Lecture #1

§1. 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 First Main Theorem 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).

§2. 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.

§3. 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).

§4. 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}] §5. 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.

§6. 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|} .

§7. 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′.

Lecture #2

§1. Conjectures

Conjecture. Let k be a number field, let S ⊇ S_{∞} be a finite set of places of k, let X be a smooth projective variety over k, let D be a normal crossings divisor on X, let A be an ample divisor on X, let 𝒦 denote the canonical line sheaf on X, and let ϵ > 0 . Then there is a proper Zariski-closed subset Z of X, depending only on the above data, such that

m_{S}(D, x) + h_{𝒦 ,k} (x) ≤ ϵ h_{A,k}(x) + O(1)

for all x ∈ (X \ Z)(k) .

Conjecture. Let X be a smooth projective complex variety, let D be a normal crossings divisor on X, let A be an ample divisor on X, and let 𝒦 denote the canonical line sheaf on X. Then there is a proper Zariski-closed subset Z of X, depending only on the above data, such that for all holomorphic maps f: ℂ → X with f(ℂ) ⊈ Z,

m_{f}(D, r) + T_{𝒦 ,f} (r) ≤_{exc} O(log^{+} T_{A,f}(r)) + o(log r) .

These are proved if dim X = 1 , or if X = ℙ^{n} and D is a sum of hyperplanes. Limited results have also been shown if X is a certain type of compactification of a semiabelian variety, or if X is a completion of a Shimura variety.

In other words, we know very little if dim X > 1 .

§2. Embeddings For example, consider ℙ^{2} , and let D be as above. If deg D ≥ 4 then ℙ^{2} \ D should not have a dense set of integral points (or admit a Zariski-dense holomorphic curve to it).

If D is smooth of degree ≥ 4 , then there’s no hope (so far). For four lines, however, we do have results (Schmidt and Cartan).

For three lines and a conic, there are some results; e.g., if L_{1}, L_{2}, L_{3} are linear forms defining the lines and Q is a quadratic polynomial defining the conic, then all L_{i}^{2}/Q must be units (or nearly so) at integral points, so we can apply the unit lemma (or Borel’s lemma) (M. Green and V.)

Besides that, how do we get any results for a conic?

Under the 2-uple embedding ℙ^{2} ↪ ℙ^{5} , the image of a conic is contained in a hyperplane. Under the 3-uple embedding ℙ^{2} ↪ ℙ^{9} , the image of a conic spans a linear subspace of codimension 3 , so we can get 3 linearly independent hyperplanes containing it.

More generally, say you have a divisor D of degree d in ℙ^{n}, and look at its image under the r-uple embedding

ℙ^{n} ↪ ℙ^{\binom{r+n}{n}−1} .

The image of the divisor spans a linear subspace of codimension \binom{r+n−d}{n}, because there are that many monomials of degree n − d (which then get multiplied by the form defining D to get homogeneous polynomials of degree n in the original variables, hence hyperplanes in the image space).

So the inequality we’d get out of applying Schmidt’s Subspace Theorem to ℙ^{\binom{r+n}{n}−1} would look like

\binom{r + n − d}{n}m(D, x) + ⋯ ≤ (\binom{r + n}{n} · r + ϵ) h_{k}(x) + O(1)

for x ∈ ℙ^{n}(k) outside of a finite union of proper subvarieties of degree ≤ r. So, as r → ∞, we’re interested in the ratio

\frac{\binom{r+n−d}{n}}{\binom{r+n}{n} · r} = \frac{(r − d + n) ⋯ (r − d + 1)}{(r + n) ⋯ (r + 1)r} = \frac{r^{n} + O(r^{n−1})}{r(r^{n} + O(r^{n−1}))} → \frac{1}{r} → 0 .

This is not very promising, but we can try harder. We can also consider that some hyperplanes can be made to contain D twice, or three times, etc. This gives something like

\begin{matrix}(\binom{r + n − d}{n} + \binom{r + n − 2d}{n} + …) m(D, x) + … \\ ≤ (\binom{r + n}{n} · r + ϵ) h_{k}(x) + O(1) \end{matrix}

To estimate the factor in front of m(D, x) :

\binom{r − kd + n}{n} = \frac{(r − kd + n) ⋯ (r − kd + 1)}{n!} = \frac{(r − kd)^{n} + O_{n}((r − kd)^{n−1})}{n!}

and therefore

∑_{k=1}^{[r/d]} \binom{r − kd + n}{n} = \frac{(r − d)^{n} + ⋯ + (r − [r/d]d)^{n} + O_{n,d}(r^{n})}{n!} .

As r → ∞, the ratio of interest now converges to

\frac{∑{[r/d] \atop k=1} \binom{r−kd+n}{n}}{r\binom{r+n}{n}} ≈ \frac{\frac{r^{n+1}}{(n+1)d·n!}}{\frac{r^{n+1}}{n!}} → \frac{1}{(n + 1)d} .

This would give us something like

d m_{S}(D, x) + ⋯ ≤ (n + 1 + ϵ)h_{k}(x) + O(1) .

This is best possible if d = 1 , but is still new and noteworthy if d > 1 .

But there is something wrong with this.

Suppose we have two divisors, and their images under the r-uple embedding span linear subspaces L_{1} and L_{2} of codimensions ρ_{1} and ρ_{2} , respectively. We have

codim(L_{1} ∩ L_{2}) ≤ ρ_{1} + ρ_{2}

(assuming L_{1} ∩L_{2} ≠ ∅). If this inequality is strict then we have problems to the extent that m(L_{1}, y) and m(L_{2}, y) come from m(L_{1} ∩ L_{2}, y) (where y is the image of x under the r-uple embedding). Say you have ρ_{1} generic hyperplanes containing L_{1} and ρ_{2} generic hyperplanes containing L_{2} . If these ρ_{1} + ρ_{2} hyperplanes are in general position then this implies

codim(L_{1} ∩ L_{2}) ≥ ρ_{1} + ρ_{2} .

So we need equality. However, the usual computation of Hilbert function using short exact sequences gives (for sufficiently large n) that the codimension of the linear span of D_{1} ∩ D_{2} is

\binom{n − d_{1} + r}{n} + \binom{n − d_{2} + r}{n} − \binom{n − d_{1} − d_{2} + r}{n} ,

which is too small by \binom{n−d_{1}−d_{2}+r}{n}.

[Lecture continued on the blackboard.]

Lecture #4

§1. More abc One can also get the abc conjecture from the earlier conjecture on rational points (for varieties of large dimension).

For n large let X_{n} be the variety in (ℙ^{2})^{n} in coordinates

([x_{1} : y_{1} : z_{1}], …, [x_{n} : y_{n} : z_{n}])

given by the equation ∏ x_{i}^{i} + ∏ y_{i}^{i} + ∏ z_{i}^{i} = 0 , and let Γ_{n} be the closure of the graph of the rational map X_{n} −−→ ℙ^{2} given by

([x_{1} : y_{1} : z_{1}], …, [x_{n} : y_{n} : z_{n}]) ↦ [∏ x_{i}^{i} : ∏ y_{i}^{i} : ∏ z_{i}^{i}] .

Let ϕ: Γ_{n} → ℙ^{2} be the resulting morphism. The image of ϕ is a line in ℙ^{2} , which we identify with ℙ^{1} .

Given relatively prime a, b, c ∈ ℤ with a + b + c = 0 , we get a point P_{a,b,c} on Γ_{n} by letting

x_{n} = ∏_{p} p^{[ord_{p} a/n]} ,  x_{i} = ∏_{ord_{p} a≡i (mod n)} p (i < n) ,

so that a = ∏ x_{i}^{i} with x_{n} as large as possible. Similarly define the y_{i} using b and the z_{i} using c. Let D be the divisor x_{1} ⋯ x_{n}y_{1} ⋯ y_{n}z_{1} ⋯ z_{n} = 0 on Γ_{n}. It is then possible to show that

N(D, P_{a,b,c}) ≤ ∑_{p|abc} log p + \frac{1}{n}N(E, ϕ(P_{a,b,c})) ,

where E is the divisor [0] + [1] + [∞] on the image ℙ^{1} of ϕ.

There are additional obstacles to overcome, but basically the conjecture for rational points applied to a certain desingularization of Γ_{n} then gives the abc conjecture.

We also note that the group 𝔾_{m}^{2n−2} acts on X_{n}, and therefore on Γ_{n}, by x_{i} ↦ tx_{i}, x_{n} ↦ t^{−i}x_{n} ( 1 ≤ i < n) for the first n − 1 factors, and similarly with the y_{i} for the remaining factors. This action preserves the divisor D and is faithful.

This implication therefore relies only on the conjecture for rational points for varieties X and divisors D for which there is a semiabelian variety G of dimension dim X − 1 acting faithfully on X in such a way as to preserve D. In this special case, the conjecture has been proved in the split function field case and in the Nevanlinna case.

It’s actually pretty easy to end up in “abc land.”

§2. The 1 + ϵ conjecture Recall that the 1 + ϵ conjecture states that if X is a smooth projective curve over a number field k then for all algebraic points x of bounded degree,

h_{K}(x) ≤ (1 + ϵ)d_{k}(x) + O(1) .

We can prove this in the “split” function field case, as follows (essentially following de Franchis):

Let C be a smooth projective curve over a ground field F of characteristic 0, and let k = K(C) . Let X_{0} be a smooth projective curve over F and D_{0} a reduced effective divisor on X_{0} . Then X := X_{0} ×_{F} C → C is a model for X_{0} ×_{F} k, and D_{0} extends to pr_{1}^{∗} D_{0} on X, where pr_{1} is the projection of X = X_{0} ×C to its first factor.

An algebraic point x on X_{k} corresponds to a nonsingular finite cover E of C of degree d and a generically injective C-morphism i: E → X. We have

d_{k}(x) = (2g(E) − 2)/d − (2g(C) − 2) ,

which is the degree of the ramification divisor divided by d. Also K_{X/C} = pr_{1}^{∗} K_{X_{0}} , and the height of x relative to this canonical divisor is

\frac{(2g(X_{0}) − 2) deg(pr_{1} ∘i)}{d} ≤ \frac{2g(E) − 2}{d} = d_{k}(x) + (2g(C) − 2) . □

A way to look at this proof is that we know the derivative of E, as an element of the absolute tangent bundle T_{X/F} . Since we’re in the “split” case, T_{X/F} splits into a product T_{X/X_{0}} ×T_{X/C} , and we can project onto the second factor, the relative tangent bundle.

As I see it, a key aspect of McQuillan’s proof of the 1+ϵ conjecture in the (general) function field case of characteristic zero is the observation that although you don’t have a canonical projection of the absolute tangent bundle to the relative tangent bundle anymore, you can choose a projection arbitrarily, and for points of large height the value of the projection doesn’t vary that much with the projection, in proportion with the tangent vector.

§3. Derivatives One of the key tools in Nevanlinna theory is the following:

Theorem (Lemma on the Logarithmic Derivative (LLD)) (Nevanlinna). Let f be a meromorphic function on ℂ. Then

∫_{0}^{2π} log^{+} |\frac{f′(re^{iθ})}{f(re^{iθ})}|\frac{dθ}{2π} ≤_{exc} O(log^{+} T_{f}(r)) + o(log r) .

Proof. Omitted (not instructive).

Theorem (Geometric LLD (GLLD) (Wong, McQuillan)). Let X be a smooth projective complex variety, let D be a normal crossings divisor on X, and let f: ℂ → X be a holomorphic curve with image not contained in Supp D. Let 𝒜 be an ample line sheaf on X. Let |·| be a hermitian metric on the log tangent bundle T_{X}(− log D) , and let d_{D}f: ℂ → T_{X}(− log D) denote the canonical lifting of f . Then

∫_{0}^{2π} log^{+}|d_{D}f(re^{iθ})|\frac{dθ}{2π} ≤_{exc} O(log^{+} T_{𝒜 ,f} (r)) + o(log r) .

Proof. Again omitted. Basically it uses the classical LLD locally and applies compactness.

Remark. The case X = ℙ^{1} , D = {0, ∞} recovers the earlier LLD.

This in turn implies:

Theorem (McQuillan’s “tautological inequality”). Let X, D, f: ℂ → X, and 𝒜 be as above. Let f′: ℂ → ℙ(Ω_{X}^{1}(log D)) be the canonical lifting of f , and let 𝒪(1) denote the tautological line sheaf on ℙ(Ω_{X}^{1}(log D)) . (Here ℙ() is as in EGA.) Then

T_{𝒪(1),f′} (r) ≤_{exc} N_{f}^{(1)}(D, r) + O(log^{+} T_{𝒜 ,f} (r)) + o(log r) .

Proof (sketch). Let V = 𝕍(Ω_{X}^{1}(log D)) , the total space of T_{X}(− log D) , let

\overline{V} = ℙ(Ω_{X}^{1}(log D) ⊕ 𝒪_{X}) ,

the projective closure of V , let [∞] = \overline{V} \ V , a divisor on \overline{V} , let p: P → \overline{V} be the blowing-up of \overline{V} along the zero section of V , let [0] ⊇ P be the exceptional divisor, and let ϕ: ℂ → P be the lifting of d_{D}f: ℂ → V .

We have a diagram ... of X-schemes and d_{D}f = p ∘ ϕ, f′ = q ∘ ϕ. In addition,

q^{∗}𝒪(1) ≅ p^{∗}𝒪(1) ⊗ 𝒪(−[0]) .

Also (on \overline{V} ) 𝒪(1) ≅ 𝒪([∞]) , since [∞] is cut out by the global section (0, 1) of Ω_{X}^{1}(log D) ⊕ 𝒪_{X} .

Therefore

T_{𝒪(1),f′} (r) = T_{q^{∗}𝒪(1),ϕ} (r) = T_{𝒪(1),d_{D}f} (r) − T_{𝒪([0]),ϕ} (r)
= N_{d_{D}f}([∞], r) + m_{d_{D}f}([∞], r) − N_{ϕ}([0], r) − m_{ϕ}([0], r)
≤_{exc} N_{f}^{(1)}(D, r) + O(log^{+} T_{𝒜 ,f} (r) + o(log r) − 0 − 0 .

Explanation: The first term ≤ N_{D}^{(1)} because d_{D}f is bounded except where f hits D, and at that point it has a simple pole ( df/f has a simple pole at worst).

The second term is bounded by the GLLD.

The third and fourth terms are ≥ 0 , and can be ignored. □

Applications:

(1) The classical SMT for curves:

N_{f}^{(1)}(D, r) ≥_{exc} T_{K+D,f}(r) + S()

Proof. Ω_{X}^{1}(log D) = 𝒪(K_{X} + D) and ℙ(Ω_{X}^{1}(log D)) = X, with 𝒪(1) = 𝒪(K_{X} + D) . Therefore we get the inequality directly. □

(2) Cartan’s theorem (using a modified version with varying D)

(3) Results on holomorphic curves in closed subvarieties of semiabelian varieties (hoped for, but not yet proved).

(4) The same for open subvarieties of semiabelian varieties (again, hoped for but not yet proved).

(5) Shimura varieties (again, not yet).

Why is this interesting to number theorists? To push the analogy further, we need to know what is the analogue of the derivative of a holomorphic function.

Comparing Schmidt’s proof with a proof of Cartan’s theorem by H. Weyl, J. Weyl, and L. Ahlfors suggests that the analogue should involve Minkowski’s theory of successive minima, applied on the relative (co)tangent bundle.

Phrased in Arakelov theory, successive minima reduces to line subbundles of maximal degree, so we want a line subbundle of T_{X/Y} (− log D) of largest Arakelov degree, and therefore a quotient subbundle of Ω_{X/Y} (log D) of small degree. This corresponds to a point in ℙ(Ω_{X/Y} (log D)) lying over P ∈ X(k) ; i.e., a Y -section in ℙ(Ω_{X/Y} (log D)) lying over a section of X → Y for which the corresponding quotient Ω_{X/Y} (log D) ↠ ℒ has small Arakelov degree. What is ℒ ? It is the restriction of 𝒪(1) , so its degree is the height of this point relative to 𝒪(1) .

This leads to...

Conjecture (“Tautological conjecture”). Let k, S, X, D, and ϵ be as usual. For all P ∈ X(k) (or X(\overline{k}) with [k(P) : k] bounded) there is a P′ ∈ ℙ(Ω_{X/Y} (log D)) lying over P with

h_{𝒪(1),k} (P′) ≤ N_{S}^{(1)}(D, P) + ϵh_{A,k}(P) + O(1) .

This makes sense only in the context of infinite sequences, and preferably:

Definition (Zhang). A generic sequence on X is an infinite sequence of points in X(\overline{k}) such that all infinite subsequences are Zariski dense.

Proposition (Arithmetic chain rule). Let f: X_{1} → X_{2} be a morphism of k-varieties. Then, for all P ∈ X(\overline{k}) where f is étale, and for all P′ ∈ ℙ(Ω_{X/Y}) lying over P , the rational map f_{∗}: ℙ(Ω_{X_{1}/Y}) −−→ ℙ(Ω_{X_{2}/Y}) takes P′ to P′′ (lying over f(P) ) for which

h_{𝒪(1),k} (P′′) ≤ h_{𝒪(1),k} (P′) + O(1)

where the constant in O(1) depends only on f .

Proof. Extend X_{1} and X_{2} to models 𝒳_{1} and 𝒳_{2} over 𝒪_{k} for which f extends as a morphism, and let i: E → 𝒳_{1} be the section corresponding to P . Then P′ corresponds to a surjection i^{∗}Ω_{X_{1}/𝒪_{k}} ↠ ℒ , and h_{𝒪(1),k} (P′) is the Arakelov degree of ℒ divided by [K(E) : k] . We also have a morphism f^{∗}Ω_{X_{2}/𝒪_{k}} → Ω_{X_{1}/𝒪_{k}} , isomorphic at P′. Therefore we get a nonzero map (f ∘i)^{∗}Ω_{X_{2}/𝒪_{k}} → ℒ , so h_{𝒪(1),k} (P′′) ≤ h_{𝒪(1),k} (P′) (with heights defined using these models). □

A similar result holds for closed immersions (without the assumption on étaleness). Both Schmidt’s proof of his Subspace Theorem and Ahlfors’ proof of Cartan’s

theorem have two parts—an old part and a new part. The old parts look like the case for dimension 1 , with some necessary modifications. The new parts both involve multilinear algebra, and derivatives of the curve in Ahlfors’ case, or successive minima in Schmidt’s case. Similarities in the latter parts is what suggested the “tautological conjecture” (as well as McQuillan’s result in Nevanlinna theory).

I’ve been trying for some time to get these two proofs to line up better, but Ahlfors’ proof considered the hyperplanes separately when going up to higher exterior powers, whereas Schmidt considered them together. In 2006 I got a variant of Ahlfors’ proof that was more like Schmidt’s in this respect.

This used a modified tautological inequality, with varying D (similar to what was done with Cartan’s theorem).

Define a function μ_{D} on X measuring the improvement when you add D to McQuillan’s inequality, so that the inequality with D is roughly comparable to

T_{𝒪(1),f′} (r) + ∫_{0}^{2π} μ_{D}(f(re^{iθ}))\frac{dθ}{2π} ≤_{exc} S() .

Then the modified McQuillan inequality is

T_{𝒪(1),f′} (r) + ∫_{0}^{2π} max_{j} μ_{D_{j}}(f(re^{iθ}))\frac{dθ}{2π} ≤_{exc} S() .

My proof of Cartan’s theorem used that as the only analysis.