CHAPTER 2

WEIL FUNCTIONS

Paul Vojta

University of California, Berkeley

1 March 2021

Weil functions are a way of giving the theory of heights more structure at each place v ∈ M_{k}. (We can do this with even more precision via models as in Chapter 3, but that will come later.) Of course the height h_{D,k} depends only on the linear equivalence class. But choosing one divisor D in the linear equivalence class allows us to define local components λ_{D,v} of the height h_{D,k} so that

h_{D,k}(P) = \frac{1}{[L : k]} ∑_{v∈M_{L}} λ_{D,v}(P)

for all finite extensions L of k and all P ∈ X(L) \ Supp D.

In contrast to other treatments of Weil functions (e.g., Lang [L 1] and [L 2], or Hindry and Silverman [H–S]), this chapter will work over the completed algebraic closures ℂ_{v} of the completed fields. Everything will still work over k̄_{v} or even just k_{v}. Recall that the algebraic closure \overline{ℚ}_{p} is not complete, but that its completion is both complete and algebraically closed ([K], Ch. III, § 4). More generally, if k is a number field, a function field, or the completion of such a field at one of its given places, and v is a non-archimedean place of k, then the methods of loc. cit. imply that the completion of the algebraic closure k̄_{v} of the completion k_{v} of k is both complete and algebraically closed; we denote this field by ℂ_{v}. If v is archimedean, then we let ℂ_{v} = ℂ. In both cases, we extend the norm ∥ · ∥_{v} to ℂ_{v}, and will often drop the subscript when working over a completed field, since in that case there is only one place v.

Throughout this chapter, k is a number field, a function field, or the completion of such a field at one of its given places. Also, all schemes are assumed to be separated and of finite type over k.

We could also have required that schemes be reduced, but this would not make any difference since we are always looking at the set of ℂ_{v}-valued points.

§1. The v-topology In this section we define the v-topology on X(ℂ_{v}) (where v ∈ M_{k}), and compare it with the Zariski topology.

Definition 1.1. Let X be a scheme (over k). Then

X(M) = ∐_{v∈M_{k}} X(ℂ_{v}).

If P ∈ X(M) then we denote by v(P) the place v ∈ M_{k} such that P ∈ X(ℂ_{v}) .

Definition 1.2. Let X be a scheme and let v ∈ M_{k}. The v-topology on X(ℂ_{v}) is the coarsest topology such that for all Zariski-open U ⊆ X and all f ∈ 𝒪(U) the subset U(ℂ_{v}) is open, and the induced function f_{v}: U(ℂ_{v}) → ℂ_{v} is continuous. The topology on X(M) is the topology such that U ⊆ X(M) is open if and only if U ∩ X(ℂ_{v}) is open for all v ∈ M_{k}.

By definition this topology is finer than the Zariski topology; if v is archimedean, then it coincides with the classical topology on the complex space X(ℂ_{v}) . If X is affine and embedded into 𝔸^{n}, then the v-topology on X(ℂ_{v}) coincides with its relative topology as a subset of ℂ_{v}^{n}.

Lemma 1.3. Let U be an open dense subset of a scheme X and let v ∈ M_{k}. Then the set U(ℂ_{v}) is dense in the v-topology on X(ℂ_{v}) .

Proof. First consider the case X = 𝔸^{n}, and assume that U(ℂ_{v}) is not dense. After shrinking U , we may assume that U is a principal open set D(f) for some nonzero f ∈ k[x_{1}, …, x_{n}] . Then there exists P ∈ ℂ_{v}^{n} and r ∈ ℝ_{>0} such that f(Q) = 0 for all Q ∈ ℂ_{v}^{n} such that ∥x_{i}(Q) − x_{i}(P)∥ < r for all i. If v is archimedean, then this already contradicts basic results of complex analysis, so we may assume that v is nonarchimedean. After a linear change of coordinates we may assume that P = (0, …, 0) and that r > 1 . We may also assume that all coefficients of f lie in the valuation ring of ℂ_{v} and that not all of them lie in the maximal ideal. Then, letting 𝔽 denote the residue field (which is algebraically closed), the polynomial f reduces to a nonzero polynomial f̄ ∈ 𝔽[x_{1}, …, x_{n}] . But the above assumptions imply that f̄ vanishes identically on 𝔽^{n}, contradicting Hilbert’s Nullstellensatz.

Now consider the general case. We may assume that X is irreducible and reduced. We may also assume that X is affine. Then X = Spec A, and A is entire. Let x_{1}, …, x_{n} ∈ A generate A over k. As before, assume that U(ℂ_{v}) is not dense in X(ℂ_{v}) . Let Z be the closed set X \ U . Then there exist P ∈ X(ℂ_{v}) and r ∈ ℝ_{>0} such that

{Q ∈ X(ℂ_{v}) : ∥x_{i}(Q) − x_{i}(P)∥ < r for all i = 1, …, n } ⊆ Z(ℂ_{v}) .

By the Noether normalization lemma there exist y_{1}, …, y_{d} ∈ A, where d = dim X, such that A is finite over the polynomial ring k[y_{1}, …, y_{d}] . This induces a finite morphism ϕ: X → 𝔸_{k}^{d}. Since roots of polynomials vary continuously in the coefficients of the polynomial, there exists r′ ∈ ℝ_{>0} such that

ϕ({Q ∈ X(ℂ_{v}) : ∥x_{i}(Q) − x_{i}(P)∥ < r for all i = 1, …, n})
⊇ {(z_{1}, …, z_{d}) ∈ ℂ_{v}^{d} : ∥z_{i} − y_{i}(ϕ(P))∥ < r′ for all i = 1, …, d}.

(1.3.1)

But dim Z < d, so dim ϕ(Z) < d and in particular the open set V := 𝔸_{k}^{n} \ ϕ(Z) is dense. But (1.3.1) implies that V (ℂ_{v}) is not dense in 𝔸^{d}(ℂ_{v}) , contradicting the special case proved earlier. Thus the lemma holds in general. □

§2. M-bounded sets The notion of M-bounded set is needed, first of all, because ℂ_{v} is not locally compact unless v is archimedean. This notion also provides a useful uniformity over all places of k.

The next two sections are quite technical. On first reading, it may be better to skip them and read the definitions of Weil functions with the case k = ℂ in mind, so that M_{k} contains only one place. In that case, an M-bounded set is just a set whose closure is compact.

The goal of this section is to define a notion of bounded set in X(M) which is useful for arithmetic applications. Definition 2.2 contains the germ of the idea; it will then be progressively refined.

Definition 2.1. Let X be an affine scheme, let {x_{1}, …, x_{n}} be a system of generators for 𝒪(X) over k, and let γ be an M_{k}-constant. Then we define

B(X, x_{1}, …, x_{n}, γ) = {P ∈ X(M) : − log ∥x_{i}(P)∥ ≥ −γ_{v(P)} for all i = 1, …, n }.

Definition 2.2. Let U be an open affine subset of a scheme X.

(a). Let x_{1}, …, x_{n} be a system of generators for 𝒪(U) over k. A subset E of X(M) is said to be affine M-bounded with respect to U and x_{1}, …, x_{n} if there exists an M_{k}-constant γ such that E ⊆ B(U, x_{1}, …, x_{n}, γ) . (b). A subset E of X(M) is said to be affine M-bounded with respect to U if it is affine M-bounded with respect to U and x_{1}, …, x_{n} for some system x_{1}, …, x_{n} of generators for 𝒪(U) over k.

In the case k = ℂ and X = 𝔸^{n}, this definition reduces to the standard notion of bounded set. If k is a number field or function field, then automorphisms of 𝔸_{k}^{n} preserve the set

(2.3)  {(x_{1}, …, x_{n}) ∈ 𝔸^{n}(ℂ_{v}) : ∥x_{i}∥ ≤ 1 for all i }

at almost all places v. In fact, Definition 2.2 requires an affine M-bounded subset of 𝔸^{n} to be bounded (in the usual sense) at all v, and to be a subset of (2.3) at almost all places. The remainder of this section generalizes this notion to k-schemes of finite type. For such schemes, if k = ℂ then a M-bounded set is one which stays away from the boundary of the complex manifold X(ℂ) . In this case a set will be M-bounded if and only if it is contained in a compact subset of X(ℂ) . This is not true for non-archimedean ℂ_{v} because these fields are not locally compact; the notion of M-boundedness provides a condition which is weaker but still has many desirable properties.

The dependence on the choice of x_{1}, …, x_{n} is not really necessary, as the following lemma shows.

Lemma 2.4. Let V ⊆ U be open affine subsets of X as above, and let {x_{1}, …, x_{n}} and {y_{1}, …, y_{m}} be systems of generators for 𝒪(U) and 𝒪(V) over k, respectively. Then any affine M-bounded set with respect to V and y_{1}, …, y_{m} is also an affine M-bounded set with respect to U and x_{1}, …, x_{n}.

Proof. We will show that there exists an M_{k}-constant γ and an integer d such that, for all M_{k}-constants γ′ ≥ 0 ,

(2.4.1)  B(V, y_{1}, …, y_{m}, γ′) ⊆ B(U, x_{1}, …, x_{n}, dγ′ + γ).

Indeed, V ⊆ U , hence 𝒪(U) ⊆ 𝒪(V) , hence there exist polynomials f_{1}, …, f_{n} ∈ k[Y_{1}, …, Y_{m}] such that x_{i} = f_{i}(y_{1}, …, y_{m}) for all i. Without loss of generality we may assume that these polynomials are all nonzero and that n ≠ 0 . For each i, let S_{i} be the set of nonzero coefficients of f_{i}, let n_{i} be the number of nonzero monomials of f_{i}, and let d_{i} be the total degree of f_{i}. Let

d = max_{1≤i≤n} d_{i},

and let γ be the M_{k}-constant defined by

γ_{v} = max_{1≤i≤n}(log max_{a∈S_{i}} ∥a∥_{v} + N_{v} log n_{i}).

Then (2.4.1) follows immediately from the fact that the inequality

∥f_{i}(η_{1}, …, η_{m})∥ ≤ n_{i}^{N_{v}} · max_{a∈S_{i}} ∥a∥_{v} · max(1, ∥η_{1}∥, …, ∥η_{m}∥)^{d_{i}}

holds for all i = 1, …, n, all v ∈ M_{k}, and all η_{1}, …, η_{m} ∈ ℂ_{v}. Thus (2.4.1) holds, which implies the lemma. □

This lemma with U = V implies that if a set is M-bounded with respect to U and some choice of generating system x_{1}, …, x_{n}, then it is true for all such choices.

We now give the main definition of this section.

Definition 2.5. Let X be a scheme and let E be a subset of X(M) .

(a). Let U_{1}, …, U_{n} be affine open subsets of X. We say that E is M-bounded with respect to U_{1}, …, U_{n} if there exist subsets E_{1}, …, E_{n} of E such that E = E_{1} ∪ ⋯ ∪ E_{n} and such that each E_{i} is affine M-bounded with respect

to U_{i}.

(b). We say that E is M-bounded if it is M-bounded with respect to U_{1}, …, U_{n} for some open affine subsets U_{1}, …, U_{n}.

It is trivial to see that M-boundedness is closed under subset and finite union.

Example 2.6. If X = ℙ_{k}^{n}, then X(M) is M-bounded. Indeed, let v ∈ M_{k} and let [x_{0} : ⋯: x_{n}] be homogeneous coordinates of a point in X(ℂ_{v}) . Let U_{0}, …, U_{n} be the standard open affine subsets. Pick j such that ∥x_{j}∥ equals max ∥x_{i}∥. Then x_{0}/x_{j}, …, x_{n}/x_{j} generate 𝒪(U_{j}) and ∥x_{i}/x_{j}∥ ≤ 1 for all i. Thus

X(K) ⊆ ⋃_{j=0}^{n} B (U_{j}, \frac{x_{0}}{x_{j}} , …, \frac{x_{n}}{x_{j}} , 0) ,

and therefore X(K) is M-bounded.

Remark 2.7. Example 2.6 is related to the fact that ℙ^{n}(ℂ) is compact. In fact, it is true more generally that if k is an archimedean local field, then a subset E ⊆ X(M) is M-bounded if and only if it is contained in a compact subset of X(M) (in the v-topology). The details are left to the reader.

We next show that Definition 2.5 holds as stated if and only if it holds with a prescribed open cover.

Theorem 2.8. Let X be a scheme, let E be a subset of X(M) , and let U_{1}, …, U_{n} and V_{1}, …, V_{M} be open affine covers of X. Then E is M-bounded with respect to U_{1}, …, U_{n} if and only if it is M-bounded with respect to V_{1}, …, V_{m}.

Proof. We first claim that if X = Spec A is affine, if U_{1}, …, U_{n} is a cover of X by principal open affine subsets D(f_{1}), …, D(f_{n}) , and if E is M-bounded with respect to X, then it is also M-bounded with respect to U_{1}, …, U_{n}.

This will be a problem on one of your homework assignments.

To prove the theorem, it suffices to prove one direction. Write E = E_{1} ∪ ⋯ ∪ E_{n}, where E_{i} is affine M-bounded with respect to U_{i} for all i. For each i and j let {W_{ijl} : l ∈ L_{ij}} be a finite cover of U_{i} ∩ V_{j} by open affine subsets which are principal subsets of U_{i}. Then, for each i, the set {W_{ijl} : 1 ≤ j ≤ m and l ∈ L_{ij} } is a finite cover of U_{i} by principal open subsets, so by the claim E_{i} is a union of affine Mbounded subsets E_{ijl} with respect to W_{ijl} for all j and l.

For all j, the sets E_{ijl} are also affine M-bounded with respect to V_{j} for all i and l (by Lemma 2.4), so their (finite) union E′_{j} is affine M-bounded with respect to V_{j} . Since E = ⋃_{j} E′_{j} , we then have that E is M-bounded with respect to V_{1}, …, V_{m}.

□ Corollary 2.9. Let X be an affine scheme. Then a subset E ⊆ X(M) is affine M-

bounded with respect to X if and only if it is M-bounded.

Corollary 2.10. Let X be a scheme, and let U_{1}, …, U_{n} be an open cover of X. Then E ⊆ X(M) is an M-bounded subset of X if and only if it can be written as a union of M-bounded subsets E_{1}, …, E_{n} of U_{1}(M), …, U_{n}(M) , respectively.

Theorem 2.8 is useful for showing that a given set is not M-bounded. For example, let k = ℂ and let X = ℙ_{ℂ}^{2} \ [1 : 0 : 0] . The set {[n : 1 : 1] : n ∈ ℕ} is not M-bounded because it is not M-bounded in the standard affine subsets U_{1} and U_{2} covering X. Thus, one cannot define boundedness of a general scheme X via the ring 𝒪(X) .

We now consider the behavior of boundedness in the context of morphisms.

Definition 2.11. Let f: X → Y be a morphism. For v ∈ M_{k} let f_{v} denote the induced morphism f_{v}: X(ℂ_{v}) → Y (ℂ_{v}) . For subsets E ⊆ X(M) we then define (with some abuse of notation) f(E) ⊆ Y (M) as:

f(E) = ⋃_{v∈M_{k}} f_{v}(E ∩ X(ℂ_{v})).

Likewise, for F ⊆ Y (M) let

f^{−1}(F) = ⋃_{v∈M_{k}} f_{v}^{−1}(F ∩ Y (ℂ_{v})).

Proposition 2.12. Let f: X → Y be a morphism and let E ⊆ X(M) be an M-bounded set. Then f(E) is also M-bounded.

Proof. By Corollary 2.10, we may assume that Y is affine, and that X is affine. By Corollary 2.9, E is an affine M-bounded set with respect to X, and it suffices to show that f(E) has the same property with respect to Y . Let x_{1}, …, x_{n} be a system of generators for 𝒪(Y) over k; we may choose a system of generators for 𝒪(X) that contains this set. The proposition then follows immediately, by Definition 2.2. □

The following proposition generalizes Example 2.6.

Proposition 2.13. Let f: X → Y be a proper morphism and let F ⊆ Y (M) be an M-bounded set. Then f^{−1}(F) is also M-bounded.

Proof. Chow’s lemma ([H], II Ex. 4.10) says that there exists a projective morphism f′: X′ → Y and a birational morphism g: X′ → X such that f′ = f ∘ g. By applying Proposition 2.12 to g, we may therefore reduce to the case in which f is projective. By Corollary 2.10, we may assume that Y is affine; the result then follows by the method of Example 2.6. □

To conclude this section, we note that boundedness is also well behaved with respect to base change. If L is a finite extension of k and if w ∈ M_{L} lies over v ∈ M_{k}, then recall that ℂ_{w} ≅ ℂ_{v} (with ∥ · ∥_{w} = ∥ · ∥_{v}^{[L_{w}:k_{v}]} ).

Proposition 2.14. Let L be a finite extension of k. Let X be a scheme over k and let X_{L} = X ×_{k} Spec L.

(a). For E ⊆ X(M) let E_{L} be the subset of X_{L}(M) defined by
E_{L} ∩ X_{L}(ℂ_{w}) = E ∩ X(ℂ_{v})

for places w ∈ M_{L} lying over v ∈ M_{k}, via the canonical isomorphism X_{L}(ℂ_{w}) ≅ X(ℂ_{v}) . If E is M-bounded, then so is E_{L}.

(b). For F ⊆ X_{L}(M) let F_{k} be the subset of X(M) defined by

F_{k} ∩ X(ℂ_{v}) = ⋃_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}} F ∩ X_{L}(ℂ_{w})

If F is M-bounded, then so is F_{k}.

The proof is trivial.

The following proposition relates the concepts of M-boundedness of subsets of X(M) for schemes X over a number field or function field k, with M-boundedness of subsets of X_{k_{v}}(M) for each v ∈ M_{k}.

Proposition 2.15. Let X be a scheme, let v be a place of k, and let X_{k_{v}} = X ×_{k} k_{v}. We may then identify X(ℂ_{v}) (as a subset of X(M) ) with X_{k_{v}}(M) . Via this identification, a subset of X(ℂ_{v}) is M-bounded as a subset of X(M) if and only if it is M-bounded as a subset of X_{k_{v}}(M) .

This again is trivial.

§3. Locally M-bounded functions The notion of M-bounded set is primarily used to define the notion of locally Mbounded function, which in turn is a key ingredient in the definition of Weil function. We let \overline{ℝ} denote the set of extended real numbers \{−∞\} ∪ ℝ ∪ \{∞\}, ordered in the usual way.

Definition 3.1. Let X be a scheme and let f: X(M) → \overline{ℝ}.

(a). We say that f is locally M-bounded from below if for all M-bounded subsets E ⊆ X(M) there is an M_{k}-constant γ such that f(P) ≥ −γ_{v(P)} for all

P ∈ E.

(b). We say that f is locally M-bounded from above if −f is locally M-bounded

from below. (c). We say that f is locally M-bounded if it is locally M-bounded from above

and from below.

If f is locally M-bounded from below, then of course it can never take on the value −∞, and similarly in the other cases.

By quasi-compactness of X, the definition of locally M-bounded function is equivalent to the more customary definition that for each P ∈ X there exists an open affine neighborhood U of P in X such that for all (affine) M-bounded subsets E ⊆ U(M) , f is bounded on E from above and below by M_{k}-constants. Because ℂ_{v} is not always locally compact, however, we cannot replace the condition on U with “there exist x_{1}, …, x_{n} and γ′ such that f is bounded on B(U, x_{1}, …, x_{n}, γ′) from above and below by M_{k}-constants.” This is because these sets B(U, x_{1}, …, x_{n}, γ′) would not necessarily cover X(M) .

Following up on Remark 2.7, we also note that if k is an archimedean local field, then any continuous function f: X(M) → ℝ is also locally M-bounded.

Proposition 3.2. Let X be a scheme.

(a). If f_{1} : X(M) → \overline{ℝ} and f_{2} : X(M) → \overline{ℝ} are locally M-bounded from below

then so are f_{1} + f_{2} , min(f_{1}, f_{2}) , and max(f_{1}, f_{2}) . (b). If f: X(M) → \overline{ℝ} is locally M-bounded from below and if g: X(M) → [0, c]

is a function for some c ≥ 0 , then g · f is locally M-bounded from below. Likewise, if f is locally M-bounded and g: X(M) → [−c, c] is any function, then g · f is locally M-bounded.

(c). If f: Y (M) → \overline{ℝ} is locally M-bounded from below and if ϕ: X → Y is a

morphism then f ∘ ϕ is locally M-bounded from below. (d). If f: X(M) → \overline{ℝ} is locally M-bounded from below and if U ⊆ X is a

subscheme, then f|_{U(M)} is locally M-bounded from below.

(e). Let f: X(M) → \overline{ℝ} be a function and let U_{1}, …, U_{n} be an open cover of X. If f|_{U_{i}(M)} is locally M-bounded from below for all i, then f is locally M-bounded from below.

(f). Let f: Y (M) → \overline{ℝ} be a function and let ϕ: X → Y be a proper surjective

morphism. If f ∘ ϕ is locally M-bounded from below, then so is f . (g). Let f: X(M) → \overline{ℝ} be a function and let X_{1}, …, X_{n} be closed subschemes

of X such that X = ⋃ X_{i}. If f|_{X_{i}(M)} is locally M-bounded from below for all i, then so is f .

(h). Let L be a finite extension of k and let X_{L} = X ×_{k} L. Let f: X(M) → \overline{ℝ}. We define f_{L}: X_{L}(M) → \overline{ℝ} via the natural isomorphisms X_{L}(ℂ_{w}) ≅ X(ℂ_{v}) for places w ∈ M_{L} and v ∈ M_{k} such that w | v. Then f is locally Mbounded from below if and only if f_{L} is.

(i). Let f: X(M) → \overline{ℝ} and let v ∈ M_{k}. In the notation of Proposition 2.15, if f is locally M-bounded from below (as a function on X(M) ), then f|_{X(ℂ_{v})} is locally bounded from below as a function on X_{k_{v}}(M) .

Similar statements hold in the context of functions that are locally M-bounded.

Proof. Parts (a) and (b) are trivial. Parts (c), (e), (f), (h), and (i) follow from Proposition 2.12, Corollary 2.10, Proposition 2.13, Proposition 2.14, and Proposition 2.15, respectively. Finally, part (d) follows from (c) by considering the inclusion map, and (g) follows by applying (f) to the natural map ∐ X_{i} → X. □

Proposition 3.3. Let X be a scheme and let f ∈ 𝒪(X) . Using the convention that − log 0 = +∞, the function − log ∥f(P)∥ on X(M) is locally M-bounded from below. Moreover, if f ∈ 𝒪(X)^{∗}, then − log ∥f(P)∥ is locally M-bounded.

Proof. First consider the first assertion. By Proposition 3.2e, we may assume that X is affine. Let x_{1}, …, x_{n} be a system of generators for 𝒪(X) over k; we may assume that x_{n} = f . Let E be an M-bounded subset of X(M) . By Corollary 2.9, there is an M_{k}-constant γ such that

E ⊆ B(X, x_{1}, …, x_{n}, γ).

Then − log ∥f(P)∥ ≥ −γ_{v(P)} for all P ∈ E. This shows that − log ∥f(P)∥ is locally M-bounded from below.

The second assertion follows immediately from the first. □

If X is normal then a converse is also true; see Lemma 5.3.

The following proposition is very easy, but it leads to one of the more useful results (Proposition 5.9) on Weil functions.

Proposition 3.4. Let X be a scheme, let v be a place of k, and let X_{k_{v}} = X ×_{k}k_{v}. Let f_{1} : X(M) → \overline{ℝ} and let f_{2} : X_{k_{v}}(M) → \overline{ℝ} be locally M-bounded from below (with respect to M_{k} and M_{k_{v}} = \{v\}, respectively). Then the function f: X(M) → \overline{ℝ} defined by

f(P) = \{ \begin{matrix}f_{1}(P), & if v(P) ≠ v; \\ f_{2}(P), & if v(P) = v \end{matrix}

is locally M-bounded from below. If f_{1} and f_{2} are locally M -bounded, then so is f .

Proof. This is an easy consequence of Proposition 2.15, together with the facts that subsets and finite unions of M-bounded sets are M-bounded. □

§4. Weil functions This section defines Weil functions and determines their principal properties.

Definition 4.1. Let X be a scheme (separated and of finite type over k, as usual). A Weil function on X is a pair (D, λ) in which D is a Cartier divisor on X and λ is a function λ: (X \Supp D)(M) → ℝ with the following property. There exists a system of representatives {(U_{i}, f_{i})}_{i∈I} for D and continuous locally M-bounded functions α_{i}: U_{i}(M) → ℝ for all i such that

(4.1.1)  λ(P) = − log ∥f_{i}(P)∥ + α_{i}(P)

for all P ∈ (U_{i} \ Supp D)(M) .

Note that α_{i} is defined on all of U_{i}(M) , so (4.1.1) imposes a condition on the growth of λ near Supp D.

Often, by abuse of notation, we will use the symbol λ to refer both to the Weil function and to the second element of the pair. It is not true, however, that a function λ: U(M) → ℝ can be part of a Weil function for at most one Cartier divisor D: see Example 5.7. In general we must therefore specify a divisor when introducing a Weil function. To shorten notation, we say that λ is a Weil function for D if λ is a Weil function whose underlying divisor is D. We also write D = (λ) .

Definition 4.2. Let λ be a Weil function. The support of λ, written Supp λ, is the support of the divisor (λ) .

Note that, in contrast to common usage in analysis, the support of a Weil function is not the set where the function is nonzero.

The following proposition shows that one can take any system of representatives {(U_{i}, f_{i})}_{i∈I} for D in the above definition.

Proposition 4.3. Let (D, λ) be a Weil function on a scheme X, let U be an open subset of X, and let f ∈ 𝒦 (U) be a function such that D|_{U} = (f) . Then the function

α := λ + log ∥f∥

extends to a continuous locally M-bounded function α: U(M) → ℝ.

Proof. Let {U_{i}}_{i∈I} , {f_{i}}_{i∈I} , and {α_{i}}_{i∈I} be as in Definition 4.1. For each i ∈ I, f_{i}/f ∈ 𝒪(U ∩ U_{i})^{∗}; hence by Proposition 3.3, − log ∥f_{i}/f∥ is a continuous locally Mbounded function on (U ∩ U_{i})(M) . Thus, α extends to (U ∩ U_{i})(M) by the formula

(4.3.1)  α = α_{i} − log ∥f_{i}/f∥.

By Proposition 3.2d and Proposition 3.2a, this extension is a locally M-bounded function on (U ∩ U_{i})(M) . It is also continuous. Moreover, by Lemma 1.3, this extension is unique. Thus, as i varies, (4.3.1) defines α on all of U(M) . By Proposition 3.2e, α is locally M-bounded, and it is obviously continuous. □

Often it will be useful to define a function on a strictly smaller open set, and show that it extends to a Weil function. The following proposition allows us to do that.

Proposition 4.4. Let X be a scheme, let D be a Cartier divisor on X, let U be a dense open subset of X, and let λ_{0} : U(M) → ℝ be a function. Let {(U_{i}, f_{i})}_{i∈I} be a system of representatives for D. Suppose also that for each i there is a continuous locally M-bounded function α_{i}: U_{i}(M) → ℝ such that

λ_{0}(P) = − log ∥f_{i}(P)∥ + α_{i}(P)

for all P ∈ (U ∩ U_{i})(M) . Then there exists a unique Weil function (D, λ) on X such that λ|_{U(M)} = λ_{0} .

Proof. Existence is immediate: define λ by (4.1.1). Uniqueness follows from the fact that U(M) is dense in X(M) (Lemma 1.3). □

We now consider some of the principal properties of Weil functions.

Additivity

Definition 4.5. Let X be a scheme and let (D_{1}, λ_{1}) and (D_{2}, λ_{2}) be two Weil functions on X. Let U = X \ (Supp D_{1} ∪ Supp D_{2}) (note that Supp D_{1} ∪ Supp D_{2} may be strictly larger than Supp(D_{1} + D_{2}) ). Let λ_{0} : U(M) → ℝ be defined by

λ_{0} = λ_{1}|_{U(M)} + λ_{2}|_{U(M)}.

Then λ_{0} extends by Proposition 4.4 to a function λ such that (D_{1} + D_{2}, λ) is a Weil function on X. We define the operation of addition of Weil functions by

(D_{1}, λ_{1}) + (D_{2}, λ_{2}) = (D_{1} + D_{2}, λ).

The set of Weil functions on X forms an abelian group under this operation, and the map (D, λ) ↦ D defines a group homomorphism to the group of Cartier divisors on X.

Functoriality

Definition 4.6. Let ϕ: X → Y be a morphism of schemes, let D be a Cartier divisor on Y such that ϕ does not map any associated point of X into Supp D (i.e., so that ϕ^{∗}D is defined), and let λ be a Weil function for D on Y . Then we let ϕ^{∗}λ be the Weil function obtained by composing λ with the map (X \ Supp ϕ^{∗}D)(M) → (Y \ Supp D)(M) induced by ϕ. Its underlying divisor is defined to be ϕ^{∗}D.

If ϕ: X → Y is a morphism, then the map λ ↦ ϕ^{∗}λ is a group homomorphism on the subgroup where it is defined.

Ordering

Definition 4.7. Let λ_{1} and λ_{2} be Weil functions on a scheme X for Cartier divisors D_{1} and D_{2} , respectively. We say that λ_{1} ≥ λ_{2} if there exists an open dense subset U of X, disjoint from Supp D_{1} and Supp D_{2} , such that λ_{1}(P) ≥ λ_{2}(P) for all P ∈ U(M) .

This definition is independent of the choice of U . We may also regard a M_{k}-constant γ as a Weil function for the trivial divisor on

any scheme X, and therefore give meaning to the statements λ ≥ γ and λ ≥ 0 .

Uniqueness

Lemma 4.8. Let X be a proper scheme over k and let λ be a Weil function for the trivial divisor. Then there exist M_{k}-constants γ_{1} and γ_{2} such that γ_{1} ≤ λ ≤ γ_{2} .

Proof. Indeed, {(X, 1)} is a system of representatives for the trivial divisor; hence the function λ + log ∥1∥ = λ extends to a continuous locally M-bounded function on X(M) . In other words, λ is locally M-bounded. But since X(M) is an M-bounded set by Proposition 2.13, M-constants γ_{1} and γ_{2} exist by Definition 3.1. □

See also Proposition 5.1.

Proposition 4.9. Let X be a proper scheme over k, let D be a Cartier divisor on X, and let λ_{1} and λ_{2} be Weil functions for D. Then there exist M_{k}-constants γ_{1} and γ_{2} such that

γ_{1} ≤ λ_{1} − λ_{2} ≤ γ_{2}.

In particular,

(4.9.1)  λ_{1}|_{(X\Supp D)(ℂ_{v})} = λ_{2}|_{(X\Supp D)(ℂ_{v})}  for almost all v ∈ M_{k} .

Proof. This follows immediately by applying Lemma 4.8 to λ := λ_{1} − λ_{2} . □

See also Proposition 5.9.

Existence Here we prove that if X is a projective scheme and if D is a Cartier divisor on X, then there exists a Weil function for D. This is true more generally if X is an arbitrary scheme (separated and of finite type over k), but this requires much more machinery. It will be proved in Section 5 of Chapter 3.

The following type of Weil function is a starting point for proving existence.

Definition 4.10. Let X be a scheme and let f be a rational function on X. Then the principal Weil function associated to f is the Weil function − log ∥f∥, with associated divisor (f) .

Lemma 4.11. Let X be a k-scheme of finite type. Let D_{0} and D_{1}, …, D_{n} be Cartier divisors on X. Suppose that D_{i} − D_{0} is effective for all i = 1, …, n, and that

(4.11.1)  ⋂_{i=1}^{n} Supp(D_{i} − D_{0}) = ∅.

Let λ_{1}, …, λ_{n} be Weil functions for D_{1}, …, D_{n}, respectively. Then

λ := min_{1≤i≤n} λ_{i}

is a Weil function on X for the divisor D_{0} .

Proof. By (4.11.1), the collection {X \ Supp(D_{i} − D_{0})}_{1≤i≤n} is an open cover of X. Let {U_{j}}_{j∈J} be a refinement of this cover such that D_{0} and D_{1}, …, D_{n} restrict to principal divisors on each U_{j} . Then D_{0}|_{U_{j}} = (f_{j}) and (D_{i} − D_{0})|_{U_{j}} = (g_{ji}) for all j ∈ J and all i = 1, …, n. By definition of Weil functions, there are continuous locally M-bounded functions α_{ji}: U_{j}(M) → ℝ such that λ_{i} = − log ∥f_{j}g_{ji}∥ + α_{ji} on (U_{j} \ Supp D_{i})(M) , for all j and all i.

Fix some j ∈ J , and let U = U_{j} \ (Supp D_{0} ∪ Supp D_{1} ∪ ⋯ ∪ Supp D_{n}) . Then U is Zariski-dense in U_{j} , and

λ(P) = min{− log ∥f_{j}(P)g_{ji}(P)∥ + α_{ji}(P) : i = 1, …, n}
= − log ∥f_{j}(P)∥ + min{− log ∥g_{ji}(P)∥ + α_{ji}(P) : i = 1, …, n}

for all P ∈ U(M) .

Since D_{i} − D_{0} is effective, each g_{ji} is regular on U_{j} . Therefore, by Propositions 3.3 and 3.2a, the functions − log ∥g_{ji}∥ + α_{ji} are continuous and locally M-bounded from below, as functions from U_{j}(M) to ℝ ∪ {∞}. Therefore the same is true of the function

α_{j} := min{− log ∥g_{ji}(P)∥ + α_{ji}(P) : i = 1, …, n} .

Since the open cover {U_{j}} is a refinement of the open cover {X \Supp(D_{i} −D_{0})}, there is some ℓ such that D_{ℓ} − D_{0} restricts to the trivial divisor on U_{j} ; hence g_{jℓ} ∈ 𝒪(U_{j})^{∗}. By Proposition 3.3 again, the function − log ∥g_{jℓ}∥ + α_{jℓ} is locally M-bounded from above; hence the same is true of α_{j} .

Thus α_{j} is a continuous, locally M-bounded function on U_{j}(M) satisfying the conditions of Definition 4.1, so λ is a Weil function for D_{0} . □

Theorem 4.12. Let X be a projective scheme over k and let D be a Cartier divisor on X. Then there exists a Weil function for D.

Proof. By Lemma 1:5.6, any Cartier divisor on a projective scheme can be written as a difference of two globally generated divisors. Therefore, by additivity of Weil functions, it suffices to show this in the case where D is globally generated. Then there exists a morphism ϕ: X → ℙ_{k}^{n} such that 𝒪(D) ≅ ϕ^{∗}𝒪(1) . Thus D = ϕ^{∗}H for some hyperplane H in ℙ_{k}^{n}. After composing ϕ with an automorphism of ℙ_{k}^{n}, we may assume that H is the hyperplane at infinity. By functoriality, we may therefore assume that X = ℙ_{k}^{n} for some n and that D is the hyperplane at infinity. Let x_{0}, …, x_{n} be the standard homogeneous coordinates on X. By Lemma 4.11 applied to −D and to the principal divisors (x_{0}/x_{0}), …, (x_{n}/x_{0}) , it then follows that

λ = − min_{0≤i≤n} − log ∥\frac{x_{i}}{x_{0}}∥

= − log \frac{∥x_{0}∥}{max(∥x_{0}∥, …, ∥x_{n}∥)}

is a Weil function for D. □

Note that the properties proved so far correspond to the “height machine,” Theorems 1:5.11 and 1:5.17.

§5. Additional properties of Weil functions

Here we show some additional properties of Weil functions.

Effective divisors

Proposition 5.1. Let X be a proper scheme over k, let D be an effective Cartier divisor on X, and let λ be a Weil function for D. Then there exists an M_{k}-constant γ such that λ ≥ γ.

Proof. Let {(U_{1}, f_{1}), …, (U_{n}, f_{n})} be a system of representatives for D. We may regard λ as a function from X(M) to ℝ ∪ {∞} by letting λ(P) = ∞ for all P ∈ (Supp D)(M) . Since λ = − log ∥f_{i}∥ + α_{i} for a locally M-bounded continuous function α_{i} for each i, on U_{i}(M) , it follows by Proposition 3.3 that λ|_{U_{i}(M)} is locally M-bounded from below for all i. By Proposition 3.2e, it therefore follows that λ is locally M-bounded from below. Since X is proper, Proposition 2.13 implies that X(M) is an M-bounded set; therefore the M_{k}-constant γ exists by Definition 3.1a. □

The following corollary is frequently useful.

Corollary 5.2. Let X be a proper scheme. Let D_{1} and D_{2} be Cartier divisors on X such that D_{2} is effective and Supp D_{2} ⊇ Supp D_{1} (or even D_{1}|_{X\Supp D_{2}} is effective). Let λ_{D_{1}} and λ_{D_{2}} be Weil functions for D_{1} and D_{2} , respectively. Then

there is an integer n and an M_{k}-constant γ such that λ_{D_{1}} + mλ_{D_{2}} ≥ γ.

Proof. Let ϕ: X′ → X be the normalization of X. Then ϕ^{∗}D_{1} and ϕ^{∗}D_{2} may be regarded as Weil divisors, and the condition ensures that, for some n ∈ ℤ, ϕ^{∗}D_{1}+nϕ^{∗}D_{2} is effective as a Weil divisor. By ([H], II Prop. 6.3A) it is therefore effective as a Cartier divisor, and by Proposition 5.1, there is an M_{k}-constant γ such that ϕ^{∗}λ_{D_{1}} +nϕ^{∗}λ_{D_{2}} , which is a Weil function for ϕ^{∗}D_{1} + nϕ^{∗}D_{2} , is bounded from below by γ. Since the map X′(M) → X(M) induced by ϕ is surjective, we obtain λ_{D_{1}} + nλ_{D_{2}} ≥ γ, as was to be shown. □

Normal schemes We show that on a normal scheme, the divisor of a Weil function is uniquely determined by its underlying function.

Lemma 5.3. Let X be a scheme, let f ∈ 𝒪(X) , and let D = (f) . If f ∉ 𝒪(X)^{∗}, then there exists an M-bounded subset E ⊆ X(M) such − log ∥f∥ is unbounded from above on E ∩ (X \ Supp D)(M) .

Proof. Let U be an open affine subset of X such that f ∉ 𝒪(U)^{∗}. Let A = 𝒪(U) , so f ∈ A but f ∉ A^{∗}. Let x_{1}, …, x_{n} be a system of generators for A over k, and fix any P_{0} ∈ X(ℂ_{v}) with f(P_{0}) = 0 . Fix v ∈ M_{k}, let γ be any M_{k}-constant with γ_{v} = log(max_{1≤i≤n} ∥x_{i}(P_{0})∥ + 1), and let

E = B(U, x_{1}, …, x_{n}, γ).

Then E is an M-bounded subset of X(M) . For any ϵ > 0 , the set

{P ∈ X(ℂ_{v}) : ∥x_{i}(P)∥ < ∥x_{i}(P_{0})∥ + 1 for all i = 1, …, n and ∥f(P)∥ < ϵ }

is an open neighborhood of P_{0} in the v-topology, so by Lemma 1.3 it contains a point P with f(P) ≠ 0 . Then we have P ∈ E but − log ∥f(P)∥ > − log ϵ. Since ϵ can be taken arbitrarily small, this shows that − log ∥f∥ is unbounded on the intersection of E with its domain, as was to be shown. □

Proposition 5.4. Let X be a scheme over k, let D be a Cartier divisor on X, and let λ be a Weil function for D. If D is effective, and if λ is bounded from above by an M_{k}-constant, then D = 0 .

Proof. Since the question is local on X, we may assume that D = (f) for some f ∈ 𝒦 (X) . Since D is effective, we have f ∈ 𝒪(X) . By Proposition 4.3, the function λ + log ∥f∥ on (X \ Supp D)(M) extends to a continuous locally M-bounded function α: X(M) → ℝ. We have − log ∥f(P)∥ = λ(P) − α(P) for all P ∈ (X \ Supp D)(M) ; both functions λ and −α are bounded on intersections of M-bounded subsets of X(M) with the domain of λ. Therefore − log ∥f∥ also has this property. If D ≠ 0 then this would contradict Lemma 5.3, so we must have D = 0 . □

Proposition 5.5. Let X be a normal scheme, let D be a Cartier divisor on X, and let λ be a Weil function for X. If there exists an M_{k}-constant γ such that λ ≥ γ, then D is effective.

Proof. Assume that λ is bounded from below by an M_{k}-constant γ, but that D is not effective. The question is local on X, so we may assume that X is affine, say X = Spec A, and that X is connected. We may also assume that X is reduced. Then X is also irreducible. (If not, then there is some point P lying in two irreducible components; its local ring will then fail to be entire, contradicting the assumption that X is normal.) Thus X is integral, and therefore A is an entire ring. After further localizing, we may assume that D is represented on X by a function f ∈ K(X) : D = (f) . Since D is not effective, f ∉ A.

Since A is integrally closed, noetherian, and entire, it is the intersection of its localizations at all prime ideals of height 1 , by ([M], Thm. 11.5(ii)). Since f ∉ A, it follows that there exists a prime 𝔭 of height 1 such that f ∉ A_{𝔭}. After further localizing (away from points Q of codimension 1 such that f lies in the maximal ideal of A_{Q}), we may assume that 1/f ∈ A.

Hence −D = (1/f) is effective, and −λ is bounded from above by an M_{k}- constant. Proposition 5.4 then implies that D = 0 , a contradiction. □

Corollary 5.6. Let X be a normal scheme and let λ_{1} and λ_{2} be Weil functions for Cartier divisors D_{1} and D_{2} , respectively. If there exists a dense open subset U ⊆ X (disjoint from Supp D_{1} and Supp D_{2} ) such that λ_{1}|_{U(M)} = λ_{2}|_{U(M)} , then D_{1} = D_{2} .

Proof. By additivity of Weil functions, we may assume that λ_{2} = 0 and that D_{2} = 0 . But then, by Proposition 5.5 (and the fact that U(M) is dense in X(M) ), both D and −D are effective. Thus D = 0 . □

This corollary provides partial justification for the convention of referring to a Weil function (D, λ) by its second coordinate: if X is normal, then λ uniquely determines D. In general, though, this is not possible:

Example 5.7. Let X be the projective closure in ℙ_{k}^{2} of the affine curve

Spec k[x, y]/(y^{2} − x^{3}) ⊆ 𝔸_{k}^{2} ,

let f = 1 + y/x ∈ K(X) , let D_{1} be the principal divisor (f) , and let λ = − log ∥f∥. Then λ is a Weil function for D_{1} (i.e., (D_{1}, λ) is a Weil function on X). Since

(1 − \frac{y}{x}) f = 1 − \frac{y^{2}}{x^{2}} = 1 − x ,

we see that f has a simple zero at the point (1, −1) , has a simple pole at the point at infinity, is undefined at (0, 0) , and is nonzero elsewhere.

Now let D_{2} be the Cartier divisor represented by the system {(U_{1}, f), (U_{2}, 1)}, where U_{1} = X \ {(0, 0)} and U_{2} = X \ {(1, −1), ∞}. Since D_{1} is represented by {(U_{1}, f), (U_{2}, f)}, we have D_{1} = D_{2} if and only if f is regular and nonzero at (0, 0) . This is not the case, so D_{1} ≠ D_{2} .

However, λ is also a Weil function for D_{2} . Indeed, to see this it suffices to show that − log ∥f∥ is a continuous locally M-bounded function on U_{2}(M) . The normalization of Spec k[x, y]/(y^{2} − x^{3}) is the curve 𝔸_{k}^{1} , via the map t ↦ (t^{2}, t^{3}) , so the normalization of U_{2} is 𝔸_{k}^{1} \ {−1}. On the normalization, f pulls back to the function 1 + t, which is continuous by definition of the v-topology, and is locally Mbounded by Proposition 3.3. Therefore the same holds true for − log ∥f∥ by the fact that the normalization map is a homeomorphism for the v-topology, and by Proposition 2.13.

Thus λ is a Weil function for both divisors D_{1} and D_{2} . Note that both D_{1} and D_{2} pull back to the same divisor on the normalization.

Similar examples are possible in higher dimensions. For example, let X be the projective closure of Spec k[x, y, z]/(y^{2} − xz^{2}) , and let f = 1 + y/z. Here X is nonsingular in codimension 1 but (necessarily) not normal.

Working locally on M_{k} We now consider the relation between Weil functions on a scheme X over k and Weil functions on the corresponding scheme X_{k_{v}} over the completion k_{v}. The two definitions are closely related.

Definition 5.8. Let X be a scheme over k, let D be a Cartier divisor on X, and let λ be a Weil function for D. Let v be a place of k. Let D_{v} denote the divisor on X ×_{k} k_{v} corresponding to D. Then the local Weil function λ_{v} is the Weil function (D_{v}, λ|_{(X\Supp D)(ℂ_{v})}).

Proposition 5.9. Let X be a scheme over k, let D be a Cartier divisor on X, and let λ be a Weil function for D. Let v_{1}, …, v_{n} be distinct places of k. For i = 1, …, n let λ_{i} be a Weil function on X ×_{k} k_{v_{i}} for the divisor D_{v_{i}} on X ×_{k} k_{v_{i}} corresponding to D. Then there exists a (unique) Weil function λ′ on X for D such that λ′_{v} = λ_{v} for all places v ∉ {v_{1}, …, v_{n}} and λ′_{v_{i}} = λ_{i} for all i = 1, …, n.

Proof. This follows immediately from Proposition 3.4 and the definition of Weil function. Uniqueness of λ′ is obvious since it is specified everywhere. □

Thus, while (4.9.1) prevents us from changing a Weil function λ at infinitely many v, this operation is possible at finitely many v.

Base change Finally, we show that Weil functions can be moved to larger and, sometimes, smaller fields.

Proposition 5.10. Let X be a scheme over k and let D be a Cartier divisor on X. Let L be a finite extension of k, let X_{L} = X ×_{k} L, and let D_{L} be the divisor on X_{L} corresponding to D. (a). If λ is a Weil function on X for D, then the condition λ′_{w} = [L_{w} : k_{v}]λ_{v}

for all w ∈ M_{L} lying over all v ∈ M_{k} defines a unique Weil function λ′ on X_{L} for D_{L}.

(b). Let λ be a Weil function on X_{L} for D_{L}. Suppose that for all v ∈ M_{k} and all w, w′ ∈ M_{L} lying over v, we have λ_{w}/[L_{w} : k_{v}] = λ_{w′}/[L_{w′} : k_{v}] . Then there exists a unique Weil function λ′ on X for D with the property that λ′_{v} = λ_{w}/[L_{w} : k_{v}] for all w | v.

Proof. This is immediate from Proposition 3.2h and Proposition 3.2b. □

§6. Weil functions and heights This section realizes one of the primary goals of Weil functions, namely, breaking down a height function at a point P into a sum over places of the field k(P) .

Definition 6.1. Let X be a scheme over k, let D be a Cartier divisor on X, and let λ be a Weil function for D. Let L be a finite extension of k, let P ∈ X(L) with P ∉ Supp D, and let w be a place of L lying over v. Then we let P_{w} denote the point in X(ℂ_{w}) obtained from P via the injection L ↪ L_{w} ↪ ℂ_{w}, and let

λ_{w}(P) = [L_{w} : k_{v}]λ(P_{w}).

Note that this is compatible with the notation of Definition 5.8. The factor [L_{w} : k_{v}] ensures that λ_{w} is compatible with − log ∥ · ∥_{w}.

Definition 6.2. Let k be a number field or function field, let X be a scheme over k, let D be a Cartier divisor on X, and let λ be a Weil function for D. Then we define the function h_{λ,k}: X(k̄) → ℝ as follows. Let L be a finite extension of k, let P ∈ X(L) , let i: X′ → X be an irreducible component of X containing P (with reduced induced subscheme structure), let f be a rational function on X′ such that P ∉ Supp(i^{∗}D + (f)) , and let λ′ = i^{∗}λ − log ∥f∥. Then λ′ is a Weil function for i^{∗}D + (f) on X′, and we can define

h_{λ,k}(P) = \frac{1}{[L : k]} ∑_{w∈M_{L}} λ′_{w}(P) .

This definition is independent of the choice of irreducible component X′ containing P since the height can actually be written in terms of the function α in Definition 4.1, which is a function on U(M) , for some open neighborhood U of P in X. The definition is also independent of the choice of f , since if f_{1} and f_{2} are two such suitable functions, then f_{2}/f_{1} is defined and nonzero at P . Letting λ_{j} = i^{∗}λ − log ∥f_{j}∥ for j = 1, 2 , we have

∑_{w∈M_{L}} λ_{2,w}(P) − ∑_{w∈M_{L}} λ_{1,w}(P) = ∑_{w∈M_{L}} − log ∥(f_{2}/f_{1})(P)∥_{w} = 0

by the product formula.

Likewise, h_{λ,k} is independent of the choice of L by (0:7.11.2). Thus h_{λ,k} is well defined. Note in particular that it is defined exactly, not up to O(1) ; the inexactness has been shifted to the choice of a Weil function λ for the divisor.

Lemma 6.3. Let k be a number field or function field.

(a). Let λ_{1} and λ_{2} be two Weil functions on a scheme X over k. Then

h_{λ_{1}+λ_{2},k} = h_{λ_{1},k} + h_{λ_{2},k}. (b). Let ϕ: X → Y be a morphism of schemes over k and let D be a Cartier divisor on Y such that ϕ does not map any associated point of X into Supp D (i.e., such that ϕ^{∗}D is defined). Let λ be a Weil function for D on Y . Then h_{ϕ^{∗}λ,k}(P) = h_{λ,k}(ϕ(P)) for all P ∈ X(k̄) .

(c). Let X = ℙ_{k}^{n} for some n ∈ ℕ, let D be the hyperplane at infinity, and let λ be the Weil function

λ = − log \frac{∥x_{0}∥}{max_{0≤i≤n} ∥x_{i}∥}.

Then h_{λ,k} = h (where h is the height defined on ℙ^{n} by Definition 1:3.3). (d). If λ_{1} and λ_{2} are two Weil functions for the same divisor D, then

h_{λ_{1},k} = h_{λ_{2},k} + O(1).

Proof. Parts (a) and (b) are trivial. Part (c) follows from the product formula:

∑_{w∈L} − log \frac{∥x_{0}∥_{w}}{max ∥x_{i}∥_{w}} = ∑_{w∈L} log max ∥x_{i}∥_{w}.

Finally, for part (d) we may assume that λ_{2} = 0 , by part (a). Therefore λ_{1} is a Weil function for the trivial divisor, so by Lemma 4.8 there exist M_{k}-constants γ_{1} and γ_{2} such that γ_{1} ≤ λ_{1} ≤ γ_{2} . Let L and P be as in Definition 6.2; we may take f = 1 there, so λ′ = λ_{1} and thus we have

\frac{1}{[L : k]} ∑_{w∈M_{L}} γ_{1,w} ≤ \frac{1}{[L : k]} ∑_{w∈M_{L}} λ′_{w}(P) ≤ \frac{1}{[L : k]} ∑_{w∈M_{L}} γ_{2,w}

and thus |γ_{1}| ≤ h_{λ_{1},k}(P) ≤ |γ_{2}| by (1:4.2). □

Theorem 6.4. Let X be a proper scheme over a number field or function field k, let D be a Cartier divisor on X, and let λ be a Weil function for D. Then h_{λ,k} is a height function relative to D.

Proof. By Chow’s lemma and functoriality of h_{λ,k} and heights, we may assume that X is projective. By Lemma 1:5.6 and by additivity of heights and of h_{λ,k}, we may assume that D is globally generated. By Lemma 1:5.2 and by functoriality, we may assume that X = ℙ^{n} and that D is the hyperplane at infinity. Then the result follows from Lemma 6.3c and Lemma 6.3d. □

§7. Integral points Throughout this section, let k, S, and R be one of the following:

(a). k is a number field, S is a finite set of places of k containing all archimedean places, and R is the ring of S-integers of k; i.e.,

(7.1)  R = {x ∈ k : ∥x∥_{v} ≤ 1 for all v ∈ M_{k} \ S } .

(b). k is a function field over a constant field F , S is a nonempty set of places of k, and R is defined by (7.1). Note that if Y is the nonsingular projective curve over F corresponding to k and if S is regarded as a subset of Y , then Y \ S is an affine variety and R is its affine ring.

(c). k is the completion of a number field or function field at a non-archimedean place, S = ∅, and R is the valuation ring of k. Then (7.1) holds in this case as well.

The possibility S = ∅ in part (b) will be addressed shortly.

Given a finite set of polynomial equations

f_{i}(x_{1}, …, x_{n}) = 0 , i = 1, …, r,

with f_{i} ∈ k[x_{1}, …, x_{n}] for all i, one often wants to find the set of common solutions (x_{1}, …, x_{n}) of such systems with x_{i} ∈ R for all i. If

X = Spec k[x_{1}, …, x_{n}]/(f_{1}, …, f_{r})

is the corresponding affine scheme in 𝔸_{k}^{n}, then this set of solutions equals X(k) ∩ R^{n}. Such solutions are one instance of integral points on X.

Unfortunately, the notion of integral point is not well behaved as a geometric property of X: it depends on the affine embedding of X. For example, if X = 𝔸_{ℤ}^{1} , the identity map and the map x ↦ 2x are both affine embeddings of X into 𝔸_{ℤ}^{1} , and the point 1/2 is an integral point relative to the second embedding but not the first. In general, dilations can introduce bounded denominators (at only finitely many places). As it turns out, this is the worst that can happen by an arbitrary change of affine embedding. Because of this, Serre [S] defined a quasi-integral set of points as a set whose denominators are bounded. Therefore (with a slight change in terminology), we define:

Definition 7.2. Let X be an affine scheme of finite type over k, with affine coordinates y_{1}, …, y_{n}. Then a set of integral points on X is a subset Σ ⊆ X(k) such that there exists an M_{k}-constant γ satisfying ∥y_{i}(P)∥_{v} ≤ e^{γ_{v}} for all v ∉ S and all P ∈ Σ .

Before showing that this notion is independent of the affine embedding of X, we first note that it can be rephrased in terms of Weil functions, leading to a more general notion of set of integral points.

Remark 7.3. Let X, y_{1}, …, y_{n}, and γ be as in Definition 7.2. We regard X as a closed subscheme of 𝔸_{k}^{n} via the functions y_{1}, …, y_{n}, and view 𝔸_{k}^{n} as the complement of the hyperplane H : x_{0} = 0 in ℙ_{k}^{n} = Proj k[x_{0}, …, x_{n}] by letting y_{i} = x_{i}/x_{0} for all i. Let

λ_{H,v}([x_{0} : ⋯: x_{n}]) = − log \frac{∥x_{0}∥_{v}}{max\{∥x_{0}∥_{v}, …, ∥x_{n}∥_{v}\}} ;

this is a Weil function for H on ℙ_{k}^{n}. Then a set Σ ⊆ X(k) satisfies the condition of Definition 7.2 for a given M_{k}-constant γ if and only if λ_{H,v}(P) ≤ γ_{v} for all v ∉ S and all P ∈ Σ .

Of course, if λ_{H} and γ are as above, then λ_{H} − γ is also a Weil function for H. This leads to the following definition.

Definition 7.4. Let X be a proper scheme over k, let D be an effective Cartier divisor on X, let λ_{D} be a Weil function for D, and let U = X \ Supp D. Then a (λ_{D}, S)-integral point on X is a point P ∈ U(k̄) such that λ_{D,w}(P) ≤ 0 for all w ∈ M_{k(P)} not lying over a place of k in S. We may also say that P is a λ_{D}-integral point if S is clear from the context.

Since the above definition does not refer to the ring R, it applies also to the case where k is a function field and S = ∅.

One can also formulate a Serre-like definition in these terms:

Definition 7.5. Let X, D, and U be as in Definition 7.4. Then a set of (D, S)-integral points on X is a subset Σ ⊆ U(k̄) (or Σ ⊆ U(k) ) for which there exists a Weil function λ_{D} for D such that all P ∈ Σ are (λ_{D}, S)-integral points. We may also say that Σ is a set of D-integral points, if S is clear from the context.

Remark 7.6. The above definition makes sense even if U is not affine. To take an extreme example, if D = 0 then we can take λ_{D} = 0 and γ = 0 , so that any algebraic (or rational) point is also an integral point. In many other cases, U is neither proper nor affine; for example U = 𝔸^{1} × ℙ^{1} . Another important family of examples is the family of moduli spaces 𝒜_{g,n} of abelian varieties, which again is neither affine nor proper.

We now show that Definition 7.5 is well-behaved with respect to change of embedding into a completion. To do that, we start by proving a more general result.

Proposition 7.7. Let X and Y be proper schemes over k; let D and E be effective Cartier divisors on X and Y , respectively; let U = X \ D and V = Y \ E; and let f: U → V be a morphism. Then f takes integral sets to integral sets; i.e., if Σ ⊆ U(k̄) is a set of D-integral points, then f(Σ) ⊆ V (k̄) is a set of E-integral points.

Proof. Since we are working with algebraic points, we may assume that X and Y are reduced. Then f defines a rational map X −−→ Y . Let X′ be the closure of the graph of f , let p: X′ → X and f′: X′ → Y be the projection morphisms, and let D′ = p^{∗}D. Then U is isomorphic to the open subscheme X′ \ D′, compatible with p. Also, if Σ is a D-integral set of points on X, then there is a Weil function λ_{D} for D such that all P ∈ Σ are λ_{D}-integral, λ_{D′} := p^{∗}λ_{D} is a Weil function for D′, and all P ∈ Σ are λ_{D′}-integral on X′; thus Σ is a D′-integral set of points there. Therefore we may replace X and D with X′ and D′, and thus reduce to the case in which f extends to a morphism X → Y .

Let Σ be a set of D-integral points, and let λ_{D} be a Weil function for D such that all P ∈ Σ are λ_{D}-integral. Let λ_{E} be a Weil function for E. Since f(U) ⊆ V , it follows that Supp D ⊇ Supp f^{∗}E. By Corollary 5.2 (applied to −f^{∗}E and D), there is an integer n and an M_{k}-constant γ such that

f^{∗}λ_{E} ≤ n · λ_{D} + γ .

After adjusting λ_{E} , we may assume that γ = 0 , so that f^{∗}λ_{E} ≤ n · λ_{D}. Then, for all P ∈ Σ , we have λ_{E,w}(f(P)) ≤ n · λ_{D,w} ≤ 0 for all w ∈ M_{k(P)} not lying over a place in S. Thus all points in f(Σ) are λ_{E}-integral, so f(Σ) is E-integral. □

Remark. In the case where both U and V are affine, this can also be proved more directly by the method of Lemma 2.4.

Corollary 7.8. The notion of set of D-integral points on X is independent of the multiplicities of the components of D, and in fact depends only on U .

Proof. Apply Proposition 7.7 to the identity map on U , using two divisors on X with support X \ U , or using two different embeddings of U into proper schemes whose images are the complements of divisors. □

Therefore we define:

Definition 7.9. Let U be a scheme of finite type over k. A subset Σ of U(k̄) is a set of S-integral points on U if Σ is (D, S)-integral for some (and hence all) pairs (X, D) , where X is a proper scheme over k containing U as an open subscheme, and D is an effective Cartier divisor on X with Supp D = X \ U . We also say that Σ is a set of integral points on U if S is clear from the context.

References

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[H–S] M. Hindry and J. H. Silverman, Diophantine geometry: an introduction, Graduate Texts in

Mathematics 201, Springer-Verlag, 2000.

[K] N. Koblitz, P-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics 58, Springer-Verlag, 1984.

[L 1] S. Lang, Fundamentals of diophantine geometry, Springer-Verlag, 1983. [L 2] , Number theory III: diophantine geometry, Encyclopaedia of Mathematical Sciences; 60,

Springer-Verlag, 1991.

[M] H. Matsumura, Commutative ring theory, Translated by M. Reid; Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1986.

[S] J.-P. Serre, Lectures on the Mordell-Weil theorem, 2nd edition, Aspects of Mathematics E 15, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1990.

Department of Mathematics, University of California, Berkeley, CA 94720