Hermite, Mordell-Weil, Siegel

Hugues Randriam

April 5, 2004

The aim of this paper is to present three finiteness or finite-generation theorems:

• Hermite’s theorem about number fields with given degree and ramification

• the Mordell-Weil theorem, concerning the group of rational points on abelian varieties

• Siegel’s theorem about integral points on curves of non-zero genus.

Our presentation follows mainly the one given in [2].

1 Hermite’s theorem

We begin with a weak form of Hermite’s finiteness theorem.

Proposition 1.1. There exists only finitely many number fields of given degree and discriminant.

Proof. Let n and D be two integers, and K a number field of degree n and discriminant D admitting r_{1} real embeddings σ_{1}, …, σ_{r_{1}} and 2r_{2} complex embeddings σ_{r_{1}+1}, …, σ_{r_{1}+r_{2}}, \overline{σ_{r_{1}+1}}, …, \overline{σ_{r_{1}+r_{2}}}, so that O_{K} can be viewed as a lattice in ℝ^{r_{1}} × ℂ^{r_{2}} via (σ_{1}, …, σ_{r_{1}+r_{2}}).

Suppose first r_{1} ≠ 0. Then by Minkowski’s theorem there exists a constant C (depending explicitely on n and D) such that there exists α ∈ O_{K} with

|σ_{1}(α)| ≤ C, |σ_{2}(α)| ≤ 1/2, …, |σ_{r_{1}+r_{2}}(α)| ≤ 1/2. (1)

Then by the product formula one also has |σ_{1}(α)| ≥ 1, so |σ_{1}(α)| ≠ |τ(α)| and hence σ_{1}(α) ≠ τ(α) for any embedding τ ≠ σ_{1}. This implies K = ℚ(α).

Now the (integral) coefficients of the minimal polynomial of such an α can be bounded in terms of n and C, so they can take only finitely many values. This in turn implies the finiteness result announced.

The case r_{1} = 0 can be treated in the same way by considering α ∈ O_{K} verifying | Re σ_{1}(α)| ≤ 1/2, | Im σ_{1}(α)| ≤ C, and |σ_{j}(α)| ≤ 1/2 for j ≥ 2 (remark that Im σ_{1}(α) is then non-zero, so that σ_{1}(α) ≠ \overline{σ_{1}(α)}).

Proposition 1.2 (Hensel). Let A be a Dedekind ring, K its quotient field, L a finite extension of K, B the integral closure of A in L, 𝔓 a non-zero prime ideal of B, and v_{𝔓} the associated discrete valuation. Let 𝔭 = 𝔓 ∩ O_{K} and e = v_{𝔓}(𝔭) the corresponding ramification index. Suppose furthermore that the residual extension is separable. Then the exponent of 𝔓 in the different 𝔇_{L/K} can be bounded as follows:

v_{𝔓}(𝔇_{L/K}) ≤ e − 1 + v_{𝔓}(e). (2)

Proof. We recall the proof from [1]. Without loss of generality one can suppose A and B complete local rings. After replacing K by its maximal unramified extension in L, one can also suppose L totally ramified over K, so [L : K] = e. In this situation there exists π ∈ B that satisfies an Eisenstein equation f(π) = 0, with

f(X) = ∑_{i=0}^{e} a_{i}X^{i}, a_{e} = 1, a_{e−1}, …, a_{0} ∈ 𝔭, a_{0} ∉ 𝔭^{2}, (3)

and such that B = A[π]. This last assertion implies that the different is generated by f′(π) = ∑_{i=1}^{e} ia_{i}π^{i−1}. Remark then that the e − 1 terms of this sum all have distinct valuation, so that

v_{𝔓}(𝔇_{L/K}) = inf_{1≤i≤e} v_{𝔓}(ia_{i}π^{i−1}) ≤ v_{𝔓}(ea_{e}π^{e−1}) = e − 1 + v_{𝔓}(e) (4)

since a_{e} = 1.

Theorem 1.3 (Hermite’s theorem). Let K be a number field, n an integer and S a finite set of places of K. Then K admits only finitely many extensions of degree n unramified out of S.

Proof. If L is such an extension, proposition 1.2 gives a bound for the different 𝔇_{L/K} at each element of S, and as the absolute discriminant of L can be expressed in terms of this different and of the absolute discriminant of K, the finiteness result follows from proposition 1.1.

2 The Mordell-Weil theorem

Theorem 2.1 (Chevalley-Weil). Let K be a number field, X and Y two smooth projective varieties over K, and h : X ⟶ Y an étale morphism. Then there exists a finite extension L of K such that Y (K) is contained in h(X(L)).

Proof. Using the openness of the étale locus and the projectivity hypothesis, one gets a finite set S of places of K, projective models 𝔛 and 𝔜 of X and Y over O_{S}, and an étale morphism 𝔥 : 𝔛 ⟶ 𝔜 that extends h. Then if Σ ⊂ 𝔜 is the Zariski closure of Q ∈ Y (K), and if P is the generic point of an irreducible component of the fiber product 𝔥^{−1}(Σ) = Σ ×_{𝔜} 𝔛, the field K(P) is an extension of K of degree less than the degree of h unramified out of S. By Hermite’s theorem there exists only finitely many such extensions, so they are all contained in their compositum. This compositum is the L we were looking for.

Remarks.

1. One in fact has proved the stronger result: h^{−1}(Y (K)) ⊂ X(L).

2. For non algebraic-geometry oriented readers, the preceding construction can be made a little more concrete by considering equations over K defining Y as a subvariety of some ℙ^{N}, and X as a subvariety of some ℙ^{M} over Y, the last ones with non-vanishing jacobian determinant. Then the set S of places to throw away are those occuring in the denominators of these equations and jacobian.

Proposition 2.2 (Weak Mordell-Weil theorem). Let K be a number field, A an abelian variety over K. Then for any integer m ≥ 1, the group A(K)/m.A(K) is finite.

Proof. Applying the Chevalley-Weil theorem with X = Y = A and h the multiplication-by-m map, one gets a finite extension L of K such that m.A(L) contains A(K). Without loss of generality one can suppose that L is Galois over K with group G. By construction the group of Galois invariants of m.A(L) is then the whole of A(K):

(m.A(L))^{G} = A(K). (5)

This and the long exact sequence in cohomology associated to the short exact sequence of G-modules

0 ⟶ A_{m} ⟶ A(L) ⟶^{m} m.A(L) ⟶ 0 (6)

gives an injection of A(K)/m.A(K) into the finite set H^{1}(G, A_{m}). This proves the proposition.

Remark. The preceding cohomological argument can be made a little more explicit as follows. Without loss of generality one can suppose that A(K) contains A_{m}. For any Q ∈ A(K), choose an m-th root P of Q in A(L). Then for any σ ∈ G, the difference P^{σ} − P lies in A_{m} and does not depend on the choice of P. The map σ ↦ P^{σ} − P is then a group homomorphism from G to A_{m}, and one easily checks that the map sending Q to this homomorphism gives a group homomorphism from A(K) to Hom(G, A_{m}) with kernel m.A(K), so that A(K)/m.A(K) embeds in the finite group Hom(G, A_{m}).

From now on we will suppose known the basic properties of the Néron-Tate height h̃ associated to a symmetric ample line bundle on A, in particular the fact that h̃ comes from a positive definite symmetric bilinear form on the (for the moment possibly infinite dimensional) real vector space A(\overline{K}) ⊗_{ℤ} ℝ.

Theorem 2.3 (Mordell-Weil). Let K be a number field, A an abelian variety over K, and Γ = A(K) its group of rational points. Then Γ is finitely generated.

Proof. Choose any symmetric ample line bundle on A, and define a norm |.| on Γ ⊗_{ℤ} ℝ by the formula |P |^{2} = h̃(P), where h̃ is the corresponding Néron-Tate height. For any real t, put

Γ_{t} = {P ∈ Γ | |P | ≤ t}, (7)

which is a finite set by Northcott’s theorem. Choose P_{1}, …, P_{n} ∈ Γ representatives of Γ/2Γ (which is finite by the weak Mordell-Weil theorem) and put

C = max{|P_{1}|, …, |P_{n}|, 1}. (8)

We claim that Γ is generated by the finite set Γ_{2C}. As Γ is the union of the Γ_{kC} for k ∈ ℕ, all we have to prove is that the subgroup < Γ_{2C} > generated by Γ_{2C} contains all the Γ_{kC}. This is obviously true for k = 2. Now we proceed by induction, supposing that < Γ_{2C} > contains Γ_{(k−1)C}, and taking P ∈ Γ_{kC}, where k ≥ 3. Now there is a unique P_{i} having the same class as P in Γ/2Γ, so that there exists Q ∈ Γ with P = 2Q + P_{i}. One then has

|Q| = \frac{1}{2}|P − P_{i}| ≤ \frac{1}{2}(|P | + |P_{i}|) ≤ \frac{k + 1}{2}C ≤ (k − 1)C. (9)

Thus Q ∈ Γ_{(k−1)C} ⊂< Γ_{2C} >, and P = 2Q + P_{i} ∈< Γ_{2C} > +Γ_{C} =< Γ_{2C} >, which proves the claim.

Remark. In fact, a careful analysis of the proof would lead to the fact that one even has Γ =< Γ_{C_{0}} > with C_{0} = max{|P_{1}|, …, |P_{n}|}.

3 Siegel’s Theorem

To begin with, we quote the following geometric reformulation of Roth’s approximation theorem:

Theorem 3.1. Let V be a smooth projective variety over a number field K, and h a height function associated to an ample line bundle L on V. Fix an embedding σ : K ↪ ℂ, choose a Riemannian metric on V_{σ}(ℂ), and denote by d_{σ} the corresponding distance. Then there exists δ > 0 such that for any α ∈ V (\overline{K}) and for any C > 0, there exists only finitely many ω ∈ V (K) with

d_{σ}(α, ω) ≤ Ce^{−δh(ω)}. (10)

Remark that the particular case V = ℙ^{1} (or more generally V = ℙ^{N}) with L = O(1) gives precisely the usual version of Roth’s theorem, where any δ > 2 is then convenient. The general case follows by using some power of L to embed V in some ℙ^{N}.

In the following theorem, we show that if V = A is an abelian variety, δ can be taken arbitrarily small.

Theorem 3.2. Let A be an abelian variety over a number field K, and h a height function associated to an ample line bundle L on A. Fix an embedding σ : K ↪ ℂ, choose a Riemannian metric on A_{σ}(ℂ), and denote by d_{σ} the corresponding distance. Then for any ε > 0, for any α ∈ A(\overline{K}) and for any C > 0, there exists only finitely many ω ∈ A(K) with

d_{σ}(α, ω) ≤ Ce^{−εh(ω)}. (11)

Proof. One can suppose that L is symmetric, that h is the associated Néron-Tate height, and that the metric is invariant by translation. Let δ > 0 be the constant given by the preceeding theorem, and choose an integer m such that εm^{2} > δ. Proceeding by contradiction, suppose there is an infinite sequence (ω_{n})_{n∈ℕ} of distinct elements of Γ = A(K) satisfying (11). By Northcott’s theorem the h(ω_{n}) tend to infinity, so that the ω_{n} converge to α (for the complex topology on A_{σ}(ℂ)).

By the weak Mordell-Weil theorem, there are infinitely many ω_{n} having the same class modulo mΓ, and after a translation one can assume this class is zero. Thus after extracting a subsequence one can suppose there exists ω′_{n} in Γ such that ω_{n} = mω′_{n} for all n. Extracting a subsequence again one can suppose the ω′_{n} converge to some α′ in A_{σ}(ℂ). One then has α = mα′, so that α′ lies in fact in A(\overline{K}). Now, since for all big enough n, d_{σ}(α′, ω′_{n}) = \frac{1}{m}d_{σ}(α, ω_{n}) and h(ω′_{n}) = \frac{1}{m^{2}} h(ω_{n}), one finds

d_{σ}(α′, ω′_{n}) = \frac{1}{m}d_{σ}(α, ω_{n}) ≤ \frac{C}{m}e^{−εh(ω_{n})} = \frac{C}{m}e^{−εm^{2}h(ω′_{n})} ≤ \frac{C}{m}e^{−δh(ω′_{n})}, (12)

which contradicts theorem 3.1.

Now we explain how this theorem of Diophantine approximation on abelian varieties can be used to get finiteness results for integral points on curves.

Theorem 3.3 (Siegel). Let K be a number field, C a smooth projective curve over K of genus g ≥ 1, and z : C ⟶ ℙ_{K}^{1} a rational function on C of degree D ≥ 1. Then there exists only finitely many P ∈ C(K) with z(P) ∈ O_{K}.

Proof. Suppose by contradiction that there is an infinite sequence P_{1}, P_{2}, … of distinct elements of C(K) with z_{i} = z(P_{i}) ∈ O_{K} for all i. This last condition implies that the height of z_{i} can be expressed as

h(z_{i}) = ∑_{τ:K↪ℂ}log^{+} |τ(z_{i})|, (13)

and then for each i there is a τ with log |τ(z_{i})| ≥ \frac{1}{[K:ℚ]}h(z_{i}). Thus after extracting a subsequence, one can suppose there is an embedding σ : K ↪ ℂ such that for all i,

log |σ(z_{i})| ≥ \frac{1}{[K : ℚ]}h(z_{i}). (14)

By compactness, after extracting again, one can suppose the P_{i} converge to some P in C_{σ}(ℂ). Remark that Northcott’s theorem implies that the terms in (14) tend to infinity, so that P is a pole of z, and hence lies in C(\overline{K}). Let e be the order of this pole.

Now choose an embedding of C in its Jacobian J, L a symmetric ample line bundle on J with associated Néron-Tate height h_{L}, and d_{σ} a distance on J_{σ}(ℂ), as in theorem 3.2. Then one has

log |σ(z_{i})| = −e log d_{σ}(P_{i}, P) + O(1) (15) and for any ε > 0,

h(z_{i}) ≥ (\frac{D}{deg L_{|C}} − ε) h_{L}(P_{i}) + O(1). (16)

Combining this with (14) one finds κ > 0 such that

− log d_{σ}(P_{i}, P) ≥ κh_{L}(P_{i}) (17)

for all i ≫ 0, which contradicts theorem 3.2.

Remark. In theorems 3.1 and 3.2 we restricted our attention to an archimedean place. However, analogous results still hold in a non-archimedean setting. Thus Siegel’s finiteness theorem generalizes to points with coordinates in O_{S} for any finite set of places S.

References

[1] J.-P. Serre. Corps Locaux. Hermann, Paris, 1968.

[2] J.-P. Serre. Lectures on the Mordell-Weil theorem (third edition). Vieweg & Sohn, Braunschweig, 1997.