Diophantine Approximation

and Nevanlinna Theory

Lecture IV

Paul Vojta University of California, Berkeley

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.”

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 Cmorphism 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.

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.