# A $p$-adic approach to rational points on curves

By Bjorn Poonen

## Abstract

In 1922 Mordell conjectured the striking statement that, for a polynomial equation , if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983 and again by a different method by Vojta in 1991. But neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of -adic Galois representations; this is the subject of the present exposition.

## 1. The Mordell conjecture

### 1.1. Rational points on curves

The equation has infinitely many solutions in integers satisfying . Equivalently, the circle has infinitely many rational points (, , etc.). This can be understood geometrically: each line through with rational slope intersects the circle at one other point, which must have rational coordinates since finding its coordinates amounts to solving a quadratic equation over for which one rational root is already known; see Figure 1. The same argument shows that any nonsingular conic section defined by a polynomial with rational coefficients having one rational point has infinitely many.

In contrast, Fermat proved that the equation has no positive integer solutions. Equivalently, the set of rational points on the plane curve is . What about ? It turns out that it too has only finitely many rational points. (They are and Reference FW01.) More generally, for any fixed and , the curve has only finitely many rational points. All these finiteness claims are instances of the Mordell conjecture, which states that a тАЬcomplicated enoughтАЭ curve has only finitely many rational points, if any at all.

In the previous paragraph, the condition is what made the curve тАЬcomplicated enoughтАЭ. To state the Mordell conjecture fully, however, we need to consider also curves defined by several polynomials in higher-dimensional space and to introduce the notion of genus to measure their geometric complexity.

### 1.2. Projective space

Let be a field and let . The set of -points on -dimensional affine space is .

Define an equivalence relation on by for all . Let denote the equivalence class of . The set of all such equivalence classes is the set

of -points on -dimensional projective space.

The points with have a unique representative of the form , so they form a copy of . For each , the same holds for the points with . Moreover, is the union of these overlapping copies of .

One advantage of projective space over affine space is that is compact for the topology coming from the Euclidean topology on each ; similarly, is compact. Related to this is that intersection theory works better in projective space: for example, two distinct lines in always meet in exactly one point.

### 1.3. Projective varieties

A finite list of polynomials defines an affine varietyтБаFootnote1 whose set of -points is

1

Some people require a variety to satisfy additional conditions, such as not being a union of two strictly smaller such varieties.

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But for a point , a polynomial condition does not necessarily make sense; to make sure that it is unchanged by scaling , we assume that is homogeneous, a sum of monomials of the same total degree, such as of degree . A finite list of homogeneous polynomials defines a projective variety whose set of -points is

The decomposition of as a union of copies of restricts to express as a union of affine varieties called affine patches. For each , dehomogenizing by setting equal to gives polynomials cutting out the th affine patch in .

### 1.4. Smooth varieties

If a variety is defined by such that for every field extension and point , the matrix has rank , then we call obviously smooth of dimension ; the rank condition is the same as the Jacobian criterion in the implicit function theorem. More generally, any affine or projective variety is called smooth of dimension if (in a sense we will not make precise) it can be covered by subvarieties isomorphic to obviously smooth varieties as above.

If is smooth of dimension over , then is a smooth -manifold of dimension . The same holds if is replaced by in all three places.

### 1.5. Genus of a curve

From now on, we consider a smooth projective curve over , that is, a projective variety over that is smooth of dimension . We assume, moreover, that is geometrically connected, meaning that the variety defined by the same polynomials over an algebraically closed extension field (such as ) is nonempty and not the disjoint union of two strictly smaller varieties. Then is a compact connected one-dimensional -manifold, that is, a compact Riemann surface. Forgetting the complex structure, we find that is a compact connected oriented two-dimensional real manifold; by the classification of such, is homeomorphic to a -holed torus for some . The integer is called the genus of . It measures the geometric complexity of .

### 1.6. The conjecture

Several proofs of the Mordell conjecture are known, none of them easy:

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Faltings Reference Fal83 proved the conjecture in 1983 using methods from Arakelov theory, a kind of arithmetic intersection theory that combines number-theoretic data with complex-analytic data.

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Vojta Reference Voj91 gave a completely different proof based on diophantine approximation, a theory whose original goal was to quantify how closely irrational algebraic numbers such as could be approximated by rational numbers with denominator of at most a certain size. For a more elementary variant of VojtaтАЩs proof due to Bombieri, see Reference Bom90 or Reference HS00.

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Lawrence and Venkatesh Reference LV19 recently gave yet another proof. Their proof shares some ingredients with FaltingsтАЩs but replaces the most difficult steps by arguments involving -adic Hodge theory. The rest of this article is devoted to explaining some of the ideas underlying their proof.

## 2. Overall strategy of the LawrenceтАУVenkatesh proof

Here let us outline the strategy of Lawrence and Venkatesh, while postponing definitions and details to later sections.

Let be a smooth projective geometrically connected curve of genus over . Lawrence and Venkatesh use two maps of sets

where each of the last two sets is really a set of isomorphism classes.

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The map sends a rational point to a curve over ; the curves are the fibers of a surjective morphism for some two-dimensional variety defined in section 3. (Fiber means the inverse image of a point. stands for Kodaira and Parshin, who constructed certain for studying the Mordell conjecture Reference Par71.)

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The map sends each curve to its ├йtale cohomology; see section 4.

Let be the composition of the two maps. To complete the proof that is finite, Lawrence and Venkatesh prove that has finite image and finite fibers; see section 5.

## 3. A family of curves

In this section, we construct the algebraic family of curves .

### 3.1. Fundamental group of a punctured Riemann surface

For now, let be a compact Riemann surface of genus . Because is homeomorphic to a -gon with edges glued appropriately, the SeifertтАУvan Kampen theorem implies that the fundamental group of (with respect to any basepoint) has a presentation

where ; that is, is the quotient of a free group on generators by the smallest normal subgroup containing the indicated product of commutators. More generally, if is a finite subset of of size , then

### 3.2. Analytic construction of a family of ramified covers

Now fix and a finite group . Let . A surjective homomorphism defines a finite covering space of , and it can be completed to a finite ramified covering , with some branches possibly coming together above .

This covering depends on , but there are only finitely many since is finitely generated. To obtain a space not depending on a choice of any one , define the finite disjoint union , which is a disconnected ramified covering of .тБаFootnote3 As varies, the vary continuously in a family. The total space of this family is a two-dimensional compact complex manifold with a proper submersion such that for each ; see Figure 2.

3

Lawrence and Venkatesh use a variant in which has trivial center and the disjoint union is over conjugacy classes of surjective homomorphisms ; this makes sense since the isomorphism type of depends only on the conjugacy class.

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### 3.3. An algebraic family of curves

The constructions above can be made algebraic, in the following sense. Suppose that is a smooth projective connected curve over . Then by the Riemann existence theorem, arises from an algebraic morphism of algebraic curves. Moreover, there is a two-dimensional variety with a morphism whose fibers are the disconnected curves .

Even better, the construction is canonical enough that if is defined over , then can be defined over . This is called a KodairaтАУParshin family; see Reference LV19, ┬з7 for details.

## 4. Galois representations

The LawrenceтАУVenkatesh proof makes essential use of -adic Galois representations. Therefore, in this section we define , define the absolute Galois group of a field, and give examples and properties of -representations of the absolute Galois group of .

### 4.1. The field of $p$-adic numbers

Let be a prime number. The ring of -adic integers is the inverse limit . Thus an element of is a sequence where the elements are compatible in the sense that the natural homomorphism maps to for each . For example,

As a ring, is a domain of characteristic . Its fraction field, denoted , is called the field of -adic numbers.

For each , the homomorphism sending to is surjective with kernel . The kernel of is the unique maximal ideal of . The collection of subsets for all and is a basis of a topology on . Equip with the unique topology making it a topological group having as an open subgroup.

Working with or amounts to working with infinitely many congruences at once, but passing to the limit has advantages. One is that one can work over a domain or field of characteristic . Another is that some ideas from analysis over have analogues for .

Whereas number fields such as are examples of what are called global fields, is an example of a local field. For a more detailed introduction to -adic numbers, see Reference Kob84.

### 4.2. The absolute Galois group of $\mathbb{Q}$

A complex number is algebraic over if it is a zero of some nonzero polynomial in . The set of all algebraic numbers is a subfield of , called an algebraic closure of .

Now let be a subfield of . Call a finite extension if is finite. Call a Galois extension if it is generated by the set of all zeros of some collection of polynomials in .тБаFootnote4 For example, is not a Galois extension of , but is. The field is the union of its finite Galois subextensions .

4

For a definition that works over an arbitrary ground field instead of , one should require each polynomial to have distinct zeros in .

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For a Galois extension , the Galois group is the set of automorphisms of that fix pointwise.тБаFootnote5 The absolute Galois group of is . Each automorphism of restricts to give an automorphism of each finite Galois subextension , and any compatible collection of such automorphisms defines an automorphism of , so

5

Fixing pointwise is automatic; this condition becomes relevant only over other ground fields.

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Just as the inverse limit had a topology, the inverse limit has a topology.

### 4.3. Global $p$-adic Galois representations

Let be a finite-dimensional -vector space. If , then , which has a topology coming from the topology of . Call a -linear action of on continuous if the homomorphism defined by the action is continuous. By a -representation of we mean a finite-dimensional -vector space equipped with a continuous action of . In the next few sections, we give examples of such representations arising in number theory and arithmetic geometry.

### 4.4. The cyclotomic character

Let be a positive integer. Define

which under multiplication is a cyclic group of order . Thus is a free -module of rank . The group acts on the group .

Now fix a prime , and let range through the powers of . Form the inverse limit

with respect to the homomorphisms sending to . Then is a free rank  module under the ring , and acts on .

Next let

Then is a one-dimensional -vector space, and acts on . It follows from the definitions that the action is continuous, so is a one-dimensional -representation of ; it is called the cyclotomic character.

### 4.5. Galois representations associated to elliptic curves

Let be an elliptic curve over . It turns out that is a group variety; in particular, there is a map of varieties making an abelian group. If , we may use this group law to define and so on. For each , it turns out that the -torsion subgroup

is a free -module of rank . Therefore the inverse limit

(with respect to the homomorphisms sending to ) is a free -module of rank , called a Tate module. Next,

is a two-dimensional -vector space. The continuous action of on induces continuous actions on , , and . Thus is a two-dimensional -representation of .

### 4.6. Galois representations associated to higher-genus curves

Let be a smooth projective geometrically connected curve of genus over . If , there is no group law , but the Jacobian of does have a group law. The construction of generalizes to produce a -dimensional -representation of .

### 4.7. Galois representations from ├йtale cohomology

If is a smooth projective variety over and , then the ├йtale cohomology group (which we will not attempt to define here) is a -representation of .

### 4.8. Semisimple representations

Let be a -representation of . Call irreducible if and there is no -invariant subspace with . Call semisimple if it is a direct sum of irreducible representations. MaschkeтАЩs theorem Reference Ser77, ┬з1.4, Theorem 2 states that any -representation of a finite group is semisimple, but this is not true for -representations of a finite group of order divisible by , and -representations of are more like the latter in this regard: they need not be semisimple.