A subfield $K$ of $\bar{\mathbb{Q}}$ is large if every smooth curve $C$ over $K$ with a $K$-rational point has infinitely many $K$-rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is big if for every positive integer $n$,$K$ contains a number field $F$ with $[F:\mathbb{Q}]$ divisible by $n$. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.
1. Introduction
Large fields (sometimes called ample fields) play a role in a number of basic conjectures regarding fields of algebraic numbers of infinite degree. For example, a conjecture of Shafarevich asserts that the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$, the maximal abelian extension of $\mathbb{Q}$, is isomorphic to the free profinite group with countably infinitely many generators. It follows from a theorem of Pop Reference 10, Theorem 2.1 that if $\mathbb{Q}^{\mathrm{ab}}$ is large, then Shafarevich’s conjecture holds. For more about large fields, see Section 5 below.
Recall that a supernatural number is a formal product
where $p$ runs through all rational primes and $0 \le n_p \le \infty$, with the obvious notion of divisibility. If $K/L$ is an algebraic extension of fields, then we define the degree of $K$ over $L$
$$\begin{equation*} [K:L] \coloneq \mathrm{lcm} \{[F:L] : \text{$F$ is a finite extension of $L$ in $K$}\}. \end{equation*}$$
The main result of this note is the following.
More precisely, we will exhibit a non-empty set $\mathcal{S}_0$ of elliptic curves over $\mathbb{Q}$ such that for every $E \in \mathcal{S}_0$, there are uncountably many big fields $K$ such that $E(K)$ has exactly one rational point.
2. $\mathcal{L}$-towers
Note that if $K_\infty /K$ is an $\mathcal{L}$-tower, then $[K_\infty :K] = \prod _i \ell _i$.
3. Selmer groups
If $F$ is a perfect field, $G_F$ will denote its absolute Galois group $\mathrm{Gal}(\bar{F}/F)$.
For this section fix a number field $K$, an elliptic curve $E$ defined over $K$, and a rational prime $\ell$ such that
Let $L$ be either $K$ itself or a cyclic extension of $K$ of degree $\ell$.
For every place $v$ of $K$, let $K_v$ denote the completion of $K$ at $v$, let $L_v$ denote the completion of $L$ at some place above $v$, and let $\mathcal{H}_\ell (L_v/K_v)$ be the “local condition” subspace of $H^1(K_v,E[\ell ])$ defined in Reference 6, Definition 7.1. We will not repeat the definition here, but the following four propositions list all the properties of the subspaces $\mathcal{H}_\ell (L_v/K_v)$ that we need.
If $v$ is nonarchimedean with residue characteristic different from $\ell$, and $E$ has good reduction at $v$, let $K_v^\mathrm{ur}$ denote the maximal unramified extension of $K_v$ and
Let $\Delta _E$ denote the discriminant of some Weierstrass model of $E$.
When $L = K$, Proposition 3.2(ii) shows that $\mathrm{Sel}(L/K,E[\ell ])$ is the standard $\ell$-Selmer group of $E/K$, and we denote it by $\mathrm{Sel}(K,E[\ell ])$.
When $\ell = [L:K] = 2$,$\mathrm{Sel}(L/K,E[\ell ])$ is the standard $2$-Selmer group of the quadratic twist $E^L/K$ (see Reference 6, Lemma 8.4).
These relative Selmer groups are useful to us because of the following proposition.
If $\Sigma$ is a finite set of places of $K$ containing all archimedean places, then the ring of $\Sigma$-integers of $K$ is
$$\begin{equation*} \mathcal{O}_{K,\Sigma } \coloneq \{x \in K : \text{$x \in \mathcal{O}_{K_v}$ for every $v \notin \Sigma $}\}. \end{equation*}$$
For $E \in \mathcal{S}$, define $T_E$ to be the compositum of all the fields $\mathbb{Q}(E[\ell ])$ for all primes $\ell$. Note that if $K$ is a number field and $K \cap T_E = \mathbb{Q}$, then for every prime $\ell$, the map $G_K \to \mathrm{Aut}(E[\ell ]) \cong \mathrm{GL}_2(\mathbb{F}_\ell )$ is surjective.
5. Additional remarks
Remark 5.3.
By Main Theorem A of Reference 12, Hilbert’s Tenth Problem has a negative answer for the ring of integers in any subfield $K$ of ${\bar{\mathbb{Q}}}$ satisfying
•
$K$ is totally real, and
•
there is an elliptic curve $E$ over $K$ such that $E(K)$ is finitely generated and has positive rank.
Applying Theorem 4.7 with the elliptic curve $E = 37.a1$ as in Remark 4.3, one can find uncountably many $\mathcal{L}$-towers$K_\infty /\mathbb{Q}$ with $K_\infty$ totally real and big such that $E(K_\infty ) = E(\mathbb{Q}) \cong \mathbb{Z}$. This gives uncountably many non-large big fields over whose ring of integers Hilbert’s Tenth Problem has a negative answer.
It is natural to ask whether there is any non-large field over whose ring of integers Hilbert’s Tenth Problem has a positive answer.
Acknowledgments
Theorem 1.3 answers a question that was brought to our attention by Arno Fehm at the American Institute of Mathematics workshop “Definability and decidability problems in number theory”, in May 2019. We are grateful to him for that and for additional helpful correspondence. We also thank the referee for very helpful comments.
There are (uncountably many) big fields that are not large.
Definition 2.1.
Suppose $K$ is a number field, and $\mathcal{L} \coloneq (\ell _1,\ell _2,\ldots )$ is a sequence of rational primes. The sequence $\mathcal{L}$ can be either infinite or finite, and the primes $\ell _i$ need not be distinct. We call an extension $K_\infty /K$ an $\mathcal{L}$-tower if there is a sequence of number fields $K = K_0 \subset K_1 \subset K_2 \subset \cdots$ such that
(i)
$K_\infty = \cup _i K_i$,
(ii)
$K_i/K_{i-1}$ is cyclic of degree $\ell _i$,
(iii)
for every $i>1$ there are primes $\mathfrak{p}_i, \mathfrak{p}'_i$ of degree $1$ of $K_i$, lying above the same prime of $K_{i-1}$, such that $\mathfrak{p}_i$ ramifies in $K_{i+1}/K_i$, but $\mathfrak{p}'_i$ does not.
Lemma 2.2.
Suppose the chain of number fields $K_{i-1} \subset K_i \subset K_{i+1}$ is part of an $\mathcal{L}$-tower$K_\infty /K$, in the notation of Definition 2.1. Then there is no Galois extension $F/K_{i-1}$ such that $FK_i = K_{i+1}$.
Lemma 2.3.
Suppose $\mathcal{L}$ is a sequence of rational primes, $K$ is a number field, and $K_\infty = \cup K_i$ is an $\mathcal{L}$-tower over $K$ as in Definition 2.1. Then:
(i)
for every $i \ge 0$, the maximal abelian extension of $K_i$ in $K_\infty$ is $K_{i+1}$,
(ii)
if $T$ is a Galois extension of $K$ (possibly of infinite degree) and $K_1 \cap T = K$, then $K_i \cap T = K$ for every $i \ge 1$.
Corollary 2.4.
Suppose $K$ is a number field and $K_\infty /K$ is an algebraic extension. There is at most one sequence of rational primes $\mathcal{L}$ and one sequence of fields $(K_0,K_1,\ldots )$ that exhibits $K_\infty /K$ as an $\mathcal{L}$-tower.
If $L_v = K_v$ then $\mathcal{H}_\ell (L_v/K_v)$ is the image of the Kummer map $E(K_v)/\ell E(K_v) \to H^1(K_v,E[\ell ]).$
Proposition 3.3.
If $v \nmid \ell \infty$,$E$ has good reduction at $v$, and $L_v/K_v$ is ramified, then $\mathcal{H}_\ell (L_v/K_v) \cap H^{1}_{\mathrm{ur}}(K_v,E[\ell ]) = 0$.
Proposition 3.4.
If $v \nmid \ell \infty$,$L_v/K_v$ is unramified, $E$ has good reduction at $v$, and $\phi _v \in \mathrm{Gal}(K_v^\mathrm{ur}/K)$ is the Frobenius generator, then:
the map $H^{1}_{\mathrm{ur}}(K_v,E[\ell ]) \to E[\ell ]/(\phi _v-1)E[\ell ]$ given by evaluating cocycles at $\phi _v$ is a well-defined isomorphism.
Proposition 3.5.
If $\ell = 2$,$v \nmid 2\infty$,$E$ has multiplicative reduction at $v$,$\mathrm{ord}_v(\Delta _E)$ is odd, and $L_v/K_v$ is unramified, then $\mathcal{H}_\ell (L_v/K_v) = \mathcal{H}_\ell (K_v/K_v)$.
Definition 3.6.
The relative Selmer group$\mathrm{Sel}(L/K,E[\ell ])$ is the subgroup of $H^1(K,E[\ell ])$ defined by
$$\begin{equation*} \mathrm{Sel}(L/K,E[\ell ]) \coloneq \{c \in H^1(K,E[\ell ]) : \text{$\mathrm{loc}_v(c) \in \mathcal{H}_\ell (L_v/K_v)$ for every $v$}\} \end{equation*}$$
where $\mathrm{loc}_v : H^1(K,E[\ell ]) \to H^1(K_v,E[\ell ])$ is the localization map at $v$.
Proposition 3.7.
(i)
If $\mathrm{Sel}(L/K,E[\ell ]) = 0$ then $\mathrm{rank}\,E(L) = \mathrm{rank}\,E(K)$.
(ii)
If $\mathrm{rank}\,E(L) = \mathrm{rank}\,E(K)$ and $L/K$ is ramified at two primes of good reduction for $E$ with different residue characteristics, then $E(L) = E(K)$.
Lemma 3.8.
Suppose that $c$ is a cocycle representing a nonzero class in $H^1(K,E[\ell ])$. Let $F = K(E[\ell ])$. The restriction of $c$ to $G_F$ induces a surjective homomorphism
The sets $\mathcal{P}_0$ and $\mathcal{P}_1$ have positive density.
(ii)
Suppose $\mathfrak{a}$ is an ideal of $\mathcal{O}_{K}$ such that $\mathrm{Sel}(K,E[\ell ])_\mathfrak{a}$ is nonzero. Then $\mathcal{P}_1(\mathfrak{a})$ has positive density, and if $\mathfrak{p}\in \mathcal{P}_1(\mathfrak{a})$ then$$\begin{equation*} \dim _{\mathbb{F}_{\ell }}\mathrm{Sel}(K,E[\ell ])_{\mathfrak{a}\mathfrak{p}} = \dim _{\mathbb{F}_{\ell }}\mathrm{Sel}(K,E[\ell ])_{\mathfrak{a}} - 1. \end{equation*}$$
Definition 3.12.
Suppose $T$ is a finite set of primes of $K$, disjoint from $\Sigma$. If $\ell =2$ let $\Sigma _0$ denote the subset of $\Sigma$ consisting of all primes $\mathfrak{p}$ where $E$ has multiplicative reduction and such that $\mathrm{ord}_\mathfrak{p}(\Delta _E)$ is odd. If $\ell >2$ let $\Sigma _0$ be the empty set. We say that an extension $L/K$ is $T$-ramified and $\Sigma$-split if
•
every $\mathfrak{p}\in T$ is ramified in $L/K$, every $\mathfrak{p} \notin T$ is unramified in $L/K$,
•
every $v \in \Sigma - \Sigma _0$ splits in $L/K$.
Proposition 3.13.
Let $r \coloneq \dim _{\mathbb{F}_{\ell }}\mathrm{Sel}(K,E[\ell ])$ and suppose $t \le r$.
(i)
There is a set of primes $T \subset \mathcal{P}_1$ of cardinality $t$ such that$$\begin{equation*} \dim _{\mathbb{F}_{\ell }}\mathrm{Sel}(K,E[\ell ])_\mathfrak{a} = r-t, \end{equation*}$$
where $\mathfrak{a} \coloneq {\prod _{\mathfrak{p}\in T}\mathfrak{p}}$.
(ii)
If $T$ satisfies (i), $T_0$ is a finite subset of $\mathcal{P}_0$, and $L/K$ is a cyclic extension of $K$ of degree $\ell$ that is $(T \cup T_0)$-ramified and $\Sigma$-split, then$$\begin{equation*} \dim _{\mathbb{F}_{\ell }}\mathrm{Sel}(L/K,E[\ell ]) = r-t. \end{equation*}$$
Suppose $T$ is a finite subset of $\mathcal{P}_0 \cup \mathcal{P}_1$. If $\ell = 2$ suppose further that $E$ has a prime $\mathfrak{q}\nmid 2$ of multiplicative reduction such that $\mathrm{ord}_\mathfrak{q}(\Delta _E)$ is odd. Then there are a finite subset $T_0 \subset \mathcal{P}_0$ and a cyclic extension $L/K$ of degree $\ell$ such that
(i)
$L/K$ is $(T \cup T_0)$-ramified and $\Sigma$-split,
(ii)
$K$ has a prime $\mathfrak{p}$ of degree $1$, unramified over $\mathbb{Q}$, such that $L/K$ is ramified at $\mathfrak{p}$ and unramified at all primes $\mathfrak{p}' \ne \mathfrak{p}$ with the same residue characteristic as $\mathfrak{p}$.
Lemma 4.2.
The set $\mathcal{S}_0$ is nonempty.
Remark 4.3.
The elliptic curve $67.a1$ in the proof of Lemma 4.2 has only one rational point. If we instead take $E$ to be the curve $37.a1$, then $E(\mathbb{Q}) \cong \mathbb{Z}$ and a similar argument shows that $E \in \mathcal{S}$.
Proposition 4.4.
Suppose $E \in \mathcal{S}$, and $K$ is a finite nontrivial extension of $\mathbb{Q}$ such that $K \cap T_E = \mathbb{Q}$ and $E(K) = E(\mathbb{Q})$. Then for each rational prime $\ell$ there are infinitely many cyclic extensions $L/K$ of degree $\ell$ such that:
(i)
$E(L) = E(K)$,
(ii)
$K$ has a prime $\mathfrak{p}$ of degree $1$, unramified over $\mathbb{Q}$, such that $L/K$ is ramified at $\mathfrak{p}$ and unramified at all primes $\mathfrak{p}' \ne \mathfrak{p}$ with the same residue characteristic as $\mathfrak{p}$,
(iii)
$L/K$ is ramified at (at least) two primes where $E$ has good reduction and that have different residue characteristics,
(iv)
every place of $K$ dividing $\ell \infty$ splits completely in $L/K$.
Lemma 4.5.
Suppose $E \in \mathcal{S}$. Then there is a real quadratic field $F$, ramified at $2$, such that $E(F) = E(\mathbb{Q})$ and $F \cap T_E = \mathbb{Q}$.
Suppose $E \in \mathcal{S}$. Then for every infinite sequence $\mathcal{L}$ of rational primes with $\ell _1 = 2$, there are uncountably many totally real $\mathcal{L}$-towers$K_\infty /\mathbb{Q}$ such that $E(K_\infty ) = E(\mathbb{Q})$.
Remark 5.1.
Let $K$ be the compositum of all $\mathbb{Z}_\ell$-extensions of $\mathbb{Q}$. Then $K$ is a big field, and we conjecture that $K$ is not large.
More precisely, let $E$ be an elliptic curve over $\mathbb{Q}$. In Reference 7, Conjecture 10.2 we conjecture that $E( K)$ is finitely generated. Thus combining Reference 7, Conjecture 10.2 with the following theorem of Fehm and Petersen leads to the conjecture that $K$ is not large.
Theorem 5.2 (Fehm and Petersen, Theorem 1.2 of Reference 8).
If $K \subset {\bar{\mathbb{Q}}}$ is a large field and $A$ is an abelian variety over $K$, then $A( K)$ has infinite rank.
Theorem 5.2 was first proved by Tamagawa in the case that $A$ is an elliptic curve (see the remark at the bottom of page 580 of Reference 8).
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