Big fields that are not large

By Barry Mazur and Karl Rubin

Abstract

A subfield of is large if every smooth curve over with a -rational point has infinitely many -rational points. A subfield of is big if for every positive integer , contains a number field with divisible by . 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

Definition 1.1.

Following Pop Reference 10, we say that a field is large if for every smooth curve defined over , if the set of -rational points is non-empty then is infinite. Equivalently (see Reference 10, Proposition 1.1), is large if for every irreducible variety defined over with a smooth -rational point, is Zariski dense in .

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 , the maximal abelian extension of , 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 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 runs through all rational primes and , with the obvious notion of divisibility. If is an algebraic extension of fields, then we define the degree of over

Definition 1.2.

We say that a field is big if . Equivalently, is big if for every positive integer , contains a number field with divisible by .

The main result of this note is the following.

Theorem 1.3.

There are (uncountably many) big fields that are not large.

More precisely, we will exhibit a non-empty set of elliptic curves over such that for every , there are uncountably many big fields such that has exactly one rational point.

2. -towers

Definition 2.1.

Suppose is a number field, and is a sequence of rational primes. The sequence can be either infinite or finite, and the primes need not be distinct. We call an extension an -tower if there is a sequence of number fields such that

(i)

,

(ii)

is cyclic of degree ,

(iii)

for every there are primes of degree of , lying above the same prime of , such that ramifies in , but does not.

Note that if is an -tower, then .

Lemma 2.2.

Suppose the chain of number fields is part of an -tower , in the notation of Definition 2.1. Then there is no Galois extension such that .

Proof.

Suppose is Galois. By Definition 2.1 there are primes of , lying above the same prime of , such that ramifies in , but does not. Note that since is cyclic of prime degree and , must split completely in . Thus the diagram

shows that

Thus , and the lemma follows.

Lemma 2.3.

Suppose is a sequence of rational primes, is a number field, and is an -tower over as in Definition 2.1. Then:

(i)

for every , the maximal abelian extension of in is ,

(ii)

if is a Galois extension of (possibly of infinite degree) and , then for every .

Proof.

Suppose that there is an abelian extension of contained in and properly containing . Without loss of generality we may assume that is prime. Choose such that , and let be such that but . Since we have .

Since and is prime, we have . This contradicts Lemma 2.2 applied with and . This contradiction proves (i).

Now suppose is Galois with group , and . We will show by induction that for every . Suppose , and consider the diagram

We either have or . Suppose first that . Let , so is abelian of degree . Since , we have . This contradicts Lemma 2.2 applied with , so we conclude that . Thus

By induction this completes the proof of (ii).

Corollary 2.4.

Suppose is a number field and is an algebraic extension. There is at most one sequence of rational primes and one sequence of fields that exhibits as an -tower.

Proof.

Suppose is an -tower for some sequence . Applying Lemma 2.3(i) inductively shows that the sequence of fields is uniquely determined, and then the fact that determines .

3. Selmer groups

If is a perfect field, will denote its absolute Galois group .

For this section fix a number field , an elliptic curve defined over , and a rational prime such that

Let be either itself or a cyclic extension of of degree .

For every place of , let denote the completion of at , let denote the completion of at some place above , and let be the “local condition” subspace of 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 that we need.

If is nonarchimedean with residue characteristic different from , and has good reduction at , let denote the maximal unramified extension of and

Proposition 3.2.
(i)

.

(ii)

If then is the image of the Kummer map

Proof.

Assertion (i) holds because is a maximal isotropic subspace of for the local Tate pairing (see for example Reference 4, Proposition 4.4). Assertion (ii) is explained in Reference 6, Definition 7.1.

Proposition 3.3.

If , has good reduction at , and is ramified, then .

Proof.

This is Reference 6, Proposition 7.8(ii).

Proposition 3.4.

If , is unramified, has good reduction at , and is the Frobenius generator, then:

(i)

,

(ii)

,

(iii)

the map given by evaluating cocycles at is a well-defined isomorphism.

Proof.

Assertions (i) and (iii) are Reference 6, Lemma 7.3, and (ii) is Reference 6, Lemma 7.2.

Let denote the discriminant of some Weierstrass model of .

Proposition 3.5.

If , , has multiplicative reduction at , is odd, and is unramified, then .

Proof.

This is Reference 5, Lemma 2.10(iii).

Definition 3.6.

The relative Selmer group is the subgroup of defined by

where is the localization map at .

When , Proposition 3.2(ii) shows that is the standard -Selmer group of , and we denote it by .

When , is the standard -Selmer group of the quadratic twist (see Reference 6, Lemma 8.4).

These relative Selmer groups are useful to us because of the following proposition.

Proposition 3.7.
(i)

If then .

(ii)

If and is ramified at two primes of good reduction for with different residue characteristics, then .

Proof.

By Reference 6, Proposition 8.8,

Assertion (i) follows directly.

Suppose now that but properly contains . Then we can fix a point such that but for some rational prime . Since and is prime, we must have . But can ramify only at places of bad reduction and primes above Reference 13, Theorem VII.7.1, so this contradicts our assumption that ramifies at good primes with two distinct residue characteristics.

Lemma 3.8.

Suppose that is a cocycle representing a nonzero class in . Let . The restriction of to induces a surjective homomorphism

Proof.

Using Equation 3.1, the kernel of restriction is . Therefore the restriction of to is a nonzero homomorphism . Since is the restriction of a class defined over , we have that is -equivariant, and in particular the image of is stable under . By Equation 3.1 it follows that is surjective.

Definition 3.9.

If is an ideal of , define relaxed-at- and strict-at- Selmer groups, respectively, by

Note that

If is a finite set of places of containing all archimedean places, then the ring of -integers of is

Definition 3.10.

From now on let be a finite set of places of containing all places where has bad reduction, all places dividing , and large enough so that

the primes in generate the ideal class group of ,

the natural map is injective

(this is possible by Reference 1, Lemma 6.1). Define a set of primes of by

and define a partition of into disjoint subsets for by

(Equivalently by Proposition 3.4, .) If is an ideal of , let be the set of all such that the localization map

is nonzero.

The next proposition is a modification of Reference 6, Proposition 9.10.

Proposition 3.11.
(i)

The sets and have positive density.

(ii)

Suppose is an ideal of such that is nonzero. Then has positive density, and if then

Proof.

Let , and fix or . It follows from the surjection Equation 3.1 that . Fix such that .

Suppose is a prime of whose Frobenius conjugacy class in is the class of . Since fixes , we have that so and therefore by definition . By Proposition 3.4 we have

so . It follows from the Cebotarev Theorem that has positive density. This is (i).

Fix an ideal of and suppose that is a cocycle representing a nonzero element of . Let be as above. By Lemma 3.8, the restriction of to induces a surjective (and therefore nonzero) homomorphism

Since , we can find such that in . Then

Let be a Galois extension of containing and such that the restriction of to factors through . If is a prime whose Frobenius conjugacy class in is the class of , then by Proposition 3.4(iii) we have , so . Now the Cebotarev Theorem shows that has positive density.

If then we have an exact sequence

where the right-hand map is surjective because it is nonzero and the target space is one-dimensional. This completes the proof of (ii).

Definition 3.12.

Suppose is a finite set of primes of , disjoint from . If let denote the subset of consisting of all primes where has multiplicative reduction and such that is odd. If let be the empty set. We say that an extension is -ramified and -split if

every is ramified in , every is unramified in ,

every splits in .

The next proposition is a modification of Reference 6, Proposition 9.17.

Proposition 3.13.

Let and suppose .

(i)

There is a set of primes of cardinality such that

where .

(ii)

If satisfies (i), is a finite subset of , and is a cyclic extension of of degree that is -ramified and -split, then

Proof.

We will prove (i) by induction on . If , then is the empty set.

Suppose satisfies (i) for , and . Let . By Proposition 3.11(ii) we can find so that

Then satisfies (i) for . This proves (i).

Now suppose that satisfies (i), and let . Consider the exact sequences

By global duality (see for example Reference 3, Theorem 2.3.4), the images of the two right-hand maps in Equation 3.14 are orthogonal complements of each other under the sum of the local Tate pairings. By our choice of the lower right-hand map is surjective, so the upper right-hand map is zero, i.e.,

Let be a finite subset of , let , and suppose is a cyclic extension of of degree that is -ramified and -split. By Propositions 3.4(i) and 3.5, we have if . Thus by Definition 3.6, is the kernel of the map

We have for every by Propositions 3.2(i) and 3.4(ii) and the definition of , so in fact is the kernel of the map

By Proposition 3.3, for every . Combining Equation 3.15 and Equation 3.16 shows that , so by our choice of we have . This proves (ii).

Proposition 3.17.

Suppose is a finite subset of . If suppose further that has a prime of multiplicative reduction such that is odd. Then there are a finite subset and a cyclic extension of degree such that

(i)

is -ramified and -split,

(ii)

has a prime of degree , unramified over , such that is ramified at and unramified at all primes with the same residue characteristic as .

Proof.

Fix a prime of degree , unramified over , whose residue characteristic is different from the residue characteristics of all primes in (this is possible by Proposition 3.11(i)). Let .

Define a set of global Galois characters

and a set of tuples of local characters

where denotes the subset of ramified characters.

Suppose first that . Restriction gives a natural map of sets , and by Reference 2, Proposition 10.7 this map is surjective. Now take an element whose -component is the trivial character if is different from , and such that is ramified. Let be any character that restricts to , and let be the set of primes (necessarily in ) not in where is ramified. If is the cyclic extension of corresponding to , then is -ramified and -split, and ramifies at but not at any other prime above . This proves the proposition when .

When the proof is similar, except that the map is not surjective. However in this case, Reference 2, Proposition 10.7 shows that the image of contains either our chosen , or else , where is if , and is the nontrivial unramified quadratic character (where is the given prime of multiplicative reduction). The proof now proceeds exactly as in the case of odd , using a character that maps to either or .

4. Proof of Theorem 1.3

Definition 4.1.

Let be the set of all elliptic curves over satisfying all of the following properties:

for every prime , the map is surjective,

has discriminant ,

has a prime of multiplicative reduction such that is odd.

Let .

Lemma 4.2.

The set is nonempty.

Proof.

Let be the elliptic curve labelled in the -functions and Modular Forms Database Reference 9. We will show that .

The surjectivity of the map for every is stated in Reference 9, curve ; alternatively this follows directly from Reference 11, Proposition 21 and the fact that has no rational isogenies Reference 9, curve . The curve has multiplicative reduction at , and its discriminant is congruent to . Thus , and so .

Remark 4.3.

The elliptic curve in the proof of Lemma 4.2 has only one rational point. If we instead take to be the curve , then and a similar argument shows that .

For , define to be the compositum of all the fields for all primes . Note that if is a number field and , then for every prime , the map is surjective.

Proposition 4.4.

Suppose , and is a finite nontrivial extension of such that and . Then for each rational prime there are infinitely many cyclic extensions of degree such that:

(i)

,

(ii)

has a prime of degree , unramified over , such that is ramified at and unramified at all primes with the same residue characteristic as ,

(iii)

is ramified at (at least) two primes where has good reduction and that have different residue characteristics,

(iv)

every place of dividing splits completely in .

Proof.

Let be a finite subset of satisfying Proposition 3.13(i) with . Let be a finite subset of containing and at least two primes in with different residue characteristics. Now apply Proposition 3.17, with the set in place of , to produce a cyclic extension that is -ramified and -split for some .

By Proposition 3.13(ii) we have , and then Proposition 3.7 shows that . Assertion (ii) is Proposition 3.17(ii), and (iii) and (iv) follow from our choice of and Definition 3.12 of -split”, since all places dividing are in .

Thus has the desired properties. By varying the set we can produce infinitely many such .

Lemma 4.5.

Suppose . Then there is a real quadratic field , ramified at , such that and .

Proof.

If is an elliptic curve over and , let denote the quadratic twist of corresponding to . Then and

Fix an elliptic curve . Apply Proposition 4.4 to the curve with and to get a quadratic extension satisfying (i) through (iv) of Proposition 4.4. Write with a squarefree integer , and put .

Applying Equation 4.6 with and yields

By Proposition 4.4(i) we have , so . Applying Equation 4.6 with and yields

so . By Proposition 4.4(iii,iv) is positive, odd, and divisible by at least primes where has good reduction. Thus is a real field, ramified at , and by Proposition 3.7(ii).

Since the discriminant , the quadratic field is unramified at , so is tamely ramified in . Since is odd, has good reduction at , so is unramified in for all odd primes Reference 13, Theorem VII.7.1. Hence is tamely ramified in , but is wildly ramified in , so .

Theorem 4.7.

Suppose . Then for every infinite sequence of rational primes with , there are uncountably many totally real -towers such that .

Proof.

We build an -tower inductively as follows. Let and let be a field as in Lemma ]4.5. Now for each apply Proposition 4.4 to produce a cyclic extension of degree such that .

Since , Lemma 2.3(ii) shows that and we can continue by induction. The result is an -tower such that . Since at each step Proposition 4.4 provides us with infinitely many possible , and different choices give rise to distinct -towers by Corollary 2.4, we obtain uncountably many -towers in this way.

Proof of Theorem 1.3.

Let be a sequence of rational primes such that

, and

every prime occurs infinitely many times in .

If is an -tower, then is a big field. We can apply Theorem 4.7, taking for the curve from the proof of Lemma 4.2, to produce uncountably many -towers with ; such a is a big field that is not large. This concludes the proof of Theorem 1.3.

5. Additional remarks

Remark 5.1.

Let be the compositum of all -extensions of . Then is a big field, and we conjecture that is not large.

More precisely, let be an elliptic curve over . In Reference 7, Conjecture 10.2 we conjecture that is finitely generated. Thus combining Reference 7, Conjecture 10.2 with the following theorem of Fehm and Petersen leads to the conjecture that is not large.

Theorem 5.2 (Fehm and Petersen, Theorem 1.2 of Reference 8).

If is a large field and is an abelian variety over , then has infinite rank.

Theorem 5.2 was first proved by Tamagawa in the case that is an elliptic curve (see the remark at the bottom of page 580 of Reference 8).

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 of satisfying

is totally real, and

there is an elliptic curve over such that is finitely generated and has positive rank.

By Theorem 5.2, such a is not a large field.

Applying Theorem 4.7 with the elliptic curve as in Remark 4.3, one can find uncountably many -towers with totally real and big such that . 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.

Mathematical Fragments

Theorem 1.3.

There are (uncountably many) big fields that are not large.

Definition 2.1.

Suppose is a number field, and is a sequence of rational primes. The sequence can be either infinite or finite, and the primes need not be distinct. We call an extension an -tower if there is a sequence of number fields such that

(i)

,

(ii)

is cyclic of degree ,

(iii)

for every there are primes of degree of , lying above the same prime of , such that ramifies in , but does not.

Lemma 2.2.

Suppose the chain of number fields is part of an -tower , in the notation of Definition 2.1. Then there is no Galois extension such that .

Lemma 2.3.

Suppose is a sequence of rational primes, is a number field, and is an -tower over as in Definition 2.1. Then:

(i)

for every , the maximal abelian extension of in is ,

(ii)

if is a Galois extension of (possibly of infinite degree) and , then for every .

Corollary 2.4.

Suppose is a number field and is an algebraic extension. There is at most one sequence of rational primes and one sequence of fields that exhibits as an -tower.

Equation (3.1)
Proposition 3.2.
(i)

.

(ii)

If then is the image of the Kummer map

Proposition 3.3.

If , has good reduction at , and is ramified, then .

Proposition 3.4.

If , is unramified, has good reduction at , and is the Frobenius generator, then:

(i)

,

(ii)

,

(iii)

the map given by evaluating cocycles at is a well-defined isomorphism.

Proposition 3.5.

If , , has multiplicative reduction at , is odd, and is unramified, then .

Definition 3.6.

The relative Selmer group is the subgroup of defined by

where is the localization map at .

Proposition 3.7.
(i)

If then .

(ii)

If and is ramified at two primes of good reduction for with different residue characteristics, then .

Lemma 3.8.

Suppose that is a cocycle representing a nonzero class in . Let . The restriction of to induces a surjective homomorphism

Proposition 3.11.
(i)

The sets and have positive density.

(ii)

Suppose is an ideal of such that is nonzero. Then has positive density, and if then

Definition 3.12.

Suppose is a finite set of primes of , disjoint from . If let denote the subset of consisting of all primes where has multiplicative reduction and such that is odd. If let be the empty set. We say that an extension is -ramified and -split if

every is ramified in , every is unramified in ,

every splits in .

Proposition 3.13.

Let and suppose .

(i)

There is a set of primes of cardinality such that

where .

(ii)

If satisfies (i), is a finite subset of , and is a cyclic extension of of degree that is -ramified and -split, then

Equation (3.14)
Equation (3.15)
Equation (3.16)
Proposition 3.17.

Suppose is a finite subset of . If suppose further that has a prime of multiplicative reduction such that is odd. Then there are a finite subset and a cyclic extension of degree such that

(i)

is -ramified and -split,

(ii)

has a prime of degree , unramified over , such that is ramified at and unramified at all primes with the same residue characteristic as .

Lemma 4.2.

The set is nonempty.

Remark 4.3.

The elliptic curve in the proof of Lemma 4.2 has only one rational point. If we instead take to be the curve , then and a similar argument shows that .

Proposition 4.4.

Suppose , and is a finite nontrivial extension of such that and . Then for each rational prime there are infinitely many cyclic extensions of degree such that:

(i)

,

(ii)

has a prime of degree , unramified over , such that is ramified at and unramified at all primes with the same residue characteristic as ,

(iii)

is ramified at (at least) two primes where has good reduction and that have different residue characteristics,

(iv)

every place of dividing splits completely in .

Lemma 4.5.

Suppose . Then there is a real quadratic field , ramified at , such that and .

Equation (4.6)
Theorem 4.7.

Suppose . Then for every infinite sequence of rational primes with , there are uncountably many totally real -towers such that .

Remark 5.1.

Let be the compositum of all -extensions of . Then is a big field, and we conjecture that is not large.

More precisely, let be an elliptic curve over . In Reference 7, Conjecture 10.2 we conjecture that is finitely generated. Thus combining Reference 7, Conjecture 10.2 with the following theorem of Fehm and Petersen leads to the conjecture that is not large.

Theorem 5.2 (Fehm and Petersen, Theorem 1.2 of Reference 8).

If is a large field and is an abelian variety over , then has infinite rank.

Theorem 5.2 was first proved by Tamagawa in the case that is an elliptic curve (see the remark at the bottom of page 580 of Reference 8).

References

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Article Information

MSC 2020
Primary: 11R04 (Algebraic numbers; rings of algebraic integers), 11U05 (Decidability (number-theoretic aspects)), 14G05 (Rational points)
Author Information
Barry Mazur
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
mazur@g.harvard.edu
ORCID
MathSciNet
Karl Rubin
Department of Mathematics, UC Irvine, Irvine, California 92697
krubin@uci.edu
MathSciNet
Communicated by
Romyar T. Sharifi
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 7, Issue 14, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , , , and published on .
Copyright Information
Copyright 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
Article References
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  • DOI 10.1090/bproc/57
  • MathSciNet Review: 4173816
  • Show rawAMSref \bib{4173816}{article}{ author={Mazur, Barry}, author={Rubin, Karl}, title={Big fields that are not large}, journal={Proc. Amer. Math. Soc. Ser. B}, volume={7}, number={14}, date={2020}, pages={159-169}, issn={2330-1511}, review={4173816}, doi={10.1090/bproc/57}, }

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