Bvx_{R})\cap\operatorname{Stab}_{L}(B\backslash Bvx_{S})=\ker(\psi)$, so
$\phi^{\prime}(t)=\phi(t)$

We are now left to compute the value of $\phi^{\prime}(t)$ where $t$ has $-1$
in position $i$ or $i+1$. This values must be concordant given that if $t_{i}$
is the diagonal matrix with $-1$ in position $i$ and $1$ everywhere else, then
$t_{i}t_{i+1}\in\operatorname{Stab}_{T}(x_{S}L/L)\cap\operatorname{Stab}_{T}(x_{R}L/L)$
and $\phi(t_{i}t_{i+1})=1$ because it is in the connected component of the
identity. On the other hand we know that there are two possible definition of
$\phi^{\prime}$, so one of them must be
$\phi^{\prime}(t_{i})=\phi^{\prime}(t_{i+1})=1$ and the other
$\phi^{\prime}(t_{i})=\phi^{\prime}(t_{i+1})=-1$. ∎

We also have the following results for other type of ascents.

######  Lemma 7.3.

Let $(v,S)$ be an admissible pair with $\\#S_{l}=r$ and let
$X=(\underbrace{1,\ldots,-1,\ldots,1}_{r})$ be the sequence associated to a
local system on $Bvx_{S}$.

Let $\alpha\in\Delta$ be an ascent for $(v,S)$. Then

  1. 1.

if $\alpha$ is of type $b1)$ then the local system extends uniquely to the
open orbit of $P_{\alpha}vx_{S}$ with sequence equal to $X$;

  2. 2.

if $\alpha$ is of type $d1)$ then the local system extends uniquely to the
open orbit of $P_{\alpha}vx_{S}$ with sequence
$(\underbrace{1,\ldots,-1,\ldots,1}_{X},\underbrace{1}_{r+1})$.

While we don’t have a complete characterization of the order in $\mathcal{D}$
as we have for the simply laced case and for type $\bf{B}$, there is something
we can say about the connected components of the Hasse diagram. This in turn
will give a necessary condition for the order in $\mathcal{D}$. To show this,
we need to study the sequences that determine possible local systems.

######  Definition 7.4.

Let $X=(a_{1},\ldots,a_{r})$ a sequence where $a_{i}=\pm 1$. Then we define
the plus number of $X$ as

$\operatorname{PL}(X)=\sum_{a_{i}=1}(-1)^{i}$

and the minus number of $X$ as

$\operatorname{MIN}(X)=\sum_{a_{i}=-1}(-1)^{i}$

The most important property of the minus and plus numbers is the following:

######  Proposition 7.5.

Let $X=\left(a_{1},\ldots,a_{r}\right)$ as above and fix
$j\in\left\\{1,\ldots,r\right\\}$ such that $a_{j}=a_{j+1}$. Denote with $Y$
the sequence
$\left(a_{1},\ldots,\widehat{a_{j}},\widehat{a_{j+1}},\ldots,a_{r}\right)$.
Then $\operatorname{PL}(Y)=\operatorname{PL}(X)$ and
$\operatorname{MIN}(Y)=\operatorname{MIN}(X)$.

###### Proof.

To avoid confusion we will write $Y=\left(b_{1},\ldots,b_{r-2}\right)$ with
$b_{i}=a_{i}$ for every $i<j$ and $b_{i}=a_{i+2}$ for every $i\geq j$.

Suppose $a_{j}=1$. We have

$\displaystyle\operatorname{PL}(X)=\sum_{a_{i}=1}(-1)^{i}$
$\displaystyle=\sum_{a_{i}=1,i<j}(-1)^{i}+\sum_{a_{i}=1,i>j+1}(-1)^{i}+(-1)^{j}+(-1)^{j+1}$
$\displaystyle=\sum_{a_{i}=1,i<j}(-1)^{i}+\sum_{a_{i}=1,i>j+1}(-1)^{i}$
$\displaystyle=\sum_{b_{i}=1,i<j}(-1)^{i}+\sum_{b_{i}=1,i\geq
j}(-1)^{i}=\operatorname{PL}(Y)$

where the last equality follows because for $(-1)^{i}=(-1)^{i+2}$. On the
other hand

$\operatorname{MIN}(X)=\sum_{a_{i}=-1}(-1)^{i}=\sum_{a_{i}=-1,i<j}(-1)^{i}+\sum_{a_{i}=-1,i>j+1}(-1)^{i}=\operatorname{MIN}(Y)$

as above. The case $a_{j}=-1$ is symmetric. ∎

We showed that the plus and minus numbers don’t change if we delete two
identical adjacent numbers.

Given a sequence, we can delete all pairs of identical adjacent numbers until
it is no longer possible to do so. Doing so, by Lemma 7.5, the plus and minus
numbers don’t change. The final sequence must be one of alternating signs and
it is easy to see that the sequences of alternating signs are uniquely
identified by their plus and minus numbers. Hence, the final sequence doesn’t
depend on the order in which we delete the pairs of identical adjacent
numbers. There is another thing we can do; if the rightmost element of the
final sequence is $1$ we delete it.

######  Definition 7.6.

Let $X=\left(a_{1},\ldots,a_{r}\right)$ be a sequence with $a_{i}=\pm 1$.
Denote with $\operatorname{r}(X)$ the sequence obtained by $X$ deleting
inductively all pairs of identical adjacent number and then, if present, the
rightmost $1$.

The sequence $\operatorname{r}(X)$ is called the reduced form of $X$.

We can finally state our last result.

######  Theorem 7.7.

Let $X$ and $Y$ be the sequences associated respectively to a local system on
$(v,S)$ and $(u,R)$. Then the corresponding elements in $\mathcal{D}$ are in
the same connected component of the Hasse diagram if and only if
$\operatorname{r}(X)=\operatorname{r}(Y)$.

###### Proof.

We will start by showing that if $\left((v,S),X\right)$ and
$\left((u,R),Y\right)$ are in the same connected component, then
$\operatorname{r}(X)=\operatorname{r}(Y)$. To see this, consider the following
order

$\left((v,S),X\right)\prec\left((u,R),Y\right)\Leftrightarrow(v,S)<(u,R)\text{
and }\operatorname{r}(X)=\operatorname{r}(Y)$

it verifies the conditions of definition 4.4 because we saw above that
extending with an ascent doesn’t change the reduced form. This means that
$\left((v,S),X\right)\leq\left((u,R),Y\right)$ implies
$\operatorname{r}(X)=\operatorname{r}(Y)$ and the claim follows.

We will now show that if $\operatorname{r}(X)=\operatorname{r}(Y)$ then
$\left((v,S),X\right)$ and $\left((u,R),Y\right)$ are in the same connected
component. Recall that every orbit that is not the open orbit in $G/L$ admits
an ascent. This implies that for every orbit $\mathcal{O}$ there is a sequence
of ascents between $\mathcal{O}$ and the open orbit. If we have an isomorphism
class of local systems on $\mathcal{O}$ we can get a sequence of ascents in
$\mathcal{D}$ by choosing one of the possible extensions in every ascent. We
know that doing this doesn’t change the reduced form, so we can suppose
without loss of generality that $Bvx_{S}=Bux_{R}$ are the open orbit of $G/L$
. More precisely, $v=\omega_{P}$ the longest element in $W^{P}$ and
$R=S=\left\\{2e_{1},\ldots,2e_{n}\right\\}$.

Now it is easy to see that $v(e_{i})=-e_{n-i}$ for every $i=1,\ldots,n$.
Hence, if we fix $j\in\left\\{1,\ldots,n-1\right\\}$ we have
$v(e_{j}-e_{j+1})=e_{n-j-1}-e_{n-j}\in\Delta$. It follows that for every $j$
there is a descent $\alpha\in\Delta$ such that
$(v,T)\in\mathcal{E}_{\alpha}(v,S)$ with $T=\left\\{2e_{i}\right\\}_{i\neq
j,j+1}\cup\left\\{e_{j}+e_{j+1}\right\\}$. Moreover, for $(v,T)$ the root
$\alpha$ is an ascent of type $c1)$. To see what all this means, take
$X=\left(a_{1},\ldots,a_{n}\right)$ and $j\in\left\\{1,\ldots,n-1\right\\}$
such that $a_{j}=a_{j+1}$. Denote with
$X^{\prime}=\left(b_{1},\ldots,b_{n}\right)$ the sequence with $b_{i}=a_{i}$
for every $i\neq j,j+1$ and $b_{j}=-a_{j}$, $b_{j+1}=-a_{j+1}$ and with
$X_{0}=\left(c_{1},\ldots,c_{n-2}\right)$ the sequence with $c_{i}=a_{i}$ for
every $i<j$ and $c_{i}=a_{i+2}$ for $i\geq j$. If we take the simple root
$\alpha$ as above we have

$\left((v,S),X\right)$$\left((v,S),X^{\prime}\right)$$\left((v,T),X_{0}\right)$$\alpha$$\alpha$

It follows that $(Bvx_{S},X)$ and $(Bvx_{S},X^{\prime})$ are in the same
connected component. This can clearly be done every time we have a pair of
identical adjacent element in $X$.

We now want to show an algorithm that takes a sequence $X$ and gives another
sequence that depends only on $\operatorname{r}(X)$. Every step of the
algorithm will be inverting the sign of two adjacent identical element, hence
the final sequence will be in the same connected component of $X$ and thus the
claim will be proved.

The algorithm is the following. If there is $k\in\left\\{1,\ldots,n\right\\}$
such that $a_{i}=1$ for every $i\leq k$ while $a_{i}=-a_{i+1}$ for every
$i\geq k$ then we are done. If not, let $k$ be the smallest number such that
either $a_{k-1}\neq a_{k}=a_{k+1}$ or $a_{k}=a_{k+1}=-1$. Change the sign of
$a_{k}$ and $a_{k+1}$ and repeat the algorithm with the new sequence.

Note that there is always a $k$ with the property above unless we are in the
first case. Moreover, the algorithm ends. To see this denote with $s_{0}$ the
length of the initial sequence of $1$ in $X$ and with $s_{i}$ the same length
after $i$ steps of the algorithm. Moreover, denote with $k_{i}$ the element
$k$ defined above for the $i$-th step. We want to show that after every step
either $k_{i}$ decreases or $s_{i}$ increases. Given that they can’t decrease
infinitely or increase infinitely we would have the claim. So, fix a sequence
$X$ and the relative $s=s_{0}$ and $k=k_{0}$. If $k$ is such that $a_{k-1}\neq
a_{k}=a_{k+1}$ we either have $s=k-1$ and $a_{k-1}=1$ or $s<k-1$. In the first
case it is clear that $s_{1}\geq k+1>s$ while in the second case $s_{1}=s$ and
$k_{1}=k-1<k$. Suppose now that $k$ is such that $a_{k}=a_{k+1}=-1$. Then it
must be $s=0$ and $k=1$ or we would be in the previous case, which means that
$s_{1}\geq 2>s$.

The last thing to note is that the algorithm doesn’t change the reduced form
of the sequence because it changes only adjacent pair of identical values and
the output of the algorithm clearly depends only on the reduced form $X$. This
proves the claim.

∎

## References

  * [1] R.W. Richardson and T.A. Springer.  Combinatorics and geometry of K-orbits on the flag manifold.  1993\.
  * [2] Jacopo Gandini and Andrea Maffei.  The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals.  2018\.
  * [3] David A. Vogan Jr.  Irreducible Characters of Semisimple Lie Groups III. Proof of Kazhdan-Lusztig Conjecture in the Integral Case.  Inventiones Mathematicae, 71:381 – 417, 1983.
  * [4] George Lusztig and David A. Vogan Jr.  Singularities of Closures of K-orbits on Flag Manifolds.  Inventiones Mathematicae, 71:365 – 379, 1983.
  * [5] R.W. Richardson and T.A. Springer.  The Bruhat order on symmetric varieties.  1990\.
  * [6] R.W. Richardson, Gerhard Rohrle, and Robert Steinberg.  Parabolic subgroups with abelian unipotent radical.  1992\.


# Hilbert’s tenth problem in Anticyclotomic towers of number fields

Anwesh Ray  Centre de recherches mathématiques, Université de Montréal,
Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal (Québec) H3T 1J4,
Canada<EMAIL_ADDRESS>and  Tom Weston  Department of Mathematics,
University of Massachusetts, Amherst, MA, USA<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits
in $K$. Let $E_{1}$ and $E_{2}$ be elliptic curves over $K$ such that the
$\operatorname{Gal}(\bar{K}/K)$-modules $E_{1}[p]$ and $E_{2}[p]$ are
isomorphic. We show that under certain explicit additional conditions on
$E_{1}$ and $E_{2}$, the anticyclotomic $\mathbb{Z}_{p}$-extension
$K_{\operatorname{anti}}$ of $K$ is integrally diophantine over $K$. When such
conditions are satisfied, we deduce new cases of Hilbert’s tenth problem. In
greater detail, the conditions imply that Hilbert’s tenth problem is
unsolvable for all number fields that are contained in
$K_{\operatorname{anti}}$. We illustrate our results by constructing an
explicit example for $p=3$ and $K=\mathbb{Q}(\sqrt{-5})$.

###### Key words and phrases:

Anticyclotomic Iwasawa theory, Selmer groups, Hilbert’s tenth problem,
congruences between elliptic curves

###### 2020 Mathematics Subject Classification:

11R23, 11U05

##  1\. Introduction

Hilbert asked whether there exists a deterministic algorithm for solving
diophantine equations with rational integer coefficients. In other words, the
problem asks whether there exists a Turing machine which takes as input a
finite set of polynomial equations over $\mathbb{Z}$ and decides whether they
have nontrivial solutions in $\mathbb{Z}$. If no such Turing machine does
exist, then it is said that Hilbert’s tenth problem is unsolvable. For further
details, we refer to [Shl07]. A celebrated result due to Matiyasevich [Mat70],
building upon previous work of Davis, Putnam and Robinson [DPR61], shows that
Hilbert’s tenth problem for diophantine equations with integer coefficients is
unsolvable. One of the main open problems in the area is to study the
generalization of Hilbert’s tenth problem to number rings. For a number field
$L$, generally this is studied by showing that $\mathbb{Z}$ is a _integrally
diophantine_ subset of $\mathcal{O}_{L}$, in the sense of Definition 2.2. It
is then known that $\mathcal{O}_{L}$ inherits the unsolvability of Hilbert’s
tenth problem from $\mathbb{Z}$. The following conjecture this predicts that
Hilbert’s tenth problem is unsolvable for all rings of integers of number
fields.

######  Conjecture 1.1 (Denef-Lipshitz).

For any number field $L$, $L/\mathbb{Q}$ is an integrally diophantine
extension.

Classical results in the field show that Conjecture 1.1 above is true for the
following classes of number fields $L$:

  * •

The conjecture holds when $L$ is either totally real or a quadratic extension
of a totally real number field (cf. [Den80, DL78]),

  * •

$L$ has exactly one complex place (cf. [Phe88, Shl89, Vid]),

  * •

$L/\mathbb{Q}$ is abelian (cf. [SS89]),

  * •

$[L\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=4$, $L$ is not totally
real and $L/\mathbb{Q}$ has a proper intermediate field (cf. [DL78]).

Poonen [Poo02], Cornelissen-Pheidas-Zahidi [CPZ05] and Shlapentokh [Shl08]
subsequently established a geometric criterion for Conjecture 1.1 to hold for
a given number field $L/\mathbb{Q}$. In establishing some new cases of
Conjecture 1.1, we use the following criterion of Shlapentokh.

######  Theorem 1.2 (Shlapentokh).

Let $L/K$ be an extension of number fields and suppose that there exists an
elliptic curve $E_{/K}$ such that
$\operatorname{rank}E(L)=\operatorname{rank}E(K)>0$. Then, $L/K$ is integrally
diophantine.

Let $K$ be a number field for which Conjecture 1.1 holds and let $L/K$ be a
number field extension. Suppose that there exists an elliptic curve such that
$\operatorname{rank}E(L)=\operatorname{rank}E(K)>0$. Then, it follows that
Conjecture 1.1 holds for $L$. Theorem 1.2 has been applied in various
contexts. For instance, Garcia-Fritz and Pasten [GFP20] established that for
certain families of primes $\mathscr{P}$ and $\mathscr{Q}$ with positive
density, Conjecture 1.1 is satisfied for $L=\mathbb{Q}(p^{1/3},\sqrt{-q})$
with $p\in\mathscr{P}$ and $q\in\mathscr{Q}$. Mazur and Rubin [MRL18] show
that if the Conjecture 1.1 holds for a given number field $K$, then it also
holds for certain infinite families of degree $p^{n}$ cyclic extensions $L/K$,
where $p$ is a prime and $n\in\mathbb{Z}_{\geq 1}$.

###  1.1. Main results

In what follows, $p$ is a fixed odd prime number and $K=\mathbb{Q}(\sqrt{-D})$
is an imaginary quadratic field in which $p$ splits into two primes
$\mathfrak{p}$ and $\mathfrak{p}^{*}$. Let $\mathbb{Z}_{p}$ denote the ring of
$p$-adic integers. A $\mathbb{Z}_{p}$-extension of $K$ is an infinite Galois
extension $K_{\infty}$ of $K$ such that $\operatorname{Gal}(K_{\infty}/K)$ is
isomorphic to $\mathbb{Z}_{p}$ as a topological group. The _anticyclotomic_
-extension $K_{\operatorname{anti}}/K$ is the unique
$\mathbb{Z}_{p}$-extension of $K$ which is Galois over $\mathbb{Q}$ such that
$\operatorname{Gal}(K_{\operatorname{anti}}/\mathbb{Q})$ is a non-abelian pro-
dihedral group. Let $K_{n}$ be the $n$-th layer, i.e., the number field such
that $K\subseteq K_{n}\subseteq K_{\operatorname{anti}}$ and
$[K\mathrel{\mathop{\mathchar 58\relax}}K_{n}]=p^{n}$. The group
$\operatorname{Gal}(K_{n}/\mathbb{Q})$ is isomorphic to the dihedral group
with $2p^{n}$ elements. The tower of fields

$K=K_{0}\subseteq K_{1}\subseteq\dots\subseteq K_{n}\subseteq
K_{n+1}\subseteq\dots$

is referred to as the _anticyclotomic $\mathbb{Z}_{p}$-tower_ over $K$. The
fields $K_{n}$ are totally imaginary non-abelian extensions of $\mathbb{Q}$,
containing $p^{n}$ complex places. Conjecture 1.1 is not known to hold for the
fields $K_{n}$, when $n>0$. We do however note that $K/\mathbb{Q}$ is known to
be integrally diophantine (by the results above). If there exists an elliptic
curve $E_{/K}$ such that
$\operatorname{rank}E(K_{n})=\operatorname{rank}E(K)>0$, then, it follows from
Slapentokh’s criterion (i.e., Theorem 1.2) that $K_{n}/K$ is integrally
diophantine. Since $K/\mathbb{Q}$ is integrally diophantine, it shall then
follow that Conjecture 1.1 is satisfied for $K_{n}/\mathbb{Q}$. We say that
$K_{\operatorname{anti}}/K$ is integrally diophantine if $K_{n}/K$ is
integrally diophantine for all $n$. We note that if
$K_{\operatorname{anti}}/K$ is integrally diophantine, it follows that
Conjecture 1.1 holds for $K_{n}$ for all $n$.

In order to study the stability of ranks of elliptic curves in anticyclotomic
towers, we use techniques from the Iwasawa theory of Selmer groups. In order
to state our main results, we introduce some further notation. Let $E_{1}$ and
$E_{2}$ be elliptic curves defined over $K$. Fix an algebraic closure
$\bar{K}$ of $K$, and set $E_{i}[p]$ to denote the $p$-torsion subgroup of
$E_{i}(\bar{K})$. We assume that $E_{1}[p]$ and $E_{2}[p]$ are isomorphic as
modules over $\operatorname{Gal}(\bar{K}/K)$. Such a pair of elliptic curves
$(E_{1},E_{2})$ shall be referred to as a _$p$ -congruent_ pair of elliptic
curves over $K$. Let $\mathfrak{T}$ be the set of primes $v$ of $K$ such that
either $E_{1}$ or $E_{2}$ (or both) have bad reduction at $v$. Let
$\mu_{p}\subset\bar{K}$ denote the $p$-th roots of unity and set
$K^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=K(\mu_{p})$, i.e.,
$K^{\prime}$ is the number field generated by $K$ and $\mu_{p}$. At a finite
prime $v$ of $K$, denote by $\mathbb{F}_{v}$ the residue field of the ring of
integers at $v$. If $E_{/K}$ is an elliptic curve with good reduction at $v$,
we set $E(\mathbb{F}_{v})$ to be denote the group of $\mathbb{F}_{v}$-points
of a regular model of $E$. For $i=1,2$, we define a subset $\Sigma(E_{i})$ of
$\mathfrak{T}$ below, which will play a role in the statement of our results.

######  Definition 1.3.

For $i=1,2$, the set $\Sigma(E_{i})$ consists of primes $v\in\mathfrak{T}$
such that the following conditions is satisfied.

  1. (1)

If the elliptic curve $E_{i}$ has good reduction at $v$, then,
$p\mid\\#E_{i}(\mathbb{F}_{v})$.

  2. (2)

If the elliptic curve $E_{i}$ has bad reduction at $v$, and $p\geq 5$, then
$E_{i}$ has split multiplicative reduction at some prime $w|v$ of
$K^{\prime}$.

  3. (3)

If the elliptic curve $E_{i}$ has bad reduction at $v$, and $p=3$, then, at
some prime $w|v$ of $K^{\prime}$, the elliptic curve $E_{i}$ has either split
multiplicative reduction or has additive reduction of type $\rm{IV}$ or
$\rm{IV}^{*}$.

When referring to a prime $v$ of $K$, it shall be understood, unless stated
otherwise, that one is referring to a finite prime. Let $v$ be a prime of $K$
and let $\ell$ be the prime number for which $v|\ell$. The prime $v$ is
finitely decomposed in $K_{\operatorname{anti}}$ if and only if $\ell$ splits
in $K$ into two distinct conjugate primes $v$ and $v^{*}$. If $v$ is finitely
decomposed in $K_{\operatorname{anti}}$ (i.e., $\ell$ splits in $K$), then we
set $s_{v}$ to denote the number of primes of $K_{\operatorname{anti}}$ that
lie above $v$. Clearly $s_{v}=s_{v^{*}}$. We note that [Bri07, Theorem 2, p.
2133] gives an explicit algorithm to compute the number $s_{v}$. For an
elliptic curve $E_{/K}$, set $V_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{T}_{p}(E)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$,
where $T_{p}(E)$ is the $p$-adic Tate module associated to $E$. Let
$\sigma_{v}$ denote the Frobenius in
$\operatorname{Gal}(K_{v}^{\operatorname{nr}}/K_{v})$ and set
$\operatorname{I}_{v}$ to denote the inertia group at $v$. Let
$V_{p}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=(V_{p})_{\operatorname{I}_{v}}$ be the maximal quotient of $V_{p}$
on which $\operatorname{I}_{v}$ acts trivially. Let
$P_{v}(X)=P_{v}^{E}(X)\in\mathbb{Z}_{p}[X]$ denote the characteristic
polynomial $\operatorname{det}\left(1-\sigma_{v|V_{p}^{\prime}}X\right)$. When
$E$ has good reduction at $v$, then, $P_{v}(X)$ is given by
$P_{v}(X)=1-a_{v}(E)X+q_{v}X^{2}$, where, $q_{v}$ is the number of elements in
$\mathbb{F}_{v}$, and $a_{v}(E)\mathrel{\mathop{\mathchar
58\relax}}=q_{v}+1-\\#E(\mathbb{F}_{v})$. When $E$ has multiplicative
reduction at $v$, then, $P_{v}(X)=1-a_{v}(E)X$, where $a_{v}(E)=+1$ (resp.
$-1$) if $E$ has split (resp. non-split) multiplicative reduction. When $E$
has additive reduction at $v$, $P_{v}(X)=1$. Let
$\widetilde{P}_{v}(X)=\widetilde{P}_{v}^{E}(X)\in\mathbb{Z}/p\mathbb{Z}[X]$ be
the mod-$p$ reduction of $P_{v}(X)$. Let $d_{v}(E)$ be the multiplicity of
$\ell^{-1}$ as a root of $\widetilde{P}_{v}(X)$. Note that if $E$ is defined
over $\mathbb{Q}$, then, $P_{v}(X)=P_{v^{*}}(X)$ and $d_{v}(E)=d_{v^{*}}(E)$.
Denote by $E_{0}(K_{v})$ the subgroup of points in $E(K_{v})$ of good
reduction. The _Tamagawa number_ $c_{v}(E)$ is the index
$[E(K_{v})\mathrel{\mathop{\mathchar 58\relax}}E_{0}(K_{v})]$. Let
$|\cdot|_{p}$ be the nonarchimedian absolute value normalized so that
$|p|_{p}=p^{-1}$. The $p$-primary part of $c_{v}(E)$ is defined as
$c_{v}^{(p)}(E)\mathrel{\mathop{\mathchar 58\relax}}=|c_{v}(E)|_{p}^{-1}$. Set
$\Omega_{K}$ to be the set of primes $v\nmid p$ of $K$ which are finitely
decomposed in $K_{\operatorname{anti}}$. Set $\Sh(E/K)[p^{\infty}]$ to denote
the $p$-primary part of the Tate-Shafarevich group of $E$.

######  Theorem 1.4.

Let $K$ be an imaginary quadratic field and let $p$ be an odd prime number
which splits into primes $\mathfrak{p}$ and $\mathfrak{p}^{*}$ in $K$. Let
$K_{\operatorname{anti}}$ be the anticyclotomic $\mathbb{Z}_{p}$-extension of
$K$, and for $n\in\mathbb{Z}_{\geq 0}$ denote by $K_{n}$, its $n$-th layer.
With respect to notation above, assume that there exist elliptic curves
$E_{1}$ and $E_{2}$ defined over $K$ such that

  1. (1)

both $E_{1}$ and $E_{2}$ have good ordinary reduction at $\mathfrak{p}$ and
$\mathfrak{p}^{*}$,

  2. (2)

$E_{1}[p]$ and $E_{2}[p]$ are isomorphic as modules over
$\operatorname{Gal}(\bar{K}/K)$,

  3. (3)

$\operatorname{rank}E_{1}(K)=0$ and $\operatorname{rank}E_{2}(K)>0$,

  4. (4)

$E_{1}(K)[p]=E_{2}(K)[p]=0$,

  5. (5)

each prime $v\in\Sigma(E_{1})\cup\Sigma(E_{2})$ is finitely decomposed in the
anticyclotomic extension $K_{\operatorname{anti}}/K$. Equivalently, let
$v\in\Sigma(E_{1})\cup\Sigma(E_{2})$ and $\ell$ be the prime number for which
$v|\ell$. Then, $\ell$ splits in $K$ into a product of two primes $v$ and
$v^{*}$.

  6. (6)

$\Sh(E_{1}/K)[p^{\infty}]=0$,

  7. (7)

$E_{1}(\mathbb{F}_{\mathfrak{p}})[p]=0$.

  8. (8)

Recall that $\Omega_{K}$ is the set of primes $v\nmid p$ of $K$ which are
finitely decomposed in $K_{\operatorname{anti}}$. Assume that
$c_{v}^{(p)}(E_{1})=1$ for all $v\in\Omega_{K}$ (at which $E_{1}$ has bad
reduction).

Furthermore, assume that the following relationship holds

(1.1)
$\operatorname{rank}E_{2}(K)=\sum_{v\in\Sigma(E_{1})\cup\Sigma(E_{2})}s_{v}\left(d_{v}(E_{1})-d_{v}(E_{2})\right).$

Then, for all $n\in\mathbb{Z}_{\geq 0}$, $\mathcal{O}_{K_{n}}$ satisfies
Conjecture 1.1, and thus Hilbert’s tenth problem has a negative answer for
$\mathcal{O}_{K_{n}}$.

Note that when the elliptic curves $E_{1}$ and $E_{2}$ are defined over
$\mathbb{Q}$, the condition (5) above implies that the sum
$\sum_{v\in\Sigma(E_{1})\cup\Sigma(E_{2})}s_{v}\left(d_{v}(E_{1})-d_{v}(E_{2})\right)$
is even, and therefore, the rank of $E_{2}(K)$ is even. The condition (6) in
the theorem above implies in particular that $\Sh(E_{1}/K)[p^{\infty}]$ is
finite. However, we do not need to assume that $\Sh(E_{2}/K)[p^{\infty}]$ is
finite. We illustrate the above theorem via an explicit example for which all
the conditions are satisfied. In what follows, let $K=\mathbb{Q}(\sqrt{-5})$,
$p=3$ and consider the pair of rational elliptic curves $E_{1}=$ 56b1 and
$E_{2}=$ 392c1 with respect to Cremona’s labelling for elliptic curves. Note
that both elliptic curves are defined over $\mathbb{Q}$. The residual
representations

$\bar{\rho}_{i}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/3\mathbb{Z})$

are surjective and $\bar{\rho}_{1}$ is isomorphic to $\bar{\rho}_{2}$. The
ranks of the Mordell Weil groups $E_{1}(\mathbb{Q})$ and $E_{2}(\mathbb{Q})$
are $0$ and $1$ respectively. One finds that $\operatorname{rank}E_{1}(K)$
remains $0$, while for $E_{2}$, there is a rank jump and
$\operatorname{rank}E_{2}(K)=2$. Computations were done on the SageMathCloud
and the Magma computational algebra system to show that the conditions of
Theorem 1.4 for $(K,p,E_{1},E_{2})$ are satisfied. We obtain the following
result, illustrating Theorem 1.4.

######  Theorem 1.5.

Let $K=\mathbb{Q}(\sqrt{-5})$ and for $n\geq 0$, let $K_{n}$ be the $n$-th
layer in the anticyclotomic $\mathbb{Z}_{3}$-extension of $K$. The curve
$E_{1}=56b1$ has rank $0$ over $K$. The function ConjecturalSha in the Magma
computational algebra system predicts that $\Sh(E_{1}/K)[3^{\infty}]=0$.
Assume that this prediction is true, then, for all $n\in\mathbb{Z}_{\geq 0}$,
$\mathcal{O}_{K_{n}}$ satisfies Conjecture 1.1. Thus, this shows that
Hilbert’s tenth problem has a negative answer for $\mathcal{O}_{K_{n}}$.

###  1.2. Method of proof

Let us briefly describe the method of proof of the Theorem 1.4. Let $E_{/K}$
be an elliptic curve and $p$ be a prime such that $E$ has good ordinary
reduction at the primes of $K$ above $p$. Let
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$ be the
$p$-primary Selmer group of $E$ over $K_{\operatorname{anti}}$. We show that
if there exists an elliptic curve $E$ for which the following conditions hold

  1. (1)

$E$ has good ordinary reduction at the primes $\mathfrak{p}$ and
$\mathfrak{p}^{*}$,

  2. (2)

$\operatorname{rank}E(K)>0$,

  3. (3)

the Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$
is a cofinitely generated $\mathbb{Z}_{p}$-module, and

  4. (4)

the $\lambda$-invariant of
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$ is equal to
$\operatorname{rank}E(K)$,

then, $K_{n}/\mathbb{Q}$ is integrally diophantine for all $n$. The proof
relies on basic properties of the $\lambda$-invariant and Theorem 1.2. We
refer to Theorem 4.1 and its proof for further details. In order to construct
an elliptic curve $E_{/K}$ such that the above conditions are satisfied, we
exploit the Iwasawa theory of congruences between elliptic curves, initiated
by Greenberg-Vatsal [GV00] and further refined by Kundu-Ray [KR22, section 6].
This refinement does not only generalize the results of Greenberg and Vatsal
to non-cyclotomic $\mathbb{Z}_{p}$-extensions, but it gives a more accurate
description of the primes that actually contribute to the change in
$\lambda$-invariant. This optimal set of primes is the set
$\Sigma(E_{1})\cup\Sigma(E_{2})$, and for the example considered in section 5,
this set is strictly smaller than the set of all primes of bad reduction for
one or both elliptic curves. Condition (LABEL:c5_of_main_theorem) stipulates
that each of primes $v\in\Sigma(E_{1})\cup\Sigma(E_{2})$ must be finitely
decomposed in $K_{\operatorname{anti}}$, which is the case for the example
considered. It is thus ideal for the set $\Sigma(E_{1})\cup\Sigma(E_{2})$ for
which this assumption is made, to be as small as possible.

We consider two elliptic curves $E_{1}$ and $E_{2}$ defined over $K$ that are
$p$-congruent satisfying the conditions of Theorem 1.4. Since
$\operatorname{rank}E_{1}(K)=0$, and the $p$-primary part of $\Sh(E_{1}/K)$ is
finite (in fact, it is assumed to be zero), it follows that the $p$-primary
Selmer group of $E_{1}$ over $K$ is finite. By a well known result of
Greenberg that
$\operatorname{Sel}_{p^{\infty}}(E_{1}/K_{\operatorname{anti}})$ is a
cotorsion module over the Iwasawa algebra (cf. Proposition 2.1). Furthermore,
the assumptions (6)-(8) imply that the $\mu$ and $\lambda$-invariants of
$\operatorname{Sel}_{p^{\infty}}(E_{1}/K_{\operatorname{anti}})$ are zero (cf.
Corollary 4.6). Such a criterion relies on a formula for the Euler
characteristic over the anticyclotomic extension, due to Van Order [vO15]. At
this step, it is crucial for $E_{1}$ to have rank $0$ over $K$. We set
$E=E_{2}$, which is an elliptic curve with positive rank over $K$, and show
that the above conditions are satisfied. The congruence with the rank $0$
elliptic curve $E_{1}$, along with the assumptions of Theorem 1.4 imply that
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$ is a cofinitely
generated $\mathbb{Z}_{p}$-module satisfying the three conditions above. At
this step we make use of Theorem 3.6, which essentially extends the results of
[KR22, section 6] to $p=3$.

###  1.3. Outlook/ future directions

Our results show that Iwasawa theory can be used as a novel tool to prove new
interesting cases of Conjecture 1.1. It is natural to extend such
investigations to more general contexts that involve $p$-adic Lie extensions
of dimension $d>1$. Let us illustrate this by considering a special case: the
$\mathbb{Z}_{p}^{2}$-extension $\widetilde{K}_{\infty}$ of an imaginary
quadratic field $K$. For $n\geq 0$, let $\widetilde{K}_{n}$ denote the $n$-th
layer, i.e., the extension of $K$, contained in $\widetilde{K}_{\infty}$ for
which
$\operatorname{Gal}(\widetilde{K}_{n}/K)\simeq(\mathbb{Z}/p^{n}\mathbb{Z})^{2}$.
Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at
$p$. When the root number of the Hasse-Weil L-function of $E_{/K}$ is $+1$,
then, a result of Cornut and Vatsal [CV07] shows that the Mordell-Weil rank of
$E(\widetilde{K}_{n})$ is bounded as $n\rightarrow\infty$. Suppose that
$\operatorname{rank}E(K)>0$ and the root number of $E_{/K}$ is equal to $+1$.
Following Mazur and Rubin, we say that the Mordell-Weil rank is _diophantine
stable_ in $\widetilde{K}_{\infty}$ if

$\operatorname{rank}E(\widetilde{K}_{n})=\operatorname{rank}E(K)$

for all $n$. Then, from Theorem 1.2, it is evident that this property implies
that Conjecture 1.1 is satisfied for all number fields $\widetilde{K}_{n}$.
Generalized upper bounds for the asymptotic growth of the rank of the
Mordell–Weil group of an abelian variety in a tower of number fields contained
in a $p$-adic Lie extension, were proven by Harris [Har00]. The precise order
of growth in Mordell-Weil ranks is however not well understood [HL21, Question
following Proposition 4.7]. It is our expectation that the search for
diophantine stability in the context of Iwasawa theory would lead to
interesting theoretical and computational developments.

###  1.4. Organization

Including the introduction, the paper consists of five sections. In section 2,
we set up notation and discuss preliminary notions from anticyclotomic Iwasawa
theory. We also discuss the notion of an integrally diophantine extension in
this section. In section 3, we discuss congruences between elliptic curves and
their effect on Selmer groups. More specifically, in this section, we recall
the results of [KR22, section 6]. These results apply only to primes $p\geq
5$, and our main task in section 3 is to extend those results to the case when
$p=3$. In section 4, we study Iwasawa theory in the context of Hilbert’s tenth
problem, and prove Theorem 1.4. In section 5, we illustrate Theorem 1.4
through an explicit example, and prove Theorem 1.5 as a consequence.

###  1.5. Acknowledgment

The first named author’s research is supported by the CRM Simons postdoctoral
fellowship. He thanks Antonio Lei for some helpful discussions.

##  2\. Preliminaries

###  2.1. Iwasawa theory of elliptic curves

In this section, we review some necessary background. For a more comprehensive
introduction to the Iwasawa theory of elliptic curves, the reader is referred
to [Coa00, Chapter 1]. Throughout, $p$ is an odd prime number and $E$ shall
denote an elliptic curve over a number field $K$. We assume that $E$ has good
ordinary reduction at the primes $v|p$. We fix an algebraic closure $\bar{K}$
of $K$. Let $\mathbb{Z}_{p}$ denote the $p$-adic integers. An infinite Galois
extension $K_{\infty}/K$ is said to be a $\mathbb{Z}_{p}$-extension if there
is a topological isomorphism of groups
$\operatorname{Gal}(K_{\infty}/K)\xrightarrow{\sim}\mathbb{Z}_{p}$. Let
$K_{\infty}/K$ be a $\mathbb{Z}_{p}$-extension, and set
$\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K_{\infty}/K)$. The _Iwasawa algebra_ is the
completed group ring defined as follows

$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Z}_{p}\llbracket\Gamma\rrbracket\mathrel{\mathop{\mathchar
58\relax}}=\lim_{n\rightarrow\infty}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}].$

Fixing a topological generator $\gamma$ of $\Gamma$ yields the usual
isomorphism $\Lambda\cong\mathbb{Z}_{p}\llbracket T\rrbracket$ with the ring
of integral $p$-adic power series. This isomorphism identifies the group-like
element $\gamma$ with $(T+1)$. For $n\in\mathbb{Z}_{\geq 0}$, set $K_{n}$ to
denote the $n$-th layer of the associated tower. This is the extension
$K_{n}/K$ such that $K_{n}\subset K_{\infty}$ and
$\operatorname{Gal}(K_{n}/K)\simeq\mathbb{Z}/p^{n}\mathbb{Z}$. Thus, there is
an infinite tower of number fields

$K_{0}\subset K_{1}\subset\dots\subset K_{n}\subset K_{n+1}\subset\dots,$

where $K=K_{0}$. We shall refer to such a tower as a $\mathbb{Z}_{p}$-tower
over $K$.

Let $F/K$ be a number field extension. Given a finite prime $v$ of $K$, set
$v(F)$ to denote the set of primes of $F$ that lie above $v$. We set

$J_{v}(E/F)\mathrel{\mathop{\mathchar 58\relax}}=\bigoplus_{w\in
v(F)}H^{1}(F_{w},E)[p^{\infty}],$

and note that there is a natural map

$\iota_{v}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(\bar{F}/F,E[p^{\infty}])\longrightarrow J_{v}(E/F)$

which factors through the restriction maps

$H^{1}(\bar{F}/F,E[p^{\infty}])\longrightarrow\bigoplus_{w\in
v(F)}H^{1}(F_{w},E[p^{\infty}]).$

The $p$-primary Selmer group of $E/F$ is defined as follows

$\operatorname{Sel}_{p^{\infty}}(E/F)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\bar{F}/F,E[p^{\infty}])\xrightarrow{\iota}\prod_{v}J_{v}(E/F)\right),$

where the map $\iota$ is induced by the maps $\iota_{v}$, and the product
ranges over all primes $v$ of $K$.

We set $\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})$ to denote the direct
limit $\lim_{n\rightarrow\infty}\operatorname{Sel}_{p^{\infty}}(E/K_{n})$,
with respect to restriction maps. Let $S$ be a set of primes of $K$ containing
the primes above $p$ and the primes at which $E$ has bad reduction. Denote by
$K_{S}$ the maximal algebraic extension of $K$ in which all primes $v\notin S$
are unramified. For $v\in S$, set $J_{v}(E/K_{\infty})$ to be the direct limit
$\lim_{n\rightarrow\infty}J_{v}(E/K_{n})$. The Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})$ is described as follows

(2.1) $\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(K_{S}/K_{\infty},E[p^{\infty}])\longrightarrow\bigoplus_{v\in
S}J_{v}(E/K_{\infty})\right).$

Given a module $M$ over $\mathbb{Z}_{p}$, we set
$M^{\vee}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$
to denote its _Pontryagin dual_. A module $M$ over $\Lambda$ is said to be
cofinitely generated (resp. cotorsion) over $\Lambda$ if $M^{\vee}$ is
finitely generated (resp. torsion) over $\Lambda$. We discuss conditions for
$\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})$ to be cotorsion module over
$\Lambda$.

######  Proposition 2.1.

Let $E$ be an elliptic curve over a number field $K$ and let $p$ be an odd
prime number. Let $K_{\infty}$ be a $\mathbb{Z}_{p}$-extension of $K$. Assume
that the following conditions are satisfied

  1. (1)

$E$ has good ordinary reduction at the primes $v|p$,

  2. (2)

the Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K)$ is finite.

Then the Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})$ is a
cotorsion module over $\Lambda$.

###### Proof.

The above result is [Gre01, Corollary 4.9]. ∎

The Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})$ is cofinitely
generated over $\Lambda$, and we may define the associated Iwasawa invariants
$\mu_{p}(E/K_{\infty})$ and $\lambda_{p}(E/K_{\infty})$ as follows. Setting
$M\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Sel}_{p^{\infty}}(E/K_{\infty})$, the module
$M^{\vee}$ is a finitely generated torsion module over $\Lambda$. A map
$M_{1}\rightarrow M_{2}$ of $\Lambda$-modules is a pseudo-isomorphism if the
kernel and cokernel are both finite. We depict this relation by writing
$M_{1}\sim M_{2}$. A polynomial $f(T)\in\Lambda$ is said to be distinguished
if it is monic and its non-leading coefficients are all divisible by $p$. By
the structure theorem for finitely generated torsion modules over $\Lambda$
[Was97, Theorem 13.12],

(2.2)
$M^{\vee}\sim\Lambda^{r}\oplus\left(\bigoplus_{i=1}^{s}\Lambda/(p^{n_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/(f_{j}(T)^{m_{j}})\right),$

where $r,s,t,n_{i},m_{j}\in\mathbb{Z}_{\geq 0}$ and $f_{j}(T)$ are all
irreducible distinguished polynomials. The $\mu$-invariant is defined as
follows

$\mu_{p}(E/K_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}&0\text{ if }s=0;\\\ &\sum_{i=1}^{s}n_{i}\text{ if
}s>0.\\\ \end{cases}$

The $\lambda$-invariant is defined as follows

$\lambda_{p}(E/K_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}&0\text{ if }t=0;\\\
&\sum_{j=1}^{t}m_{j}\operatorname{deg}f_{j}(T)\text{ if }t>0.\\\ \end{cases}$

Suppose that $M$ is cotorsion over $\Lambda$, i.e., $r=0$. It is easy to see
that $\mu_{p}(E/K_{\infty})=0$ if and only if $M^{\vee}$ is finitely generated
as a $\mathbb{Z}_{p}$-module. Furthermore, if these equivalent conditions are
satisfied, then,

(2.3)
$\lambda_{p}(E/K_{\infty})=\operatorname{rank}_{\mathbb{Z}_{p}}\left(M^{\vee}\right).$

From here on in, we assume that $K=\mathbb{Q}(\sqrt{-D})$ is an imaginary
quadratic field in which $p$ splits into primes $\mathfrak{p}$ and
$\mathfrak{p}^{*}$. Here, $D>0$ is a squarefree integer. There are precisely
two $\mathbb{Z}_{p}$-extensions of $K$ that are Galois over $\mathbb{Q}$,
namely, the cyclotomic $\mathbb{Z}_{p}$-extension $K_{\operatorname{cyc}}$ and
the anticyclotomic $\mathbb{Z}_{p}$-extension $K_{\operatorname{anti}}$. The
cyclotomic $\mathbb{Z}_{p}$-extension is abelian over $\mathbb{Q}$, while the
anticyclotomic $\mathbb{Z}_{p}$-extension is pro-dihedral over $\mathbb{Q}$.
Given a finite prime $v$, set $v(K_{\operatorname{anti}})$ to denote the set
of primes $w$ of $K_{\operatorname{anti}}$ that lie above $v$.

It conveniences us to work with Selmer groups over $K_{\operatorname{anti}}$
defined by Greenberg’s local conditions. It follows (for instance) from [KR22,
Lemma 6.5] that this definition coincides with that of (2.1). Given a finite
prime $v\nmid p$ of $K$, set

$\mathcal{H}_{v}(E/K_{\operatorname{anti}})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{w\in
v(K_{\operatorname{anti}})}H^{1}(K_{\operatorname{anti},w},E)[p^{\infty}].$

Here, $K_{\operatorname{anti},w}$ is taken to be the union of local fields
$\bigcup_{n\geq 0}K_{n,w}$. Let $v|p$ be a prime, and
$T\mathrel{\mathop{\mathchar 58\relax}}=T_{p}(E)$ be the $p$-adic Tate-module
of $E$, viewed as a module over
$\operatorname{G}_{v}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\bar{F}_{v}/F_{v})$. Since $E$ is assumed to
have good ordinary reduction at $v$, $T$ fits into a short exact sequence of
$\operatorname{G}_{v}$-modules

$0\rightarrow T^{+}\rightarrow T\rightarrow T^{-}\rightarrow 0.$

The modules $T^{\pm}$ are free $\mathbb{Z}_{p}$-modules of rank $1$, and the
action of $\operatorname{G}_{v}$ on $T^{-}$ is unramified. We set $D_{v}$ to
denote the $p$-primary $\operatorname{G}_{v}$-module defined by
$D_{v}\mathrel{\mathop{\mathchar
58\relax}}=T^{-}\otimes_{\mathbb{Z}_{p}}\left(\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$.
For $w\in v(K_{\operatorname{anti}})$, set

$L_{w}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(K_{\operatorname{anti},w},E[p^{\infty}])\longrightarrow
H^{1}(\operatorname{I}_{w},D_{v})\right),$

where $\operatorname{I}_{w}$ is the inertia subgroup of the absolute Galois
group of $K_{\operatorname{anti},w}$. The map above is induced from the
natural map $E[p^{\infty}]\rightarrow D_{v}$ obtained by tensoring
$T\rightarrow T^{-}$ with $\mathbb{Q}_{p}/\mathbb{Z}_{p}$. With respect to the
above notation, for $v|p$, we define the local Greenberg Selmer condition as
follows

$\mathcal{H}_{v}(E/K_{\operatorname{anti}})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{w\in
v(K_{\operatorname{anti}})}H^{1}(K_{\operatorname{anti},w},E[p^{\infty}])/L_{w}.$

Note that this product can be infinite. The Selmer group is then described as
follows

(2.4)
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(K_{S}/K_{\operatorname{anti}},E[p^{\infty}])\rightarrow\bigoplus_{v\in
S}\mathcal{H}_{v}(E/K_{\operatorname{anti}})\right\\}.$

###  2.2. Integrally diophantine extensions and Shlapentokh’s criterion

In this section, we briefly recall the notion of an integrally diophantine
extension. For further details, the reader is referred to [Shl07, Section
1.2]. Let $A$ be a commutative ring with identity, and $A^{n}$ be the free
$A$-module of rank $n$, consisting of tuples $a=(a_{1},\dots,a_{n})$ with
entries in $A$. Let $m$ and $n$ be positive integers and let
$a=(a_{1},\dots,a_{n})\in A^{n}$ and $b=(b_{1},\dots,b_{m})\in A^{m}$. Denote
by $(a,b)\in A^{n+m}$ the tuple $(a_{1},\dots,a_{n},b_{1},\dots,b_{m})$. Given
a finite set of polynomials, $F_{1},\dots,F_{k}$, we set

$\mathcal{F}(a;F_{1},\dots,F_{k})\mathrel{\mathop{\mathchar 58\relax}}=\\{b\in
A^{m}\mid F_{i}(a,b)=0\text{ for all }i=1,\dots,k\\}.$

######  Definition 2.2.

A subset $S$ of $A^{n}$ is a _diophantine subset_ of $A^{n}$ if for some
$m\geq 1$, there are polynomials $F_{1},\dots,F_{k}\in
A[x_{1},\dots,x_{n},y_{1},\dots,y_{m}]$ such that $S$ consists of all $a\in
A^{n}$ for which the set $\mathcal{F}(a;F_{1},\dots,F_{k})$ is nonempty.

######  Definition 2.3.

An extension of number fields $L/K$ is said to be _integrally diophantine_ if
$\mathcal{O}_{K}$ is a diophantine subset of $\mathcal{O}_{L}$.

Let $L^{\prime},L$ and $K$ be number fields such that

$L^{\prime}\supseteq L\supseteq K.$

Suppose that $L^{\prime}/L$ and $L/K$ are integrally diophantine extensions.
Then, it is a well known fact that $L^{\prime}/K$ is an integrally diophantine
extension. Indeed, this is a special case of [Shl07, Theorem 2.1.15].

##  3\. Iwasawa theory of congruences

Building on the results of Greenberg-Vatsal [GV00] and Kundu-Ray [KR22], we
describe the effect of congruences between elliptic curves on Selmer groups
over anticyclotomic $\mathbb{Z}_{p}$-extensions. The results apply in what is
called the _definite case_ , i.e., the setting in which the Selmer groups are
cotorsion over the Iwasawa algebra. The results shall be applied to a
congruence between an elliptic curve of rank $0$ and an elliptic curve of
positive rank. In the setting in which we work, both the Selmer groups over
$K_{\operatorname{anti}}$ shall be cofinitely generated as
$\mathbb{Z}_{p}$-modules. We extend the results of [KR22, section 6] (which
are proven for $p\geq 5$) to include the case $p=3$ as well. Let $E_{1}$ and
$E_{2}$ be elliptic curves defined over $K$. We let $E_{i}[p]$ denote the
$p$-torsion subgroup of $E_{i}(\bar{K})$, equipped with the natural action of
the absolute Galois group $\operatorname{G}_{K}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\bar{K}/K)$. We say that $E_{1}$ and $E_{2}$ are
$p$-congruent if $E_{1}[p]$ is isomorphic to $E_{2}[p]$ as modules over
$\operatorname{G}_{K}$. Let $\bar{\rho}_{i}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
denote the Galois representation associated to $E_{i}[p]$. Assume that $p$
splits in $K$, and let $\Sigma_{p}=\\{\mathfrak{p},\mathfrak{p}^{*}\\}$ be the
primes of $K$ that lie above $p$. We assume that $E_{1}(K)[p]=E_{2}(K)[p]=0$.
In other words, the Galois modules $E_{i}[p]$ do not contain the trivial
module as a submodule. Note that this condition is automatically satisfied
when the Galois representation $\bar{\rho}_{i}$ is irreducible. Since it is
assumed that $\bar{\rho}_{1}$ is isomorphic to $\bar{\rho}_{2}$, it follows
that $E_{1}(K)[p]=0$ if and only if $E_{2}(K)[p]=0$. We assume that both
elliptic curves $E_{1}$ and $E_{2}$ have good ordinary reduction at all primes
$v\in\Sigma_{p}$. Let $\mathfrak{T}$ be the set of primes $v$ of $K$ such that
either $E_{1}$ or $E_{2}$ (or both) have bad reduction at $v$. Set
$K^{\prime}$ to denote $K(\mu_{p})$.

######  Definition 3.1.

For $i=1,2$, the set $\Sigma(E_{i})$ consists of primes $v\in\mathfrak{T}$
such that the following conditions is satisfied.

  1. (1)

If the elliptic curve $E_{i}$ has good reduction at $v$, then,
$p\mid\\#E_{i}(\mathbb{F}_{v})$.

  2. (2)

If the elliptic curve $E_{i}$ has bad reduction at $v$, and $p\geq 5$, then
$E_{i}$ has split multiplicative reduction at some prime $w|v$ of
$K^{\prime}$.

  3. (3)

If the elliptic curve $E_{i}$ has bad reduction at $v$, and $p=3$, then, at
some prime $w|v$ of $K^{\prime}$, the elliptic curve $E_{i}$ has either split
multiplicative reduction or has additive reduction of type $\rm{IV}$ or
$\rm{IV}^{*}$.

We set $\Sigma\mathrel{\mathop{\mathchar
58\relax}}=\Sigma(E_{1})\cup\Sigma(E_{2})$ and $S\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{T}\cup\Sigma_{p}$.

Note that $\Sigma$ is a subset of $S$. For $i=1,2$, we define the imprimitive
Selmer group as follows

(3.1)
$\operatorname{Sel}_{p^{\infty}}^{\Sigma}(E_{i}/K_{\operatorname{anti}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(K_{\operatorname{anti}},E_{i}[p^{\infty}])\longrightarrow\bigoplus_{v\in
S\backslash\Sigma}\mathcal{H}_{v}(E_{i}/K_{\operatorname{anti}})\right).$

In other words, we relax the local conditions at the primes $v\in\Sigma$.
There is one crucial difference between the cyclotomic and anticyclotomic
extensions. It is easy to see that all primes are finitely decomposed in the
cyclotomic $\mathbb{Z}_{p}$-extension (cf. [Bri07, Proposition 2]). This is
not the case for other $\mathbb{Z}_{p}$-extensions. Let $\ell$ be a prime
number and $v|\ell$ be a prime of $K$ that lies above $\ell$. By an
application of class field theory, $v$ is infinitely decomposed in
$K_{\operatorname{anti}}$ if and only if $\ell\neq p$ and $\ell$ is inert or
ramified in $K$ (cf. [Bri07, p.2132, ll. 8-10]). Throughout this section, we
make the following assumption.

######  Assumption 3.2.

Assume that all primes $v\in\Sigma$ are finitely decomposed in
$K_{\operatorname{anti}}$. Equivalently, we assume that $\ell$ splits into a
product of two primes of $K$ for every prime $\ell$ divisible by some
$v\in\Sigma$.

For $v\in S$, we refer to [KR22, section 6] for the definition of the local
condition $\mathcal{H}_{v}(E[p]/K_{\operatorname{anti}})$. The _residual
imprimitive Selmer group_ is defined as follows

$\operatorname{Sel}_{p^{\infty}}^{\Sigma}(E[p]/K_{\operatorname{anti}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(K_{S}/K_{\operatorname{anti}},E[p^{\infty}])\rightarrow\bigoplus_{v\in
S\backslash\Sigma}\mathcal{H}_{v}(E[p]/K_{\operatorname{anti}})\right\\}.$

######  Lemma 3.3.

For $v\in S\backslash\Sigma$, the natural map

(3.2)
$\mathcal{H}_{v}(E[p]/K_{\operatorname{anti}})\rightarrow\mathcal{H}_{v}(E[p^{\infty}]/K_{\operatorname{anti}})[p]$

is injective.

###### Proof.

When $p\geq 5$, the above result is [KR22, Lemma 6.10]. Therefore, we consider
the case when $p=3$.

First, we assume that $v|p$. In this case, the result follows from part (1) in
the proof of Lemma 6.10 from _loc. cit_. Next, suppose that $v\nmid p$ is a
prime at which $E$ has good reduction. The injectivity of the map follows from
part (2) in the proof of the above mentioned Lemma. The only case to consider
therefore is when $v\nmid p$ is a prime at which $E$ has bad reduction. Recall
that $K^{\prime}=K(\mu_{p})$, and set
$K^{\prime}_{\operatorname{anti}}=K^{\prime}\cdot K_{\operatorname{anti}}$. It
follows from the proof of (3) of Lemma 6.10 of _loc. cit._ that the map (3.2)
is injective if $E(K^{\prime}_{\operatorname{anti},w^{\prime}})[p^{\infty}]=0$
for all primes $w^{\prime}|v$ of $K^{\prime}_{\operatorname{anti}}$. Let $w$
be a prime of $K^{\prime}$ such that $w|v$. Let $w^{\prime}|w$ be a prime of
$K^{\prime}_{\operatorname{anti}}$. Since
$K^{\prime}_{\operatorname{anti},w^{\prime}}/K^{\prime}_{w}$ is a pro-$p$
extension, it follows from [KR22, Lemma 6.9] that

$E(K^{\prime}_{w})[p^{\infty}]=0\Rightarrow
E(K^{\prime}_{\operatorname{anti},w^{\prime}})[p^{\infty}]=0.$

Therefore, it suffices to show that $E(K^{\prime}_{w})[p^{\infty}]=0$ for all
$w|v$ of $K^{\prime}$. We recall part (3) of the Definition 3.1. Note that in
this case $p=3$, $v\notin\Sigma$ and $E$ is assumed to have bad reduction at
$v$. Therefore, $E$ has non-split multiplicative reduction or additive
reduction of Kodaira type $I\notin\\{\rm{IV},\rm{IV}^{*}\\}$ at all primes
$w|v$ of $K^{\prime}$. Let $E_{0}(K^{\prime}_{w})$ be the subgroup of $E$
consisting of points of non-singular reduction. It follows from the proof of
[HM99, Proposition 5.1 (iii)] that $E_{0}(K^{\prime}_{w})[p^{\infty}]=0$. It
suffices to show that $p\nmid[E(K^{\prime}_{w})\mathrel{\mathop{\mathchar
58\relax}}E_{0}(K^{\prime}_{w})]$ for all primes $w|v$ of $K^{\prime}$. Since
$p=3$, and $E$ has non-split multiplicative reduction or additive reduction of
Kodaira type $I\notin\\{\rm{IV},\rm{IV}^{*}\\}$, the result follows from
[Sil94, 4th row of Table 4.1, p. 365]. ∎

######  Proposition 3.4.

Let $E$ be either $E_{1}$ or $E_{2}$ and recall that
$\Sigma=\Sigma(E_{1})\cup\Sigma(E_{2})$. Assume that $E(K)[p]=0$. Then, the
natural map

$\operatorname{Sel}_{p^{\infty}}^{\Sigma}(E[p]/K_{\infty})\xrightarrow{\sim}\operatorname{Sel}_{p^{\infty}}^{\Sigma}(E/K_{\infty})[p]$

is an isomorphism.

###### Proof.

The result follows verbatim from the proof of [KR22, Proposition 6.11] and
Lemma 3.3. ∎

We introduce some further notation. Let $E$ be either $E_{1}$ or $E_{2}$.
Recall that by the Assumption 3.2, each prime $v\in\Sigma$ is finitely
decomposed in $K_{\operatorname{anti}}$. For $v\in\Sigma$, let $\ell$ be the
prime number for which $v|\ell$. Note that $\ell\neq p$, since both elliptic
curves $E_{1}$ and $E_{2}$ have good reduction at all primes $v\in\Sigma_{p}$.
Denote by $v^{*}$ the conjugate prime, so that $\ell\mathcal{O}_{K}=vv^{*}$.
We identify $K_{v}$ with $\mathbb{Q}_{\ell}$, and set
$\mathbb{Q}_{\ell}^{\operatorname{nr}}$ to denote the unramified
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}_{\ell}$. Recall that $s_{v}$ is the
number of primes of $K_{\operatorname{anti}}$ that lie above $v$. We note that
[Bri07, Theorem 2, p. 2133] gives an explicit algorithm to compute $s_{v}$.
Let $\sigma_{E}^{(v)}$ denote the $\mathbb{Z}_{p}$-corank of
$\mathcal{H}_{v}(E/K_{\operatorname{anti}})$, for $v\in\Sigma$. Recall that
$d_{v}(E)$ is the multiplicity of $\ell^{-1}$ as a root of the local
$L$-factor $P_{v}(X)$.

######  Proposition 3.5.

For $v\in\Sigma$, the $\mu$-invariant of
$\mathcal{H}_{v}(E/K_{\operatorname{anti}})$ is $0$ and
$\sigma_{E}^{(v)}=s_{v}d_{v}(E)$.

###### Proof.

The above result follows from the proof of [GV00, Proposition 2.4]. ∎

######  Theorem 3.6.

Let $p$ be an odd prime and $K$ be an imaginary quadratic field in which $p$
splits. Let $E_{1}$ and $E_{2}$ be two elliptic curves defined over $K$.
Assume that the following conditions hold

  1. (1)

Both elliptic curves have good ordinary reduction at the primes
$v\in\Sigma_{p}$.

  2. (2)

The elliptic curves $E_{1}$ and $E_{2}$ are $p$-congruent. In other words,
$E_{1}[p]$ and $E_{2}[p]$ are isomorphic as modules over
$\operatorname{G}_{K}$.

  3. (3)

We require that $E_{1}(K)[p]=E_{2}(K)[p]=0$. (This assumption is clearly
satisfied when the residual representations $\bar{\rho}_{i}$ on $E_{i}[p]$ are
irreducible as modules over $\operatorname{G}_{K}$.)

  4. (4)

Assumption 3.2 is satified. In other words, all primes $v\in\Sigma$ (cf.
Definition 3.1) are finitely decomposed in $K_{\operatorname{anti}}$.

Then, the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E_{1}/K_{\operatorname{anti}})$ is cotorsion
over $\Lambda$ with $\mu=0$ if and only if
$\operatorname{Sel}_{p^{\infty}}(E_{2}/K_{\operatorname{anti}})$ is cotorsion
over $\Lambda$ with $\mu=0$. Furthermore, suppose that indeed both Selmer
groups $\operatorname{Sel}_{p^{\infty}}(E_{i}/K_{\operatorname{anti}})$ are
cotorsion over $\Lambda$ with $\mu=0$. Then, we have that

$\lambda_{p}(E_{2}/K_{\operatorname{anti}})-\lambda_{p}(E_{1}/K_{\operatorname{anti}})=\sum_{v\in\Sigma}\left(\sigma_{E_{1}}^{(v)}-\sigma_{E_{2}}^{(v)}\right)=\sum_{v\in\Sigma}s_{v}\left(d_{v}(E_{1})-d_{v}(E_{2})\right).$

###### Proof.

When $p\geq 5$, the above result follows from [KR22, Theorem 6.14, Proposition
6.19]. When $p=3$, the same proof applies verbatim from Proposition 3.4, the
key ingredient being Lemma 3.3. ∎

##  4\. A general criterion

Let $p$ be an odd prime, $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary quadratic
field in which $p$ splits, and $K_{\operatorname{anti}}$ denote the
anticyclotomic $\mathbb{Z}_{p}$-extension of $K$. Given $n\in\mathbb{Z}_{\geq
1}$, we recall that $K_{n}$ is $n$-th layer in $K_{\operatorname{anti}}$. Let
$\mathfrak{p}$ and $\mathfrak{p}^{*}$ be the primes of $K$ that lie above $p$.

######  Theorem 4.1.

With respect to notation above, suppose that there exists an elliptic curve
$E_{/K}$ such that all of the following conditions are satisfied

  1. (1)

the elliptic curve $E$ has good ordinary reduction at the primes
$\mathfrak{p}$ and $\mathfrak{p}^{*}$,

  2. (2)

the Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$
is cotorsion over $\Lambda$ with $\mu_{p}(E/K_{\operatorname{anti}})=0$,

  3. (3)

the rank of the Mordell-Weil group $E(K)$ is positive,

  4. (4)

$\lambda_{p}(E/K_{\operatorname{anti}})=\operatorname{rank}E(K)$.

Then, for all $n\in\mathbb{Z}_{\geq 0}$, the Hilbert’s tenth problem has a
negative answer for $\mathcal{O}_{K_{n}}$.

###### Proof.

It follows from the structure theory of $\Lambda$-modules that condition (2)
above implies that
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})^{\vee}$ is a
finitely generated $\mathbb{Z}_{p}$-module and

$\lambda_{p}(E/K_{\operatorname{anti}})=\operatorname{rank}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})^{\vee}\right).$

The kernel of the natural restriction map

$\operatorname{Sel}_{p^{\infty}}(E/K_{n})\rightarrow\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$

is finite (cf. [Gre01, Theorem 4.1]), and therefore,

$\operatorname{rank}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}_{p^{\infty}}(E/K_{n})^{\vee}\right)\leq\lambda_{p}(E/K_{\operatorname{anti}}).$

Note that $\operatorname{Sel}_{p^{\infty}}(E/K_{n})$ fits into a short exact
sequence

$0\rightarrow
E(K_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow\operatorname{Sel}_{p^{\infty}}(E/K_{n})\rightarrow\Sh(E/K_{n})[p^{\infty}]\rightarrow
0,$

where $\Sh(E/K_{n})[p^{\infty}]$ is the $p$-primary part of the Tate-
Shafarevich group of $E/K_{n}$. Therefore, we deduce that for all $n\geq 0$,

$\operatorname{rank}E(K_{n})\leq\operatorname{rank}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}_{p^{\infty}}(E/K_{n})^{\vee}\right)\leq\lambda_{p}(E/K_{\operatorname{anti}}).$

On the other hand, it is assumed that
$\lambda_{p}(E/K_{\operatorname{anti}})=\operatorname{rank}E(K)$. Therefore,
we find that $\operatorname{rank}E(K_{n})=\operatorname{rank}E(K)$. Since,
$\operatorname{rank}E(K)>0$, it follows from Theorem 1.2 that $K_{n}/K$ is
integrally diophantine. Since $K/\mathbb{Q}$ is integrally diophantine, it
follows that for all $n$, $K_{n}/\mathbb{Q}$ is integrally diophantine.
Therefore, for all $n$, Hilbert’s tenth problem has a negative answer for
$\mathcal{O}_{K_{n}}$. ∎

In order to construct an elliptic curve $E$ that satisfies the above
conditions, we begin with two elliptic curves $E_{1}$ and $E_{2}$ that are
both defined over the rational numbers. We assume that $E_{1}$ and $E_{2}$ are
$p$-congruent, i.e., the Galois modules $E_{1}[p]$ and $E_{2}[p]$ are
isomorphic.

######  Proposition 4.2.

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number which
splits into primes $\mathfrak{p}$ and $\mathfrak{p}^{*}$ in $\mathcal{O}_{K}$.
Assume that there exist elliptic curves $E_{1}$ and $E_{2}$ defined over $K$
such that

  1. (1)

both $E_{1}$ and $E_{2}$ have good ordinary reduction at $\mathfrak{p}$ and
$\mathfrak{p}^{*}$,

  2. (2)

$E_{1}[p]$ and $E_{2}[p]$ are isomorphic as modules over
$\operatorname{Gal}(\bar{K}/K)$,

  3. (3)

$\operatorname{rank}E_{1}(K)=0$ and $\operatorname{rank}E_{2}(K)>0$,

  4. (4)

$E_{1}(K)[p]=E_{2}(K)[p]=0$,

  5. (5)

each prime $v\in\Sigma$ is finitely decomposed in the anticyclotomic extension
$K_{\operatorname{anti}}/K$. Equivalently, let $v\in\Sigma$ and $\ell$ be the
prime number for which $v|\ell$. Then, $\ell$ splits in $K$ into a product of
two primes $v$ and $v^{*}$.

  6. (6)

the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E_{1}/K_{\operatorname{anti}})$ is cotorsion
over $\Lambda$ with $\mu_{p}(E_{1}/K_{\operatorname{anti}})=0$ and
$\lambda_{p}(E_{1}/K_{\operatorname{anti}})=0$.

Suppose that the following relationship holds

(4.1)
$\operatorname{rank}E_{2}(K)=\sum_{v\in\Sigma}\left(\sigma_{E_{1}}^{(v)}-\sigma_{E_{2}}^{(v)}\right),$

where we recall that $\sigma_{E}^{(v)}$ is the $\mathbb{Z}_{p}$-corank of
$\mathcal{H}_{v}(E/K_{\operatorname{anti}})$,

Then, for all $n\in\mathbb{Z}_{\geq 0}$, Hilbert’s tenth problem has a
negative answer for $\mathcal{O}_{K_{n}}$.

###### Proof.

Let $E$ be the elliptic curve $E_{2}$. We show that the conditions of Theorem
4.1 is satisfied. The conditions (1) and (3) of the Theorem are satisfied by
assumption. The condition (2) of the Theorem requires that the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$ is cotorsion over
$\Lambda$ with $\mu_{p}(E/K_{\operatorname{anti}})=0$. We note that it is
assumed that $\operatorname{Sel}_{p^{\infty}}(E_{1}/K_{\operatorname{anti}})$
is cotorsion over $\Lambda$ with $\mu_{p}(E_{1}/K_{\operatorname{anti}})=0$.
Therefore, the condition (2) follows from Theorem 3.6. Furthermore, it follows
from Theorem 3.6 that

$\lambda_{p}(E/K_{\operatorname{anti}})-\lambda_{p}(E_{1}/K_{\operatorname{anti}})=\sum_{v\in\Sigma}\left(\sigma_{E_{1}}^{(v)}-\sigma_{E}^{(v)}\right).$

Since it is assumed that

$\lambda_{p}(E_{1}/K_{\operatorname{anti}})=0$

and that

$\operatorname{rank}E(K)=\sum_{v\in\Sigma}\left(\sigma_{E_{1}}^{(v)}-\sigma_{E}^{(v)}\right),$

we deduce that

$\operatorname{rank}E(K)=\lambda_{p}(E/K_{\operatorname{anti}}).$

The assumptions of Theorem 4.1 are satisfied and the result follows. ∎

The condition (6) of Proposition 4.2 is the most difficult to verify and
warrants further explanation. We derive explicit conditions on the elliptic
curve $E_{1}$ for this condition to be satisfied. These conditions are
obtained by studying the Euler characteristic of the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$, where $E=E_{1}$.

Let $M$ be a cofinitely generated cotorsion $\Lambda$-module. Since the
$p$-cohomological dimension of $\Lambda$ is $0$, it follows that
$H^{i}(\Gamma,M)=0$ for all values $i\geq 2$. Note that $H^{1}(\Gamma,M)$ is
identified with $M_{\Gamma}$. The module $H^{0}(\Gamma,M)$ is finite if and
only if $H^{1}(\Gamma,M)$ is finite (cf. [KR21, Lemma 2.1]). The Euler
characteristic of $M$ is then defined as follows

$\chi(\Gamma,M)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\\#H^{0}(\Gamma,M)}{\\#H^{1}(\Gamma,M)}.$

Let $f_{M}(T)=\sum_{i}c_{i}T^{i}$ be a generator of the characteristic ideal
of $M^{\vee}$. Given $a,b\in\mathbb{Z}_{p}$, we write $a\sim b$ if $a=ub$ for
some unit $u\in\mathbb{Z}_{p}^{\times}$.

######  Lemma 4.3.

Let $M$ be a cofinitely generated cotorsion $\Lambda$-module such that the
cohomology groups $H^{0}(\Gamma,M)$ and $H^{1}(\Gamma,M)$ are finite. Then,
$\chi(\Gamma,M)\sim c_{0}$.

###### Proof.

This is a well known result, we refer for instance to [HKR21, Lemma 3.4]. ∎

######  Proposition 4.4.

Let $M$ be a cofinitely generated and cotorsion $\Lambda$-module for which the
cohomology groups $H^{i}(\Gamma,M)$ are finite. Then, the following are
equivalent

  1. (1)

$\chi(\Gamma,M)=1$,

  2. (2)

$f_{M}(T)=1$,

  3. (3)

$\mu(M)=0$ and $\lambda(M)=0$,

  4. (4)

$M$ has finite cardinality.

###### Proof.

The above result is a direct consequence of Lemma 4.3. We refer to [HKR21,
Proposition 3.6] for the proof. ∎

######  Theorem 4.5.

Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$.
Set $\Gamma$ to denote the Galois group
$\operatorname{Gal}(K_{\operatorname{anti}}/K)$. Assume that the following
conditions are satisfied

  1. (1)

$E(K)[p]=0$,

  2. (2)

$E$ has good ordinary reduction at each prime that lies above $p$.

  3. (3)

$\operatorname{rank}E(K)=0$,

  4. (4)

$\Sh(E/K)[p^{\infty}]$ is finite.

Then, the following assertions hold.

  1. (1)

the Selmer group $M\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$ is a
cotorsion module over $\Lambda$,

  2. (2)

the cohomology groups $H^{0}(\Gamma,M)$ and $H^{1}(\Gamma,M)$ are finite,

  3. (3)

the Euler characteristic is given by the following formula

$\chi(\Gamma,E[p^{\infty}])\sim\\#\Sh(E/K)[p^{\infty}]\times\left(\prod_{\mathfrak{p}|p}\\#E(\mathbb{F}_{\mathfrak{p}})\right)^{2}\times\prod_{v\in\Omega_{K}}c_{v}^{(p)}(E).$

###### Proof.

That $M$ is a cotorsion over $\Lambda$ follows from Proposition 2.1. That the
cohomology group $H^{i}(\Gamma,M)$ are finite, follows from [HKR21, Lemma
3.3]. The formula for the Euler characteristic is due to J. Order [vO15].
There is a correction to this formula, for which we refer to [HKR21, Theorem
4.4]. ∎

######  Corollary 4.6.

Let $E$ be an elliptic curve satisfying the above conditions of Theorem 4.5.
Furthermore, assume that the following additional conditions are satisfied

  1. (1)

$\Sh(E/K)[p^{\infty}]=0$,

  2. (2)

$E(\mathbb{F}_{\mathfrak{p}})[p]=0$,

  3. (3)

$c_{v}^{(p)}(E)=1$ for all $v\in\Omega_{K}$.

Then, the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})$ is a cotorsion
module over $\Lambda$, with
$\mu_{p}(E/K_{\operatorname{anti}})=\lambda_{p}(E/K_{\operatorname{anti}})=0$.

###### Proof.

It follows from Theorem 4.5 that
$\chi\left(\Gamma,\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{anti}})\right)=1$.
Thus the result follows from Proposition 4.4. ∎

######  Theorem 4.7.

Suppose that $E_{1}$ and $E_{2}$ satisfy the conditions (1)-(5) of Proposition
4.2, and that

  1. (1)

$\Sh(E_{1}/K)[p^{\infty}]=0$,

  2. (2)

$E_{1}(\mathbb{F}_{\mathfrak{p}})[p]=0$,

  3. (3)

$c_{v}^{(p)}(E_{1})=1$ for all $v\in\Omega_{K}$,

  4. (4)

(4.2)
$\operatorname{rank}E_{2}(K)=\sum_{v\in\Sigma}\left(\sigma_{E_{1}}^{(v)}-\sigma_{E_{2}}^{(v)}\right),$

Then, for all $n\in\mathbb{Z}_{\geq 0}$, the Hilbert’s tenth problem has a
negative answer for the ring of integers of $K_{n}$.

###### Proof.

The result is deduced from Proposition 4.2. It follows from Corollary 4.6 that
the condition (6) of this Proposition is satisfied. We assume that the
conditions (1)-(5) of Proposition 4.2 are satisfied. Hence, the result
follows. ∎

###### Proof of Theorem 1.4.

Let $E_{1}$ and $E_{2}$ be elliptic curves which satisfy the conditions of the
Theorem 1.4. Note that by Proposition 3.5,
$\sigma_{E_{i}}^{(v)}=s_{v}d_{v}(E_{i})$. The conditions of Theorem 4.7 are
satisfied, and the result follows. ∎

##  5\. An explicit example

In this section, we construct an explicit congruence between two elliptic
curves $E_{1}$ and $E_{2}$ defined over $\mathbb{Q}$ for which the conditions
of Theorem 1.4 are satisfied. It suffices to check that the conditions of
Theorem 4.7 are satisfied. We shall take $K=\mathbb{Q}(\sqrt{-5})$ and $p=3$
and show that the Hilbert’s tenth problem has a negative answer in
$\mathcal{O}_{K_{n}}$ for all $n\geq 0$, in the anticyclotomic
$\mathbb{Z}_{3}$-extension of $K$. This will thus prove Theorem 1.5. Let
$E_{1}$ be 56b1 and $E_{2}$ be 392c1 with respect to Cremona’s labelling for
elliptic curves. Both elliptic curves are defined over $\mathbb{Q}$, with good
ordinary reduction at $3$. The elliptic curves $E_{1}$ and $E_{2}$ are
associated to normalized newforms
$f_{1}(z)=\sum_{n=1}^{\infty}a_{n}(f_{1})q^{n}$ and
$f_{2}(z)=\sum_{n=1}^{\infty}a_{n}(f_{2})q^{n}$ respectively of weight $2$ on
$\Gamma_{0}(392)$. Here, $q$ is taken to be $e^{2\pi iz}$. At $p=3$, the
residual representations for both $E_{1}$ and $E_{2}$ are surjective.

This can be checked using the following code on the SageMathCloud:

    E1=EllipticCurve(‘56b1’)
    E2=EllipticCurve(‘392c1’)
    rho1=E1.galois_representation()
    rho2=E2.galois_representation()
    rho1.image_type(3)
    rho2.image_type(3)

Next, we show that the elliptic curves $E_{1}$ and $E_{2}$ are $3$-congruent,
i.e., $E_{1}[3]\simeq E_{2}[3]$ as modules over
$\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. Since the residual Galois
representations at $p=3$ are irreducible, by the Brauer-Nesbitt theorem, it
suffices to show that

$a_{n}(f_{1})\equiv a_{n}(f_{2})\mod{3}$

for all $n$ coprime to $392$ ($=2^{3}\times 7^{2}$). We wish to apply a result
of Sturm on congruences between modular forms, and it conveniences us to
modify $f_{1}$ slightly. We set $g_{1}(z)\mathrel{\mathop{\mathchar
58\relax}}=f_{1}(z)-f_{1}(7z)$, and $g_{2}(q)\mathrel{\mathop{\mathchar
58\relax}}=f_{2}(z)$. Let $a_{n}(g_{i})$ denote the $n$-th Fourier coefficient
of $g_{i}$. Note that $a_{n}(g_{1})=a_{n}(f_{1})$ for all $n$ coprime to $7$.
Therefore, in order to show that

$a_{n}(f_{1})\equiv a_{n}(f_{2})\mod{3}$

for all $n$ coprime to $392$, it suffices to show that

$a_{n}(g_{1})\equiv a_{n}(g_{2})\mod{3}$

for all $n$. It follows from the result of Sturm [Stu06] that if this
congruence holds for all
$n\leq\frac{[\operatorname{SL}_{2}(\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}\Gamma_{1}(392)]}{12}+1$, then, it holds for all $n$. The index of
$\Gamma_{1}(392)$ is given by

$[\operatorname{SL}_{2}(\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}\Gamma_{1}(392)]=392^{2}\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{7^{2}}\right)=112896.$

It suffices to check the congruence $a_{n}(f_{1})\equiv a_{n}(f_{2})\mod{p}$
for all $n\leq 112897$. Continuing with the code on SageMath, the following
returns True for the value of the boolean variable v.

    v=True
    for n in range(112898):
        a=E1.an(n)
        b=E2.an(n)
        if n%7==0:
            m=n/7
            a2=E1.an(m)
            a=E1.an(n)-a2
        c=a-b
        d=c%3
        if d!=0:
            v=False

Note that $3$ splits in $K$. We check the conditions (1)-(5) of Proposition
4.2.

  1. (1)

Both $E_{1}$ and $E_{2}$ have good ordinary reduction at $\mathfrak{p}$ and
$\mathfrak{p}^{*}$,

  2. (2)

$E_{1}$ and $E_{2}$ are $3$-congruent, as has been checked by the code above.

  3. (3)

The following SageMath code verifies that $\operatorname{rank}E_{1}(K)=0$,
$\operatorname{rank}E_{2}(K)=2$:

        K = QuadraticField(-5, ‘a’)
    A1=E1.base_extend(K)
    A1.rank()
    A2=E2.base_extend(K)
    A2.rank()

  4. (4)

Since the residual representations

$\bar{\rho}_{i}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{Aut}(E_{i}[3])\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{F}_{3})$

are surjective, they remain irreducble when restricted to the index-$2$
subgroup $\operatorname{Gal}(\bar{\mathbb{Q}}/K)$. In particular, $E_{i}[p]$
does not contain a trivial submodule for the action of
$\operatorname{Gal}(\bar{\mathbb{Q}}/K)$. It follows that for $i=1,2$,
$E_{i}(K)[3]=0$.

  5. (5)

Both $E_{1}$ and $E_{2}$ have additive reduction at the prime above $2$ in
$K^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=K(\mu_{3})$, of Kodaira type
$\operatorname{I}_{1}^{*}$. Therefore, in accordance with Definition 3.1, the
prime above $2$ is not in $\Sigma$. We find that
$\Sigma\mathrel{\mathop{\mathchar
58\relax}}=\Sigma(E_{1})\cup\Sigma(E_{2})=\\{v_{7},v_{7}^{*}\\}$, where
$v_{7}$ and $v_{7}^{*}$ are the primes of $K$ that lie above $7$. The prime
$7$ splits in $K$, and hence the primes $v_{7}$ and $v_{7}^{*}$ are finitely
decomposed in $K_{\operatorname{anti}}$. The code is given below, and is a
continuation of our SageMath code. The field $L$ in the code below is
$K^{\prime}$.

        K.<a> = NumberField(x^2+5)
    R.<t> = K[]
    L.<b> = K.extension(t^2+t+1)
    A1=E1.base_extend(L)
    A1.local_data()
    A2=E2.base_extend(L)
    A2.local_data()

The conditions of Theorem 4.7 are satisfied, in greater detail,

  1. (1)

$\Sh(E_{1}/K)=0$ is predicted by the ConjecturalSha function in Magma. The
method for computing the order of $\Sh(E_{1}/K)$ relies on techniques for
approximating the L-value $L(E_{1/K},1)$, which is non-vanishing, and the
strong Birch and Swinnerton-Dyer formula relating this value to the order of
$\Sh(E_{1}/K)$. Although $E_{1/K}$ has rank $0$, and all that is required is
that $\Sh(E_{1}/K)[3^{\infty}]=0$, and not the vanishing of the full Tate-
Shafarevich group over $K$. Methods involving $3$-descent have been
implemented into code for elliptic curves over the rationals, but not for more
general number fields. The Magma code is given below:

        E:=EllipticCurve("56b1");
    R<x> := PolynomialRing(Integers());
    f := x^2+5;
    K<y> := NumberField(f);
    A:=BaseExtend(E,K);
    ConjecturalSha(A,Generators(A))

Let $K(E_{1}[3])$ be the number field obtained by adjoining the coordinates of
all points in $E_{1}[3]$ to $K$, or equivalently, the fixed field over $K$ of
the Galois representation on $E_{1}[3]$. We remark here that one possible
alternative approach to proving that $\Sh(E_{1}/K)[3^{\infty}]=0$ for a rank
$0$ elliptic curve $E_{1/K}$ would be to study the relationship with the
$3$-part of the class group of the field $K(E_{1}[3])$. The relationship
between Tate-Shafarevich groups and such class groups is studied in [PS21],
however, such results would potentially require some modification to our
setting. We do not pursue this approach in this paper. It would be challenging
to compute the $3$-primary class group of a large number field like
$K(E_{1}[3])$, within a reasonable period of time. However, class group
computations on Sage are unconditional.

  2. (2)

The prime $p=3$ splits in $K$, hence,
$\mathbb{F}_{\mathfrak{p}}\simeq\mathbb{F}_{3}$. We find that
$a_{3}(E_{1})=2$, hence, $\\#E_{1}(\mathbb{F}_{3})=3+1-2=2$. Thus,
$E_{1}(\mathbb{F}_{3})[3]=0$ and the condition is satisfied.

  3. (3)

The Tamagawa product for $E_{1}$ is $2$ and hence, no Tamagawa number is
divisible by $3$.

  4. (4)

The values $d_{v}(E_{2})=0$ for all $v\in\Sigma=\\{v_{7},v_{7}^{*}\\}$. This
is because $E_{2}$ has additive reduction at the primes $v_{7}$ and
$v_{7}^{*}$, and the polynomial $P_{v}^{E_{2}}(X)=1$ for
$v\in\\{v_{7},v_{7}^{*}\\}$. Thus, the quantity $\sigma_{E_{2}}^{(v)}=0$ for
$v\in\Sigma$. The same reasoning shows that $\sigma_{E_{2}}^{(v_{2})}=0$,
where $v_{2}$ is the prime of $K$ above $2$.

We now show that the algorithm [Bri07, Theorem 2, p. 2133] implies that the
primes $v_{7}$ and $v_{7}^{*}$ are totally inert in $K_{\operatorname{anti}}$.
This will show that $s_{v_{7}}=s_{v_{7}^{*}}=1$. In accordance with notation
from _loc. cit._ , set $\Delta\mathrel{\mathop{\mathchar 58\relax}}=5$, note
that $K=\mathbb{Q}(\sqrt{-\Delta})$. The number $h$ is the class number of
$K$, which we express as $h=3^{\mu}u$, where $3\nmid u$. The quantity $\mu$
should not be confused with the $\mu$-invariant. We thus find that $h=u=2$ and
$\mu=0$. Since $\Delta\not\equiv 3\mod{4}$, the algorithm [Bri07, Theorem 2,
p. 2133] asks that write $7^{2}=a^{2}+5b^{2}$, where $a$ and $b$ are non-zero
integers that are relatively prime. We find that $a=2$ and $b=3$. Set
$\omega\mathrel{\mathop{\mathchar 58\relax}}=\sqrt{-\Delta}=\sqrt{-5}$ and
find that

$(a+b\omega)^{3-1}=(2+3\sqrt{-5})^{2}=-41+12\sqrt{-5}=a^{*}+\omega b^{*},$

where $a^{*}=-41$ and $b^{*}=12$. The number $\nu$ is defined such that
$K_{H}\cap K_{\operatorname{anti}}=K_{\nu}$, where $K_{H}$ is the Hilbert
class field of $K$. Since the class number is prime to $p$, we find that
$\nu=0$. Part (a) of the aforementioned algorithm tells us that a prime
$v\in\\{v_{7},v_{7}^{*}\\}$ splits in $K_{n}$ if and only if

$b^{*}\equiv 0\mod{3^{(n+1+\mu-\nu)}}.$

Since both $\mu$ and $\nu$ are $0$, and $b^{*}=12$ is not divisible by
$3^{2}$, we find that $v$ must be inert in every field $K_{n}$, and thus,
inert in $K_{\operatorname{anti}}$.

Note that $E_{1}$ has split multiplicative reduction at both $v_{7}$ and
$v_{7}^{*}$. For $v\in\\{v_{7},v_{7}^{*}\\}$, we find that
$\widetilde{P_{v}^{E_{1}}}(X)=1-X\in\mathbb{Z}/3\mathbb{Z}[X]$, and thus,
$7^{-1}=1$ is a simple root in $\mathbb{Z}/3\mathbb{Z}$. As a result,
$d_{v}(E_{1})=1$, and we find that
$\sigma_{E_{1}}^{(v_{7})}=\sigma_{E_{1}}^{(v_{7}^{*})}=1$. We find that

$2=\operatorname{rank}E_{2}(K)=\sum_{v\in\Sigma}\left(\sigma_{E_{1}}^{(v)}-\sigma_{E_{2}}^{(v)}\right).$

###### Proof of Theorem 1.5.

We have checked that all conditions of Theorem 4.7 are satisfied for
$(E_{1},E_{2},K,p)$, and thus, the result follows. ∎

## References

  * [Bri07] David Brink.  Prime decomposition in the anti-cyclotomic extension.  Mathematics of Computation, 76(260):2127–2138, 2007.
  * [Coa00] John Henry Coates.  Galois cohomology of elliptic curves.  Lecture Notes at the Tata Institute of Fundamental Research No. 88, 2000.
  * [CPZ05] Gunther Cornelissen, Thanases Pheidas, and Karim Zahidi.  Division-ample sets and the diophantine problem for rings of integers.  Journal de théorie des nombres de Bordeaux, 17(3):727–735, 2005\.
  * [CV07] Christophe Cornut and Vinayak Vatsal.  Nontriviality of rankin-selberg l-functions and cm points.  London Mathematical Society Lecture Note Series, 320:121, 2007.
  * [Den80] Jan Denef.  Diophantine sets over algebraic integer rings. II.  Transactions of the American Mathematical Society, 257(1):227–236, 1980.
  * [DL78] Jan Denef and Leonard Lipshitz.  Diophantine sets over some rings of algebraic integers.  Journal of the London Mathematical Society, 2(3):385–391, 1978\.
  * [DPR61] Martin Davis, Hilary Putnam, and Julia Robinson.  The decision problem for exponential diophantine equations.  Annals of Mathematics, pages 425–436, 1961.
  * [GFP20] Natalia Garcia-Fritz and Hector Pasten.  Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and heegner points.  Mathematische Annalen, 377(3):989–1013, 2020.
  * [Gre01] Ralph Greenberg.  Introduction to Iwasawa theory for elliptic curves.  Arithmetic algebraic geometry (Park City, UT, 1999), 9:407–464, 2001.
  * [GV00] Ralph Greenberg and Vinayak Vatsal.  On the Iwasawa invariants of elliptic curves.  Inventiones Mathematicae, 142(1):17, 2000.
  * [Har00] Michael Harris.  Correction to p-adic representations arising from descent on abelian varieties.  Compositio Mathematica, 121(1):105–108, 2000.
  * [HKR21] Jeffrey Hatley, Debanjana Kundu, and Anwesh Ray.  Statistics for anticyclotomic iwasawa invariants of elliptic curves.  arXiv preprint arXiv:2106.01517, 2021.
  * [HL21] Pin-Chi Hung and Meng Fai Lim.  On the growth of mordell–weil ranks in $p$-adic lie extensions.  Asian Journal of Mathematics, 24(4):549–570, 2021.
  * [HM99] Yoshitaka Hachimori and Kazuo Matsuno.  An analogue of Kida’s formula for the selmer groups of elliptic curves.  Journal of Algebraic Geometry, 8(3):581–601, 1999.
  * [KR21] Debanjana Kundu and Anwesh Ray.  Statistics for Iwasawa invariants of elliptic curves.  Trans. Amer. Math. Soc., 374(11):7945–7965, 2021.
  * [KR22] Debanjana Kundu and Anwesh Ray.  Iwasawa invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and Kida’s formula.  Forum Math., 34(4):945–967, 2022.
  * [Mat70] Yuri Vladimirovich Matiyasevich.  The Diophantineness of enumerable sets.  In Doklady Akademii Nauk, volume 191, pages 279–282. Russian Academy of Sciences, 1970.
  * [MRL18] Barry Mazur, Karl Rubin, and Michael Larsen.  Diophantine stability.  American Journal of Mathematics, 140(3):571–616, 2018.
  * [Phe88] Thanases Pheidas.  Hilbert’s tenth problem for a class of rings of algebraic integers.  Proceedings of the American Mathematical Society, 104(2):611–620, 1988.
  * [Poo02] Bjorn Poonen.  Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers.  In International Algorithmic Number Theory Symposium, pages 33–42. Springer, 2002.
  * [PS21] Dipendra Prasad and Sudhanshu Shekhar.  Relating the Tate–Shafarevich group of an elliptic curve with the class group.  Pacific Journal of Mathematics, 312(1):203–218, 2021.
  * [Shl89] Alexandra Shlapentokh.  Extension of Hilbert’s tenth problem to some algebraic number fields.  Communications on Pure and Applied Mathematics, 42(7):939–962, 1989\.
  * [Shl07] Alexandra Shlapentokh.  Hilbert’s tenth problem: Diophantine classes and extensions to global fields.  Number 7. Cambridge University Press, 2007.
  * [Shl08] Alexandra Shlapentokh.  Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers.  Transactions of the American Mathematical Society, 360(7):3541–3555, 2008.
  * [Sil94] Joseph H Silverman.  Advanced topics in the arithmetic of elliptic curves, volume 151\.  Springer Science & Business Media, 1994.
  * [SS89] Harold N Shapiro and Alexandra Shlapentokh.  Diophantine relationships between algebraic number fields.  Communications on Pure and Applied Mathematics, 42(8):1113–1122, 1989.
  * [Stu06] Jacob Sturm.  On the congruence of modular forms.  In Number Theory: A Seminar held at the Graduate School and University Center of the City University of New York 1984–85, pages 275–280. Springer, 2006.
  * [Vid] C Videla.  Sobre el décimo problema de Hilbert.  Atas da Xa Escola de Algebra, Vitoria, ES, Brasil. Colecao Atas, 16:95–108.
  * [vO15] Jeanine van Order.  On the dihedral Euler characteristics of Selmer groups of abelian varieties.  Arithmetic and Geometry, 420:458, 2015.
  * [Was97] Lawrence C Washington.  Introduction to cyclotomic fields, volume 83.  Springer Science & Business Media, 1997.