Differential algebraic geometry and abc

Alexandru Buium

Lecture 1. Motivation: abc on affine varieties.

We start by explaining a generalization of abc that makes sense for any affine variety over a global field. The usual abc is simply the case of P^{1}\{0, 1, ∞}. We then state our main result that says that this generalized abc holds, under certain trace conditions, for affine open sets of abelian varieties over function fields.

Let K be a field equipped with a family of absolute values | |_{v} : K → [0, ∞), v ∈ M, all of which, except finitely many, are non-archimedian. Set v(x) = −log|x|_{v} for x ∈ K^{×}. Assume (m_{v}) is a collection of positive integers such that the “product formula”

∑_{v} m_{v}v(x) = 0, x ∈ K^{×}

holds. Set

γ_{v} := inf{v(K^{×}) ∩ (0, ∞)}

Also, for any η = (η_{1}, ..., η_{N}) ∈ K^{N} and v ∈ M set

v(η) = min_{j} v(η_{j})

Define the (affine, logarithmic) height

height_{A^{N}} : A^{N}(K) = K^{N} → [0, ∞)

by the formula

height_{A^{N}} (η) = − ∑_{v(η)≤0} m_{v}v(η) = ∑_{v} m_{v} max_{j} log^{+} |η_{j}|_{v}

where log^{+} x := max{log x, 0}, x ∈ [0, ∞). Note that

height_{A^{N}} (η) = height_{P^{N}} (1 : η_{1} : ... : η_{N})

where

height_{P^{N}} (x_{0} : ... : x_{N}) = ∑_{v} m_{v} max_{j} log |x_{j}|_{v}

is the usual height in projective space. On the other hand define the (logarithmic) conductor

cond_{A^{N}} : A^{N}(K) = K^{N} → [0, ∞)

by the formula

cond_{A^{N}} (η) = ∑_{v(η)≤0} m_{v}γ_{v}

Clearly, by the very definition of γ_{v} we have

cond_{A^{N}} (η) ≤ height_{A^{N}} (η), η ∈ K^{N}

We need the following piece of notation. Let f, g : S → [0, ∞) be two real functions on a set S; we write

f ≤ g + O(1)

if there exists a positive real constant C such that

f(P) ≤ g(P) + C, P ∈ S

and we write f = g + O(1) if f ≤ g + O(1) and g ≤ f + O(1). We write

f << g + O(1)

if there exist two real positive constants C_{1}, C_{2} such that

f(P) ≤ C_{1}g(P) + C_{2}, P ∈ S

and we write f ≡ g + O(1) if f << g + O(1) and g << f + O(1). Coming back to height and cond one easily checks that if P : A^{N}(K) → A^{n}(K) a map given by an n−tuple of polynomials in N variables with K−coefficients then we have

height_{A^{n}} ∘ P << height_{A^{N}} + O(1)
cond_{A^{n}} ∘ P ≤ cond_{A^{N}} + O(1)

This permits to define the height and the conductor for any affine variety as follows. Let U be an affine variety over K. Let i : U → A^{N} be a closed immersion and define

height_{U} : U(K) → [0, ∞)
cond_{U} : U(K) → [0, ∞)

by the formulae

height_{U}(P) := height_{A^{N}} (i(P)), P ∈ U(K)
cond_{U}(P) := cond_{A^{N}} (i(P)), P ∈ U(K)

By the above discussion, if height_{U} and cond_{U} are defined by a closed immersion i and height′_{U} and cond′_{U} are defined by a closed immersion i′ then

height_{U} ≡ height′_{U} + O(1)
cond_{U} = cond′_{U} + O(1)

In particular we have

cond_{U} << height_{U} + O(1)

Here is our basic definition.

Definition. We say that the abc estimate holds on U if

height_{U} << cond_{U} + O(1)

In this lecture we would like to understand what are, conjecturally, the affine varieties U on which the abc estimate holds. To tackle this question, and establish the link with the “usual” abc we need to be more specific about our field K. Assume, in what follows, that we are in one of the following situations:

1) Number field case. K is a number field equipped with its standard family of absolute values (| |_{v}) (normalized in such a way that they extend the standard absolute values of Q; in particular, if v divides a rational prime p, then |p|_{v} = p^{−1}.) We take m_{v} = [K_{v} : Q_{v}]. So, for any non-archimedian v, dividing an unramified rational prime p, we have γ_{v} = v(p) = log p so m_{v}γ_{v} = log Nv where Nv is the norm of v (cardinality of the residue field of v). Since there are only finitely many ramified primes, we have

cond_{A^{N}} (η) = ∑_{v(η)<0} log Nv, η ∈ K^{N}

2) Function field case. K is a function field of one variable over an algebraically closed field k of characteristic zero. We equip K with the absolute values | |_{v} arrising from the k−rational points v of the smooth projective model V of K/k; we normalise them via the condition γ_{v} = 1 for all v, and we take m_{v} = 1 for all v. So in this case cond_{A^{N}} (η), for η ∈ K^{N}, is simply the number of points v ∈ V which are poles for at least one of the rational functions η_{j}.

It is a trivial exercise to show that if, say, K = Q or K = k(t), and if there exists a non-constant morphism of K−varieties

P^{1}\{0, ∞} → U

into an affine K−variety U then the abc estimate fails on U. On the other hand the optimist would be tempted to believe that the presence of such morphisms is the only obstruction to the abc estimate; we make, optimistically, the following:

Conjecture. Assume U is a smooth affine variety over K and assume that any morphisms of K−varieties P^{1}\{0, ∞} → U is constant. Then the abc estimate holds on U.

Our Conjecture should be viewed as an afine analogue of “Lang’s conjecture” saying that if X is a smooth projective variety over K and if any morphism from an algebraic group G to X is constant then the points of X(K) have bounded height; hence, in the number field case X(K) is finite. (In Lang’s conjecture one allows morphisms G → X defined over the algebraic closure K̄ of K; it might be reasonable to allow this in our Conjecture as well.)

In case K = Q and U = P^{1}\{0, 1, ∞}, the Conjecture above simply says that the abc estimate holds on this particular U; note that this is equivalent to the following:

Variant of abc conjecture. For any relatively prime integers a, b, c with a + b = c there exist real numbers C > 1 and μ > 1 such that

max{|a|, |b|, |c|} ≤ C · rad(abc)^{μ}

(As usual rad(n), where n is an integer, is defined as the product of all primes dividing n; this Variant is weaker than the abc of Masser and Oesterlé which predicts that μ can be made as close to 1 as we want. However this Variant still implies, say, the assymptotic Fermat.)

To see the equivalence between the Conjecture and the Variant of abc for K = Q and U = P^{1}\{0, 1, ∞}, embed U into A^{2} via the map

(1 : x) ↦ (x, \frac{1}{x(x − 1)})

Then, for coprime integers a, b, c with a + b = c, we have that height_{U}(1 : \frac{c}{a}) equals

height_{P^{2}}(1 : \frac{c}{a} : \frac{a^{2}}{cb}) = log max(|abc|, |b^{2}c|, |a^{3}|) ≡ log max(|a|, |b|, |c|) + O(1)

On the other hand

cond_{U}(1 : \frac{c}{a}) = cond_{A^{2}}(\frac{c}{a}, \frac{a^{2}}{cb}) = log rad(abc)

Note that, by a result due (independently) to Mason and Silverman, the abc estimate holds for the projective line minus 3 points. On the other hand results of Voloch [V], Brownawell-Masser [BM], and Wang [W] can suitably be interpreted as abc estimates “outside some exceptional loci” for projective spaces minus unions of hyperplanes.

Theorem. (abc for abelian varieties with trace zero [B94’]) In the function field case the abc estimate holds for any affine open set of an abelian variety with trace zero.

Theorem. (abc for isotrivial abelian varieties [B98]) Let K be a function field over k, let A_{k} be an abelian variety over k and D_{k} a divisor in A which does not contain any translate of a non-zero abelian subvariety (in particular D_{k} is ample so U_{k} := A_{k}\D_{k} is affine). Let U := U_{k} ⊗_{k} K. Then the abc estimate holds on U.

The above Theorems are immediate consequences of the following stronger results:

Theorem. (Bounded Multiplicity Theorem, trace zero case [B94’]) Assume A/K is an abelian variety with trace zero and let f ∈ K(A) be a rational function. Then there exists a constant C depending only on K, A, f with the following property. For any point P ∈ A(K) where f is defined and does not vanish, all zeroes and poles of f(P) ∈ K^{×} have multiplicity at most C.

Theorem. (Bounded Multiplicity Theorem, isotrivial case [B98]) Let X be a smooth projective curve over k, A an abelian variety over k, and D an effective divisor on A. Assume that D contains no translate of a non zero abelian subvariety. Then there exists a real constant C > 0, depending only on X, A, and D with the property that for any morphism f : X → A, with f(X) ⊄ D, all points of the divisor f^{∗}D have multiplicity at most C.

The proofs of the results above are based on “differential algebraic geometry”. A characteristic p version of these results was proved by Scanlon [Sc].

Lecture 2. Differential algebraic geometry.

Let F be a field of characteristic zero equipped with a derivation δ. We let C be the constant field. (If K is a function field we will always assume we have fixed a non-zero k−derivation on it and (K, d) is embedded into (F, δ).)

Following the classical work of Ritt and Kolchin one defines the δ−polynomial ring F{T }, where T is an n−tuple of variables, as the (usual) polynomial ring over F in the variables T^{(i)}, i ≥ 0. The order of A ∈ F{T } is the highest i such that a variable T_{j}^{(i)} is present in A. F is called δ−closed if for any A, B ∈ F{T }, T a variable, such that ord B < ord A, there exists a ∈ F such that A(a) = 0 and B(a) ≠ 0.

We need more definitions. A D−scheme is simply an F−scheme V with a given derivation on O_{V} that lifts δ. D−schemes form a category (morphisms are required to commute with the derivations). A D−group scheme is a D−scheme which is also an F−group scheme such that the multiplication, inverse, and unit are morphisms of D−schemes. A D−variety is a D−scheme which is also a variety over F. An algebraic D−group is a D−group scheme which is also an algebraic group over F.

One has a forgetful functor

{D − schemes} → {F − schemes}, V ↦ V^{!}

It has a right adjoint

{F − schemes} → {D − schemes}, X ↦ X^{∞}
Hom_{F}(V^{!}, X) = Hom_{D}(V, X^{∞})

defined as follows. If X = Spec F[T]/I set X^{∞} := Spec F{T }/[I], [I] := (I, δI, δ^{2}I, ...). In the non-affine case one glues the affine pieces. (Note that X^{∞} is the inverse limit of a system X^{n} of varieties obtained by truncating everything to order n. One has a natural map ∇ : X(F) → X^{∞}(F) which in coordinates sends a ∈ F into (a, δa, δ^{2}a, ...). Pull backs via this map of Zariski closed sets are called δ−closed sets. (Kolchin Topology). If F is δ−closed there is a bijection between delta closed sets and reduced closed D−subschemes of X^{∞}. If Σ ⊂ X(F) is δ−closed, corresponding to a D−subscheme H ⊂ X^{∞} then one defines the absolute dimension a(Σ) as the maximum of the transcendence degrees over F of the function fields of the irreducible components of H. Example: a(X(F)) = ∞ if dim X > 0. If X descends to C (i.e. comes from a variety X_{C} over C) then a(X_{C}(C)) = dim X. For G an algebraic group over F, the above bijection induces a bijection between the δ−closed subgroups of finite absolute dimension of G(F) and algebraic D-subgroups of G^{∞}.

By a δ−function on X(F), where X is a variety, we mean a map X(F) → F obtained by composing ∇ with a regular map on X^{∞}. A δ−character on an algebraic group will mean a δ−function which is also an additive homomorphism.

Theorem. (δ−density Theorem [B93]) If X is a smooth projective unirational F−variety then any δ−function on X(F) is constant. Moreover, if X defined over an intermediate field L between C and F, L ≠ C, then X(L) is δ−dense in X(F).

Theorem. (Finiteness of absolute dimension [B92]) Let A be an abelian F−variety of dimension g. Then the intersection A^{♯} of the kernels of all δ−characters of A has finite absolute dimension g ≤ a(A^{♯}) ≤ 2g. Consequently, the δ−closure of any finite rank subgroup of A(F) has finite absolute dimension ≤ 2g + r, where r is the rank. In particular, in the trace zero case, the δ−closure of the group of division points of A(K) has finite absolute dimension.

The δ−characters are the incarnation, in differential algebraic geometry, of the “Manin homomorphisms”; but our finiteness result (a(A^{♯}) < ∞) is quite different in nature from the Manin-Chai “Theorem of the kernel”. What the latter says, in the trace zero case, in our terminology is the following:

Theorem of the Kernel. [Manin-Chai] Let A be an abelian F−variety with F/C−trace zero. Then for any intermediate δ−field C ⊂ L ⊂ F of definition for A which is finitely generated over C we have A^{♯} ∩ A(L) = A(L)_{tors}.

Theorem. (δ−maps on curves [B94]) Let X be a smooth projective curve over F of genus g ≥ 2, that does not descend to C. Then there exists an injective δ−map ϕ : X → A^{n}.

In a certain precise sense one can actually prove more namely that projective curves of genus at least 2 are affine in this geometry.

Half way towards our abc for abelian varieties we have the following differential algebraic generalisation of the geometric Lang conjecture on subvarieties of abelian varieties:

Theorem. (Differential Algebraic Lang [B92]) Let A be an abelian variety over F with trace zero over C. Let Σ ⊂ A(F) be a δ−closed subgroup of finite absolute dimension and X ⊂ A a closed subvariety. Then there exist in X finitely many translates of abelian subvarieties whose union contains X(F) ∩ Σ.

The above Theorem is indeed a generalisation of the geometric Lang conjecture because, together with the Theorem on finite absolute dimension, it formally implies the

Theorem. (Geometric Lang Conjecture [B92].) Let A be an abelian variety over K with trace zero over k. Let Γ ⊂ A(K̄) be a subgroup of finite rank and X ⊂ A a closed subvariety. Then there exist in X(K̄) finitely many translates of abelian subvarieties whose union contains X(K̄) ∩ Γ.

Lecture 3. Description of proofs.

a. Descent results.

An F−variety (resp. an algebraic group over F) is said to descend to constants if it comes, via base change, from a variety (resp. an algebraic group) over C.

The next four theorems are proved via complex analytic arguments; for the first theorem Gillet showed me an argument based on formal schemes which should be also considered in some sense analytic; this kind of argument, in formal geometry, does not seem to apply to the other results.

Theorem. (Descent of projective varieties [B87]) Any projective D−variety descends to constants.

Theorem. (Descent of linear algebraic groups [B92]) Any linear algebraic D-group descends to constants.

Remark: the above fails for non-linear groups. A VERY interesting question: Let G be an algebraic D-group and H be its maximum connected linear algebraic subgroup; does H descend to constants ?

We won’t need the previous two Theorems for our diophantine purposes. For the abc Theorem for abelian varieties with trace zero we will need:

Theorem. (Descent via D-groups [B92]) Let G be an algebraic D-group, V ⊂ G a closed D−subvariety, and V → W a dominant morphism to a projective variety W of general type. Then the Albanese variety Alb(W) descends to constants.

For the abc Theorem for isotrivial abelian varieties we need:

Theorem. (Descent via D-groups, split case [B98]) Let W be a projective variety of general type over K. Assume W is a closed subvariety of A_{K}, where A is an abelian k−variety. Let G be any algebraic D−group, V ⊂ G a D−subvariety and u : V → W be a dominant morphism. Then, after replacing K by a finite extension of it, one may find a closed k−subvariety Z ⊂ A and a point Q ∈ A(K) such that W = Z_{K} +Q in A_{K}. Moreover, if we view W as a D−scheme by trivially lifting δ from K to W ≃ Z_{K} = Z ⊗_{k} K, then u : V → W is necessarily a morphism of D−schemes.

Let us sketch the proof of the Theorem on “descent via D-groups”. We may assume C is the field of complex numbers. We may further reduce ourselves to the case when (V, W, G, Spec F, δ) comes, via base change, from a “situation in complex algebraic geometry” (V, W, G, S, δ_{S}) where

S is an affine complex curve with a non-vanishing vector field δ_{S}

W is projective over S with integral geometric fibres of general type

G is a group scheme of finite type over S equiped with a vector field δ_{G}, lifting δ_{S}, such that the inverse and the multiplication on G are δ_{G}−equivariant (call such a δ_{G} “group compatible”)

V is a closed subvariety of G, horizontal with respect to δ_{G}, with a dominant map to W.

A lemma of Hamm says that an “analytic 1-parameter family of complex Lie groups” whose total space is equipped with a “group-compatible” analytic vector field lifting a non-vanishing vector field on the base, is locally analytically trivial. So locally analitically G = G_{0} × S, δ_{G} = (0, δ_{S}), where G_{0} is some Lie group. (So we have an analytic splitting; note that there is no algebraic splitting in general!) It follows that V → S is locally analytically trivial. Fix s_{0} ∈ S. For any s ∈ S consider the analytic map

ϕ_{s} : V_{s_{0}} ≃ V_{s} → W_{s}

Its image contains a Zariski open set. The Big Picard Theorem [Kobayashi-Ochiai 75] says that any analytic map from an algebraic variety to a projective variety of general type, whose image contains a complex open set, is in fact algebraic. So our maps ϕ_{s} are algebraic. So they induce maps from the Albanese variety of (a smooth projective model of) V_{s_{0}} into Alb(W_{s}). So all Alb(W_{s}) are isomorphic to each other and we are done.

b. Sketch of proof of the Finite Absolute Dimension Theorem .

Set C = Spec O(A^{∞}). One easily checks that U := Ker(A^{∞} → A) is unipotent. Then one proves that whenever one has an extension

0 → U → G → A

with A an abelian variety and U unipotent (infinite dimensional) then one also has an exact sequence

0 → Ã → G → Ũ → 0

with Ã finite dimensional and Ũ unipotent (infinite dimensional). (This is an exercise in the theory of proalgebraic groups.) The one is done by noting that Ã can be chosen to be a D−subgroup and that A^{♯} is the δ−closed subset correspoding to Ã.

c. Proof of Differential Algebraic Lang.

Assume we are in the hypothesis of Differential Algebraic Lang. Then Σ corresponds to an algebraic D−subgroup G of A^{∞}. Let V be an irreducible component of X^{∞} ∩ G and let Y ⊂ X be the Zariski closure of the image of V in X. We claim Y is a translate of an abelian subvariety, and we shall be done. Assume it’s not. By a result of Ueno there is an abelian subvariety A_{1} ⊂ A such that the image W of Y in A/A_{1} is positive dimensional, of general type. By the Theorem on “descent via D-groups” it follows that Alb(W) descends to constants. But A was assumed to have trace zero, a contradiction.

d. Proof of abc for abelian varieties.

Preparation. Let A be an abelian variety over K, with K/k−trace zero and let U be an affine open set. The closed set A\U is the support of a very ample effective divisor D on A. Note that D is neither irreducible nor reduced apriori. Embed A into P_{K}^{N} such that D is given by x_{0} = 0, where x_{0}, ..., x_{N} are a basis of H^{0}(P_{K}^{N}, O(1)). Let U_{j} ⊂ A be the open sets defined by x_{j} ≠ 0. So U_{0} = U. We need to show that there is a constant C such that for any point η = (η_{1}, ..., η_{N}) ∈ U(K) and for any place v we have v(η) ≥ C. This actually easily implies the stronger Bounded Multiplicity Theorem as well.

For simplicity we shall assume A is simple.

Constructing the finite set Y. Let A^{∞}, D^{∞} be attached to A and D respectively (the whole construction being made over K, rather than over F). Then D^{∞} appears as a closed subscheme of A^{∞}. Exactly as in the proof of Differential Algebraic Lang, there exists an algebraic D−subgroup G ⊂ A^{∞} such that for all P ∈ A(K) we have ∇(P) ∈ G(K). Set V := D^{∞}∩G (scheme theoretic intersection) and let Y ⊂ A be the Zariski closure of the image of V via the map A^{∞} → A; we view Y with its structure of reduced subscheme of A. As in the proof of Differential Algebraic Lang Y is isomorphic, over the algebraic closure of K, with a finite union of translates of abelian subvarieties. Since A was assumed simple, Y is finite.

For simplicity we shall assume the finitely many points of Y are rational over K.

“Uniform discreteness”: ϕ_{ij} and γ. It is an easy consequence of the Manin-Chai Theorem of the Kernel that one can find, for each j, a finite family (ϕ_{ij})_{i∈I_{j}}, ϕ_{ij} ∈ J (Y ∩ U_{j}) and there exists a real number γ ≥ 0 such that for any v and any P ∈ U_{j}(K)\Y, there exists an index i ∈ I_{j} such that v(ϕ_{ij}(P)) ≤ γ. (Here J of a subscheme denotes the ideal defining that subscheme.) Identify ϕ_{ij} ∈ O(U_{j}) with their pull-backs in O(U_{j}^{∞}). Then, for a suitable q, we have

ϕ_{ij}^{q} ∈ J (V ∩ U_{j}^{∞}) = J (D^{∞} ∩ U_{j}^{∞}) + J (G ∩ U_{j}^{∞})

Clearly

J (D ∩ U_{j}) = \frac{x_{0}}{x_{j}} · O(U_{j})

So we get

O(D^{∞} ∩ U_{j}^{∞}) = (\frac{x_{0}}{x_{j}} , δ(\frac{x_{0}}{x_{j}}), δ^{2}(\frac{x_{0}}{x_{j}})...)O(U_{j}^{n})

We may write, for all j and i ∈ I_{j}:

(1) ϕ_{ij}^{q} = ∑_{t=0}^{n} F_{ijt} (\frac{x_{0}}{x_{j}} , ..., \frac{x_{N}}{x_{j}} ) δ^{t} (\frac{x_{0}}{x_{j}} ) + g_{ij}

with g_{ij} ∈ J (G ∩ U_{j}^{∞}) ⊂ O(U_{j}^{∞}) and F_{ijt} differential polynomials with K−coefficients.

Defining β. Clearly we may find a real number β ≥ 0 such that for any η ∈ K^{N} and any v with v(η) ≥ 0 we have v(F_{ijt}(η)) ≥ −β for all i, j, t.

Fixing P and v. Now we fix a point P ∈ U(K) = A(K)\D with coordinates ξ ∈ K^{N+1} and fix a place v. Estimate equation (1) at ∇(P) ∈ A^{∞}(K). Since ∇(P) ∈ G(K) we have g_{ij}(∇(P)) = 0. So by further taking v we get, for each j, and each i ∈ I_{j},:

(2) q · v(ϕ_{ij}(P)) ≥ min_{t}\{v (F_{ijt} (\frac{ξ_{0}}{ξ_{j}} , ..., \frac{ξ_{N}}{ξ_{j}} ))\} + min_{t}\{v (δ^{t} (\frac{ξ_{0}}{ξ_{j}} ))\}

Choosing j. Now for our fixed P and v there exists j such that ξ_{j} ≠ 0 (hence P ∈ U_{j}(K)) and v(ξ_{i}/ξ_{j}) ≥ 0 for all i, i.e. v(ξ_{j}) = v(ξ). Fix such a j.

Choosing i. For our fixed P, v, j there exists, by “uniform discreteness”, an index i ∈ I_{j} such that

(3) v(ϕ_{ij}(P)) ≤ γ

Fix such an index i.

Conclusion. Now, by the choice of β, we have, for all t,

(4) v(F_{ijt} (\frac{ξ_{0}}{ξ_{j}} , ..., \frac{ξ_{N}}{ξ_{j}} )) ≥ −β

It is trivial to see that there exists a real number θ ≥ 0 such that for all x ∈ K and all v we have v(δx) ≥ v(x) − θ. Then putting together (2), (3), and (4), we get

q · γ ≥ q · v(ϕ_{ij}(P)) ≥ −β + (v(ξ_{0}) − v(ξ_{j}) − n · θ)

If η = (ξ_{1}/ξ_{0}, ..., ξ_{N}/ξ_{0}) then we get

v(η) = v(ξ) − v(ξ_{0}) ≥ C := −(β + q · γ + n · θ)

and we are done.

References

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Dept. Math. and Stat. University of New Mexico, Albuquerque