)n}$ by the uniformality of H.

Proposition 5.3

Let $M=\Lambda (G)/I$ be a $\Lambda (G)$-module that is finitely generated over $\Lambda (H)$ such that I contains a polynomial f in X of degree $\tau$. Then, $M_{H_n}$ is a finitely generated $\Lambda (\Gamma )$-torsion module. Furthermore, its characteristic power series factorizes into polynomial of degree $\le \tau$.

Proof

Note that $\Lambda (G)_{H_n}$ can be identified with $\Lambda (H/H_n\rtimes \Gamma )=\mathbb Z_p[H/H_n]\llbracket X;\sigma ,\delta \rrbracket$. We fix a set of coset representatives of $H/H_n$, say $\{h_i:i=1,\ldots , p^{(d-1)n}\}$. Each element of $\Lambda (G)_{H_n}$ can be written as $\sum _{i}f_ih_i$, where $f_i$ are some elements of $\Lambda (\Gamma )$. As $\Lambda (\Gamma )$-modules, we have the isomorphism

$$\begin{aligned} s_\Gamma :\Lambda (G)_{H_n}\rightarrow&\Lambda (\Gamma )^{\oplus p^{dn}}\\ \sum _{i}f_ih_i\mapsto&(f_i)_{i=1,\ldots ,p^{(d-1)n}}. \end{aligned}$$

We know that $M_{H_n}$ is a quotient of $\Lambda (G)_{H_n}/I_{H_n}$, so it suffices to prove our result for the latter. Since, in particular, $s_\Gamma (\Lambda (G)_{H_n})/s_\Gamma (I_{H_n})$ is a quotient of $\Lambda (\Gamma )^{\oplus p^{dn}}$, it is finitely generated over $\Lambda (\Gamma )$.

By Lemma 5.2, the image of f under $s_\Gamma$ is a polynomial of degree $\tau$ at each component. Therefore, $s_\Gamma (\Lambda (G)_{H_n})/s_\Gamma (I_{H_n})$ may be decomposed as a direct sum of $\Lambda (\Gamma )$-modules where each summand is killed by a polynomial of degree d. $\square$

Corollary 5.4

Let $M=\Lambda (G)/I$ be a $\Lambda (G)$-module that is finitely generated over $\Lambda (H)$. Then, $M_{H_n}$ is a finitely generated $\Lambda (\Gamma )$-torsion module for all $n\ge 1$. Furthermore, there exists an integer $\tau$, independent of n, such that the characteristic power series of $M_{H_n}$ factorizes into polynomial of degree $\le \tau$.

Proof

By Lemma 5.1, I contains an element of f that is of finite reduced order. Venjakob’s Weierstrass Preparation Theorem tells us that we may replace f by a polynomial in X, say of degree $\tau$. The result now follows from Proposition 5.3. $\square$