Diophantine Approximation

and Nevanlinna Theory

Paul Vojta University of California, Berkeley

Nevanlinna Theory

Nevanlinna theory is part (most) of value distribution theory of holomorphic functions

Consider the function e^{z}. It has no zeroes or poles, so as a map ℂ → ℙ^{1}(ℂ) it omits the values 0 and ∞.

Theorem (Picard). There is no non-constant holomorphic function ℂ → ℙ^{1}(ℂ) omitting three or more values.

[draw (e^{z})^{−1}(2) ]

Note that (e^{z})^{−1}(w) = \{ \begin{matrix}\{log w + 2πin : n ∈ ℤ\} & w ∈ ℂ \ \{0\}, \\ ∅ & w = 0, ∞ \end{matrix}

∴ \{z ∈ ℂ : e^{z} = w and |z| ≤ r \} = \{ \begin{matrix}\frac{r}{π} + O_{w}(1) & w ∈ ℂ \ \{0\}, \\ 0 & w = 0, ∞ \end{matrix}

From now on assume f(0) ≠ 0, ∞.

Let log^{+} x = max{0, log x}. Also let f: ℂ → ℂ be meromorphic.

Definition. The proximity function is

m_{f}(r) = ∫_{0}^{2π} log^{+} |f(re^{iθ})|\frac{dθ}{2π} , and

m_{f}(a, r) = m_{1/(f−a)}(r) = − ∫_{0}^{2π} log^{−} |f(re^{iθ}) − a|\frac{dθ}{2π} a ∈ ℂ .

Also let m_{f}(∞, r) = m_{f}(r) .

Definition. The counting function is

N_{f}(r) = ∑_{|z|<r} ord_{z}^{+}(1/f) · log \frac{r}{|z|} , and

N_{f}(a, r) = N_{1/(f−a)}(r) = ∑_{|z|<r} ord_{z}^{+}(f − a) · log \frac{r}{|z|} .

Also let N_{f}(∞, r) = N_{f}(r) .

Finally, we define the height function by

T_{f}(r) = m_{f}(r) + N_{f}(r) .

If f(z) = e^{z} then N_{f}(∞, r) = 0 and

m_{f}(∞, r) = ∫ log^{+} e^{r cos θ} \frac{dθ}{2π} = r ∫_{−π/2}^{π/2} cos θ \frac{dθ}{2π} = \frac{r}{π} .

Theorem (First Main Theorem (FMT)). For all a ∈ ℂ,

m_{f}(a, r) + N_{f}(a, r) = T_{f}(r) + O_{f,a}(1) .

Since m_{f}(a, r) ≥ 0 , this gives an upper bound on N_{f}(a, r) . Compare with Jensen’s formula

log |c_{λ}| = ∫_{0}^{2π} log |f(re^{iθ})|\frac{dθ}{2π} + N_{f}(∞, r) − N_{f}(0, r) .

Theorem (Second Main Theorem (SMT)). Let a_{1}, …, a_{q} ∈ ℙ^{1}(ℂ) be distinct. Then

(*)  ∑_{i=1}^{q} m_{f}(a_{i}, r) ≤_{exc} 2 T_{f}(r) + O(log^{+} T_{f}(r)) + o(log r) ,

where O( ) and o( ) depend only on f and a_{1}, …, a_{q}, and ≤_{exc} means that the inequality holds for all r ∈ (0, ∞) outside of a set of finite Lebesgue measure.

Corollary (Picard). If f: ℂ → ℙ^{1}(ℂ) \ {a_{1}, a_{2}, a_{3}} is holomorphic with a_{1}, a_{2}, a_{3} distinct, then f is constant.

Proof. Since f never equals a_{i}, we have N_{f}(a_{i}, r) = 0 , so the FMT gives m_{f}(a_{i}, r) = T_{f}(r) + O(1) . The left-hand side of (*) is therefore 3 T_{f}(r)+O(1) , so (*) becomes T_{f}(r) ≤_{exc} O(log^{+} T_{f}(r))+ o(log r) . But, if f is nonconstant then T_{f}(r) ≥ O(log r) , a contradiction. Therefore f is constant. □

One can view the SMT as a lower bound on N_{f}(a, r) : the left-hand side of (*) is q T_{f}(r) − ∑ m_{f}(a_{i}, r) , so (*) is equivalent to

∑_{i=1}^{q} N_{f}(a_{i}, r) ≥_{exc} (q − 2)T_{f}(r) + O(log^{+} T_{f}(r)) + o(log r) .

Advantages:

(1). q − 2 = χ(ℙ^{1} \ q points ) ,

(2). The left-hand side is independent of metrics; and (3). One can phrase it using truncated counting functions

(abc conjecture).

Number Theory

For a number field k, let M_{k} be its set of places. This is in one-to-one correspondence with the disjoint union

{nonzero primes in 𝒪_{k} } ∐{σ: k ↪ ℝ}
∐{unordered pairs (σ, σ̄): σ ≠ σ̄: k ↪ ℂ } .

For v ∈ M_{k} we define norms ∥ · ∥_{v} by

∥x∥ = \{ \begin{matrix}(𝒪_{k} : 𝔭)^{− ord_{𝔭}(x)} & if v ∤ ∞, x ≠ 0, \\ |σ(x)| & if v is real , \\ |σ(x)|^{2} & if v is complex . \end{matrix}

We then have a product formula ∏_{v∈M_{k}} ∥x∥_{v} = 1 for all x ∈ k, x ≠ 0 .

Let S_{∞} denote the set of archimedean (real or complex) places.

Let S ⊇ S_{∞} be a finite set of places of k; for x ∈ k we then define

m_{S}(x) = m_{S}(∞, x) = ∑_{v∈S} log^{+} ∥x∥_{v} ,

m_{S}(a, x) = m_{S}(\frac{1}{x − a}) = ∑_{v∈S} log^{+} ∥\frac{1}{x − a}∥_{v} ,

N_{S}(x) = N_{S}(∞, x) = ∑_{v∉S} log^{+} ∥x∥_{v} = ∑_{v∉S} ord_{v}^{+}(1/x) log(𝒪_{k} : 𝔭) ,

N_{S}(a, x) = N_{S}(\frac{1}{x − a}) = ∑_{v∉S} log^{+} ∥\frac{1}{x − a}∥_{v} .

h_{k}(x) = m_{S}(x) + N_{S}(x) = ∑_{v∈M_{k}} log^{+} ∥x∥_{v} = log ∏_{v} max{1, ∥x∥_{v}} .

Corresponding to the FMT, we have

m_{S}(a, x) + N_{S}(a, x) = h_{k}(\frac{1}{x − a}) = h_{k}(x) + O_{a,k}(1) ,

a property of heights.

Theorem (Roth). Let k and S be as above, and for all v ∈ S let α_{v} ∈ \overline{ℚ}. Let ϵ > 0 . Then the inequality

∏_{v∈S} min\{1, ∥x − α_{v}∥_{v}\} ≤ \frac{1}{H_{k}(x)^{2+ϵ}} .

holds for only finitely many x ∈ k.

This is equivalent to the same statement with α_{v} ∈ k for all v (expand k).

Equivalently, given k, S, ϵ, and a_{1}, …, a_{q} ∈ k, then the inequality

∏_{i=1}^{q} ∏_{v∈S} min\{1, ∥x − a_{i}∥_{v}\} ≤ \frac{1}{H_{k}(x)^{2+ϵ}}

holds for only finitely many x ∈ k.

Taking − log of both sides, and rearranging the logic, we then have that

∑_{i=1}^{q} m_{S}(a_{i}, x) ≤ (2 + ϵ)h_{k}(x) + O(1)

for almost all x ∈ k.

Schmidt’s Subspace Theorem and a Theorem of Cartan We start with Nevanlinna theory. Let H be a hyperplane in ℙ^{n}(ℂ) ( n > 0 ), defined by a_{0}x_{0} + ⋯ + a_{n}x_{n} = 0 . For P ∈ ℙ^{n}(ℂ) , P ∉ H, with homogeneous coordinates [z_{0} : ⋯: z_{n}] , let

λ_{H}(P) = −\frac{1}{2} log \frac{|a_{0}z_{0} + ⋯ + a_{n}z_{n}|^{2}}{|z_{0}|^{2} + ⋯ + |z_{n}|^{2}} ,

and for a holomorphic curve f: ℂ → ℙ^{n}(ℂ) with image not contained in H, define the proximity function

m_{f}(H, r) = ∫_{0}^{2π} λ_{H}(f(re^{iθ}))\frac{dθ}{2π} .

For the counting function, note that f^{∗}H is an analytic divisor on ℂ; we can then define

N_{f}(H, r) = ∑_{|z|<r} ord_{z} f^{∗}H · log \frac{r}{|z|} .

For the height function, we note that for any hyperplane H not containing the image of f ,

T_{f}(r) := m_{f}(H, r) + N_{f}(H, r)

is independent of H up to O(1) , with an implicit constant depending only on H.

Theorem (Cartan). Let n > 0 and let H_{1}, …, H_{q} be hyperplanes in ℙ^{n}(ℂ) in general position (i.e., every intersection of r ≤ n of them has codimension r). Let f: ℂ → ℙ^{n}(ℂ) be a holomorphic map whose image is not contained in any hyperplane. Then

∑_{i=1}^{q} m_{f}(H_{i}, r) ≤_{exc} (n + 1)T_{f}(r) + O(log^{+} T_{f}(r)) + o(log r) .

In number theory, things are very similar. Let k and S be as above, and let H be a hyperplane in ℙ_{k}^{n} given by a_{0}x_{0}+⋯+a_{n}x_{n} = 0 (with a_{i} ∈ k for all i). For P ∈ P^{n}(k) \ H with homogeneous coordinates [x_{0} : ⋯: x_{n}] (again with x_{i} ∈ k for all i), and v ∈ M_{k} we define

λ_{H,v}(P) = − log \frac{∥a_{0}x_{0} + ⋯ + a_{n}x_{n}∥_{v}}{max\{∥x_{0}∥_{v}, …, ∥x_{n}∥_{v}\}} ,

and define a proximity function m_{S}(H, P) = ∑_{v∈S} λ_{H,v}(P) , a counting function N_{S}(H, P) = ∑_{v∉S} λ_{H,v}(P) , and a height function

h_{k}(P) = m_{S}(H, P) + N_{S}(H, P) = ∑_{v∈M_{k}} − log \frac{∥a_{0}x_{0} + ⋯ + a_{n}x_{n}∥_{v}}{max\{∥x_{0}∥_{v}, …, ∥x_{n}∥_{v}\}}

= ∑_{v} log max ∥x_{i}∥_{v}

by the product formula.

Theorem (Schmidt’s Subspace Theorem). Let k, S, ϵ > 0 , and n > 0 be as above, and let H_{1}, …, H_{q} be hyperplanes in ℙ_{k}^{n} in general position. Then

∑_{i=1}^{q} m_{S}(H_{i}, x) ≤ (n + 1 + ϵ)h_{k}(x) + O(1)

for all x ∈ ℙ^{n}(k) outside of a finite union of proper linear subspaces of ℙ_{k}^{n}.

Actually, Schmidt had a collection H_{v,0}, …, H_{v,n} for each v ∈ S, in general position for each v. The union, however, did not need to be in general position. This is equivalent to:

Theorem (Schmidt’s Subspace Theorem). Let k, S, ϵ > 0 , and n > 0 be as above, and let H_{1}, …, H_{q} be hyperplanes in ℙ_{k}^{n}. Then the inequality

∑_{v∈S} max_{J} ∑_{i∈J} λ_{H_{i},v}(x) ≤ (n + 1 + ϵ)h_{k}(x) + O(1)

holds for all x ∈ ℙ^{n}(k) outside of a finite union of proper linear subspaces. Here J varies over a given collection of subsets of {1, …, q} for which {H_{i} : i ∈ J} lie in general position.

Theorem (Cartan). Let H_{1}, …, H_{q} be hyperplanes in ℙ^{n}(ℂ) and let f: ℂ → ℙ^{n} be a holomorphic curve whose image is not contained in any hyperplane. Then

∫_{0}^{2π} max_{J} ∑_{i∈J} λ_{H_{i}}(f(re^{\sqrt{−1}θ}))\frac{dθ}{2π} ≤_{exc} (n+1)T_{f}(r)+O(log^{+} T_{f}(r))+o(log r)

(where J is as before).

The Dictionary

• One holomorphic map corresponds to an infinite set of rational points.

• One rational point may correspond to f|_{\overline{𝔻}_{r}} .

• r ∈ (0, ∞) corresponds to x ∈ k

• θ ∈ [0, 2π] corresponds to v ∈ S

• |f(re^{iθ})| corresponds to ∥x∥_{v}, v ∈ S

• ord_{z} f ( |z| < r) corresponds to ord_{v} x, v ∉ S

• log \frac{r}{|z|} corresponds to log(𝒪_{k} : 𝔭)

[draw 𝔻_{r}]

The Borel Lemma/Unit Theorem

Lemma (Borel). If g_{1}, …, g_{n} are entire functions such that

e^{g_{1}} + ⋯ + e^{g_{n}} = 1

then some g_{i} is constant.

Proof. [e^{g_{1}} : ⋯: e^{g_{n}}] gives a holomorphic map f: ℂ → ℙ^{n−1} missing the n coordinate hyperplanes and also the hyperplane x_{0} + ⋯ + x_{n} = 0 . Therefore m(H_{i}, r) = T_{f}(r) + O(1) for each of these n + 1 hyperplanes, so f must be linearly degenerate. We can then eliminate one of the g_{i} and then use induction on n.

□

Lemma (Schlickewei, van der Poorten). Given a collection of n-tuples (u_{1}, …, u_{n}) of units in 𝒪_{k} satisfying u_{1} +⋯+u_{n} = 1 , all but finitely many have the property that there is a proper subset I of {1, …, n} with at least two elements and having the property that ∑_{i∈I} u_{i} = 0 .

Proof. N_{S}(H_{i}, [u_{1} : ⋯: u_{n}]) = 0 for the same hyperplanes as before, so the given points all lie in a finite union of proper linear subspaces. Therefore there is an infinite subset lying in a hyperplane. □

Why do these theorems correspond?

Proofs of results relying on these two theorems also correspond.

Weil Functions

There is a unique way (up to a rather involved definition of O(1) ) to assign to each pair (X, D) , where X is a complete k-variety and D is a Cartier divisor on X, a Weil function

λ_{D}: ∐_{v∈M_{k}}(X \ D)(\overline{k}_{v}) → ℝ ,

such that

(i). normalization: if H ⊆ ℙ^{n} is the hyperplane at infinity, then

λ_{H,v}([x_{0} : ⋯: x_{n}]) = − log \frac{∥x_{0}∥_{v}}{max ∥x_{i}∥_{v}} + O(1) ,

(ii). additivity: λ_{D+D′} = λ_{D} + λ_{D′} + O(1) , and (iii). functoriality: if f: X → Y is a morphism and D is a divisor on Y whose support does not contain the image of X, then λ_{f^{∗}D}(x) = λ_{D}(f(x)) + O(1) . (iv). continuity: λ_{D,v} is continuous in the v-topology.

Here λ_{D,v} means the restriction of λ_{D} to (X \ D)(\overline{k}_{v}) , and additivity is only assumed to hold on the intersection of the domains.

We also have h_{𝒪(D),k} (x) = ∑_{v∈M_{k}} λ_{D,v}(x)+O(1) for all x ∉ Supp D. This also works for points over \overline{k}, with suitable adjustments.

We can then define a proximity function m_{S}(D, P) = ∑_{v∈S} λ_{D,v}(P) and a counting function N_{S}(D, P) = ∑_{v∉S} λ_{D,v}(P) for all P ∈ (X \ D)(k) .

Again, m_{S}(D, P) + N_{S}(D, P) = h_{𝒪(D),k} (P) + O(1) .

In Nevanlinna theory itself: Let X be a complete complex variety and let D be a Cartier divisor on X. Choose a hermitian metric on 𝒪(D) , and let 1_{D} denote the canonical section. Then λ_{D} := −\frac{1}{2} log |1_{D}|^{2} gives a Weil function (which is a much simpler notion in this context).

For a holomorphic map f: ℂ → X whose image is not contained in the support of D, we may then define a proximity function

m_{f}(D, r) = ∫_{0}^{2π} λ_{D}(f(re^{iθ}))\frac{dθ}{2π}

and counting function N_{f}(D, r) = ∑_{|z|<r} ord_{z} f^{∗}D · log \frac{r}{|z|} .

Integral Points

Let k and S be as above, let X be a complete variety over k, and let D be an effective divisor on X such that X \ Supp D is affine. Naïvely, a point P ∈ X(k) is a (D, S)-integral point on X if there is a rational map i: X −−→ ℙ^{n} for some n, defined and injective on X \ Supp D, such that Supp D = i^{−1}({x_{0} = 0}) and i(P) ∈ 𝔸^{n}(𝒪_{k,S}) .

However, this notion is useful only for infinite sets (for any finite set you can clear the denominators) (Serre). This definition can be expressed using Weil functions:

Definition. A subset Σ of X(k) is (D, S)-integral if there is a Weil function λ_{D} for D and constants c_{v} ∈ ℝ for all v ∉ S such that c_{v} = 0 for almost all v and λ_{D,v}(P) ≤ c_{v} for all v ∉ S and all P ∈ Σ .

If D = i^{∗}({x_{0} = 0}) , then this is easily seen to be equivalent to the earlier naive definition if i is an actual morphism, by functoriality and normalization of Weil functions. Moreover, by additivity of Weil functions this notion depends only on Supp D. Indeed, if D and D′ are effective divisors with the same support, then D ≤ mD′ for some m, and so λ_{D} ≤ mλ_{D′} + O(1) . It also depends only on the quasi-projective variety X \ Supp D.

This definition does not assume that X \D is affine. This is useful, e.g., for 𝒜_{g,n} (the moduli space of principally polarized abelian varieties of given dimension and level structure). Also, if D = 0 then the condition is vacuous and all sets of rational points are integral.

Also, note that this definition is well-behaved with respect to morphisms of X \ D: given morphisms ϕ: X → X′ and effective divisors D on X and D′ on X′ such that ϕ(X\Supp D) ⊆ X′\Supp D′, then Supp D ⊇ ϕ^{−1}(Supp D′) , so ϕ^{∗}D′ ≤ nD. Therefore, by functoriality and linearity of Weil functions, (D, S)-integral points on X are mapped to (D′, S)-integral points on X′.

Of course, (D, S)-integrality implies that N_{S}(D, P) is bounded, and this corresponds to N_{f}(D, r) being bounded in Nevanlinna theory. One important case in which this holds is when f misses the support of D, and one notes of course that maps to X \ Supp D are also well-behaved with respect to morphisms ϕ: X → X′ taking X \ D to X′ \ D′.