2.1 Integrally Diophantine extensions

Let A be a commutative ring and $n>0$ be an integer. Let $A^n$ be a free A-module of rank n consisting of elements of the form $a=(a_1, \dots , a_n)$ , with entries $a_i\in A$ . For $m\geq 0$ , $a=(a_1,\dots , a_n)\in A^n$ , $b=(b_1, \dots , b_m)\in A^m$ , the element $(a,b)\in A^{n+m}$ is given by $(a_1, \dots , a_n, b_1, \dots , b_m)$ . Given a set of polynomials $F_1, \dots , F_k\in A[x_1,\dots , x_n, y_1, \dots , y_m]$ and $a\in A^n$ , set

$$ \begin{align*}\mathcal{F}(a;F_1, \dots, F_k):=\{b\in A^m\mid F_i(a,b)=0\text{ for }1\leq i\leq k\}.\end{align*} $$

Given a number field K, denote by $\mathcal {O}_K$ its ring of integers.

Definition 2.1. Let S be a subset of $A^n$ . The set S is Diophantine in $A^n$ if for some $m\geq 0$ , there are polynomials $F_1, \dots , F_k\in A[x_1,\dots , x_n, y_1, \dots , y_m]$ such that

$$ \begin{align*}S=\{a\in A^n\mid \mathcal{F}(a;F_1, \dots, F_k)\text{ is not empty}\}.\end{align*} $$

An extension of number fields $L/K$ is said to be integrally Diophantine if $\mathcal {O}_K$ is a Diophantine subset of $\mathcal {O}_L$ .

It follows from standard arguments that if $L/\mathbb {Q}$ is an integrally Diophantine extension of number fields, then Hilbert’s tenth problem has a negative answer for $\mathcal {O}_L$ (see [Reference Denef and Lipshitz3, page 385]). Moreover, if $L/F$ and $F/K$ are integrally Diophantine extensions of number fields, then so is $L/K$ .

We recall Shlapentokh’s criterion for an extension of number fields to be integrally Diophantine.

2.2 Iwasawa theory of elliptic curves

Fix an algebraic closure $\bar {\mathbb {Q}}$ of $\mathbb {Q}$ . Let p be an odd prime number and K be a number field. Throughout, $\mathbb {Z}_p$ is the ring of p-adic integers. Let E be an elliptic curve defined over K with good ordinary reduction at the primes above p. Denote by $\mu _{p^n}$ the group of $p^n$ th roots of unity in $\bar {\mathbb {Q}}$ , and set $\mu _{p^\infty } = \bigcup _n \mu _{p^n}$ . We denote by $K(\,\mu _{p^n})$ (respectively $K(\,\mu _{p^\infty })$ ) the cyclotomic extension of K in $\bar {\mathbb {Q}}$ generated by $\mu _{p^n}$ (respectively $\mu _{p^\infty }$ ). The cyclotomic $\mathbb {Z}_p$ -extension of K is the unique $\mathbb {Z}_p$ -extension of K which is contained in $K(\,\mu _{p^\infty })$ and is denoted by $K_{\operatorname {cyc}}$ . Given a number field extension $\mathcal {F}$ of K, let $\operatorname {Sel}_{p^\infty }(E/\mathcal {F})$ denote the p-primary Selmer group of E over $\mathcal {F}$ (see [Reference Coates and Sujatha1, page 9] for the definition).

We let denote the Tate–Shafarevich group of E over K and its p-primary part. The p-primary Selmer group fits into a short exact sequence:

(2.1)

Note that $\operatorname {Sel}_{p^\infty }(E/\mathcal {F})$ is a module over $\mathbb {Z}_p[G]$ , when $\mathcal {F}$ is a finite Galois extension of K with Galois group $G=\operatorname {Gal}(\mathcal {F}/K)$ . For $n\geq 0$ , let $K_n$ be the unique extension of K in $K_{\operatorname {cyc}}$ such that $\operatorname {Gal}(K_n/K)\simeq \mathbb {Z}/p^n\mathbb {Z}$ . The Selmer group of E over $K_{\operatorname {cyc}}$ is taken to be the natural direct limit with respect to restriction maps

$$ \begin{align*}\operatorname{Sel}_{p^\infty}(E/K_{\operatorname{cyc}}):=\varinjlim_n \operatorname{Sel}_{p^\infty}(E/K_{n}).\end{align*} $$

Setting $\Gamma :=\operatorname {Gal}(K_{\operatorname {cyc}}/K)$ , note that $\Gamma ^{p^n}\!=\operatorname {Gal}(K_{\operatorname {cyc}}/K_n)$ . The Iwasawa algebra is the completed group ring

$$ \begin{align*}\Lambda(\Gamma):=\varprojlim_n \mathbb{Z}_p[\Gamma/\Gamma^{p^n}].\end{align*} $$

Let denote the formal power series ring over $\mathbb {Z}_p$ in the variable T. Fix a topological generator $\gamma $ of $\Gamma $ and consider the isomorphism sending $\gamma -1$ to the formal variable T. The Selmer group $\operatorname {Sel}_{p^\infty }(E/K_{\operatorname {cyc}})$ is naturally a module over the Iwasawa algebra $\Lambda (\Gamma )$ .

Given a module M over $\Lambda (\Gamma )$ , let $M^{\vee } = \operatorname {Hom}_{\mathbb {Z}_p}(M, \mathbb {Q}_p/\mathbb {Z}_p)$ be its Pontryagin dual. It is conjectured by Mazur that the dual Selmer group $\operatorname {Sel}_{p^\infty }(E/K_{\operatorname {cyc}})^{\vee }$ is a finitely generated and torsion module over $\Lambda $ . This is known to be true in the following special cases:

1. (1) the p-primary Selmer group $\operatorname {Sel}_{p^\infty }(E/K)$ (over K) is finite (see [Reference Coates and Sujatha1, Theorem 2.8]);

2. (2) K is an abelian extension of $\mathbb {Q}$ . (This is a result of Kato [Reference Kato8] and [Reference Hachimori and Matsuno6, Theorem 2.2]. Rubin proved the result for CM elliptic curves.)

We say that a polynomial in

is distinguished if it is monic and all its nonleading coefficients are divisible by p. A map of

-modules $M_1\rightarrow M_2$ is said to be a pseudo-isomorphism if the kernel and cokernel have finite cardinality. One associates a characteristic element and Iwasawa invariants to a finitely generated and torsion module M over $\Lambda (\Gamma )$ as follows. By the structure theorem of finitely generated and torsion

-modules (see [Reference Washington22, Ch. 13]), there is a pseudo-isomorphism

where $f_1, \dots , f_t$ are nonzero elements of

. The characteristic element is the product $\prod _j f_j$ and is denoted by $f_M(T)$ . It is well defined up to multiplication by a unit in

. According to the Weierstrass preparation theorem, we may factor $f_M(T)$ as $p^{\mu }P(T)u(T)$ , where $\mu \geq 0$ , $P(T)$ is a distinguished polynomial and $u(T)$ is a unit in

. The Iwasawa $\mu $ invariant $\mu (M)$ is the quantity $\mu $ that appears in this factorisation. The $\lambda $ -invariant $\lambda (M)$ is the degree of the polynomial $P(T)$ .

Assume that Mazur’s conjecture is satisfied for the Selmer group of E over $K^{\operatorname {cyc}}$ , that is, $\operatorname {Sel}_{p^\infty }(E/K^{\operatorname {cyc}})^\vee $ is finitely generated and torsion as a $\Lambda (\Gamma )$ -module. Then, we denote by $\mu _p(E/K)$ and $\lambda _p(E/K)$ the associated $\mu $ and $\lambda $ -invariants for the dual Selmer group $\operatorname {Sel}_{p^\infty }(E/K^{\operatorname {cyc}})^{\vee }$ .

Proof. From (2.1), we arrive at

$$ \begin{align*}\operatorname{rank} E(K_n)\leq \operatorname{rank}_{\mathbb{Z}_p} (\operatorname{Sel}_{p^\infty}(E/K^{n})^{\vee}).\end{align*} $$

According to [Reference Greenberg and Viola5, Theorem 1.9],

$$ \begin{align*}\operatorname{rank}_{\mathbb{Z}_p} (\operatorname{Sel}_{p^\infty}(E/K^{n})^{\vee})\leq \lambda_p(E/K),\end{align*} $$

provided $\operatorname {Sel}_{p^\infty }(E/K^{\operatorname {cyc}})^{\vee }$ is finitely generated and torsion as a module over $\Lambda (\Gamma )$ . The result follows.

3.1 Rank constancy in cyclotomic towers

We shall study the following property in the context of the cyclotomic $\mathbb {Z}_p$ -extension of a number field K.

We specialise the above notion to $K_\infty =K_{\operatorname {cyc}}$ . Thus, K is integrally Diophantine in $K_{\operatorname {cyc}}$ precisely when $K_n/K$ is an integrally Diophantine extension for all $n\geq 0$ . Thus, if $K/\mathbb {Q}$ is integrally Diophantine and $K_{\operatorname {cyc}}/K$ is integrally Diophantine, then it will follow that $K_n/\mathbb {Q}$ is intergrally Diophantine for all n. In particular, this shall imply that Hilbert’s tenth problem has a negative solution for the ring of integers of all number fields $K_n$ in the infinite cyclotomic tower.

We shall use methods from the Iwasawa theory of elliptic curves to derive conditions for a number field K to be integrally Diophantine in $K^{\operatorname {cyc}}$ .

Theorem 3.3. Let p be an odd prime and E an elliptic curve over a number field K with good ordinary reduction at all primes $v \mid p$ . Assume that $\operatorname {Sel}_{p^\infty }(E/K_{\operatorname {cyc}})^\vee $ is finitely generated and torsion as a module over $\Lambda (\Gamma )$ as conjectured by Mazur. Suppose that

$$ \begin{align*}\lambda_p(E/K)=\operatorname{rank} E(K)>0.\end{align*} $$

Then, for all $n\geq 0$ , $K_n/K$ is a Diophantine extension. Therefore, if Conjecture 1.1 is satisfied for K, then it is satisfied for $K_n$ for all $n\geq 0$ .

Proof. Suppose $K_n/K$ is an integral Diophantine extension and $K/\mathbb {Q}$ is an integral Diophantine extension. Then, $K_n/\mathbb {Q}$ is an integral Diophantine extension and thus Hilbert’s tenth problem is negative for $\mathcal {O}_{K_n}$ . We show that the above hypotheses imply that $K_n/K$ is an integral Diophantine extension for all $n\geq 0$ .

According to Proposition 2.3, $\operatorname {rank} E(K_n)\leq \lambda _p(E/K)$ . Since it is assumed that $\operatorname {rank} E(K) = \lambda _p(E/K)>0$ , it follows that

$$ \begin{align*}\operatorname{rank} E(K_n)=\operatorname{rank} E(K)>0.\end{align*} $$

Therefore, by Theorem 2.2, $K_n/K$ is an integral Diophantine extension for all $n\geq 0$ .

3.2 An Euler characteristic formula

Let E be an elliptic curve over a number field K and let p be a prime number. Assume that E has good ordinary reduction at all primes of K that lie above p. Assume that the dual Selmer group $\operatorname {Sel}_{p^\infty }(E/K_{\operatorname {cyc}})^{\vee }$ is a finitely generated torsion $\Lambda (\Gamma )$ -module and denote by $f_{E,p}(T)$ its characteristic element. Note that $f_{E,p}(T)$ is well defined up to multiplication by a unit in $\Lambda (\Gamma )$ . We may express $f_{E,p}(T)$ as a formal power series in T:

$$ \begin{align*}f_{E,p}(T)=\sum_{i=r}^{\infty}a_i T^i,\end{align*} $$

where $a_r\neq 0$ . The quantity $r\geq 0$ is the order of vanishing of $f_{E,p}(T)$ at zero, thus we may denote it by $\operatorname {ord}_{T=0} f_{E,p}(T)$ . We call $a_r$ the leading coefficient. Perrin-Riou [Reference Perrin-Riou13] and Schneider [Reference Schneider16] provide an explicit formula for the leading coefficient $a_r$ . Note that $a_r$ is well defined up to a unit in $\mathbb {Z}_p$ . Given two elements $a,b\in \mathbb {Q}_p$ , we write $a\sim b$ to mean that $a=ub$ for a unit $u\in \mathbb {Z}_p^\times $ . The p-adic regulator $\mathcal {R}_p(E/K)$ is the determinant of the p-adic height pairing on the Mordell–Weil group of E (see [Reference Mazur, Tate and Teitelbaum11, Reference Schneider15, Reference Schneider16] for further details). Schneider has conjectured that the p-adic regulator $\mathcal {R}_p(E/K)$ is always nonzero, that is, the p-adic height pairing is always nondegenerate. Let $R_p(E/K)$ be the normalised p-adic regulator, defined by $R_p(E/K):=p^{-\operatorname {rank}E(K)}\mathcal {R}_p(E/K)$ . At each prime v of K, let $c_v(E/K)$ be the Tamagawa number of E at v (see [Reference Coates and Sujatha1, Ch. 3] for the definition). If v is a prime at which E has good reduction, then $c_v(E/K)=1$ . At any prime v of K, let $\mathbb {F}_v$ denote the residue field of $\mathcal {O}_v$ at v. For each prime $v \mid p$ , let $\widetilde {E}$ be the reduction of E to an elliptic curve over $\mathbb {F}_v$ .

The above result shows that given a number field K, if there exists an elliptic curve $E_{/K}$ satisfying the conditions of Theorem 3.6, then, $K_n/K$ is integrally Diophantine for all n. In the next section, we shall explain the conditions of Theorem 3.6.