The Andr´E–Oort Conjecture

Bas Edixhoven, Lenny Taelman The Andr´e–Oortconjecture NAW 5/15 nr. 4 december 2014 279

Bas Edixhoven Lenny Taelman Mathematical Institute Korteweg-de Vries Institute for Mathematics Leiden University University of Amsterdam [email protected] [email protected]

The Solution The Andr´e–Oort conjecture

The Andre–Oort´ conjecture is a problem in algebraic geometry from around 1990, with arith- C/ ∼ E. −→ metic, analytic and differential geometric aspects. Klingler, Ullmo and Yafaev, as well as Pila and Tsimerman have now shown that the Generalized Riemann Hypothesis implies the Andre–´ Λ Each morphism of elliptic curves a: E1 E2 → Oort conjecture. Both proofs appeared in the Annals of Mathematics in 2014. In this article Bas corresponds to a homothety z αz such that Edixhoven and Lenny Taelman describe the conjecture and these recent solutions. 7→ α 1 2. ⊂ The complex number λ is called special The story of the Andre–Oort´ conjecture starts and [1, λ] form circles C1 and C2 on E. These ifΛ the latticeΛ has complex multiplications, in 1988, with a question, posed by Yves Andre´ two circles intersect transversally, see Figure meaning that there are non-real complex [1, X.4.3] at the end of his book on solutions of 1, in a unique point, and therefore generate numbers α suchΛ that α (such an α ⊂ differential equations coming from algebraic the fundamental group of E. Integrating the defines an endomorphism of C/ ). For ex- 1 varieties defined over Q. algebraic differential form ω = y− dx along ample, λ = 2 is special asΛ the correspondingΛ the two circles gives the periods (defined up lattice, Λ Elliptic integrals and complex multiplication to sign): The simplest such differential equation is the = Z2.622 ... + Zi 2.622 ..., 1 · equation dx η1 = ω = 2 , C 0 x(x 1)(x λ) Λ Z 1 Z − − has multiplication by i. This gives the map p λ(λ 1)η′′(λ) + (2λ 1)η′(λ) − − (1) 1 and E E, (x, y) (2 x, iy). + 4 η(λ) = 0, → 7→ − λ dx The special λ form a countable subset of C. η2 = ω = 2 . C 1 x(x 1)(x λ) which was already studied by Gauss [9]. It Z 2 Z − − arises from the Legendre family of elliptic p curves: For varying λ, these form a basis of the complex vector space of solutions of (1). The inte- y2 = x(x 1)(x λ), λ C 0, 1 . − − ∈ − { } gral linear combinations of η1 and η2 form a lattice For each λ, the set of solutions (x, y) C2 is ∈ = k1η1 + k2η2 : k1, k2 Z C. a Riemann surface that can be compactified { ∈ } ⊂ by adding one point. The compactification E Λ is an elliptic curve. It is homeomorphic to a The Weierstrass P-function and its derivative, torus S1 S1. If λ is not in ( , 1), then the suitably normalised, give an isomorphism of × −∞ Figure 1 The intersection of C1 and C2 near the point solutions with x in the real segments [0, 1] complex analytic manifolds: (1,0), projected to the complex y-coordinate (for λ=1+i).

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Andre–Oort´ conjecture holds for trivial rea- Manin–Mumford for C C sons. × × × Let C C C be the set of zeroes of an irreducible complex polynomial f in two The simplest non-trivial case of the conjec- ⊂ × × × variables. Assume that C contains inﬁnitely many torsion points (pairs (x, y) C C ture is for the Shimura variety ∈ × × × with both x and y a root of unity). The Manin–Mumford conjecture for C C predicts that × × × f = aXnY m b with n and m coprime, and b/a a root of unity. Equivalently, it predicts A1 A1 = (GL2(Z) GL2(Z)) H± H± . − × × \ × that C is the image of a complex line The special points are the pairs (x, y) with x (x, y) C2 : αx + βy + γ = 0 (α, β, γ Q) { ∈ } ∈ and y special. There are three types of one- dimensional Shimura subvarieties: A1 y × { } under the exponential map with y special, x A1 with x special, and { } × the image of 2πix 2πiy C C C× C×, (x, y) (e , e ). × → × 7→ (ατ, βτ): τ H± H± H±, { ∈ } ⊂ × In fact the Manin–Mumford conjecture for C C is not hard to prove, and would be × × × suitable for the problem section in this journal. with α, β GL2(Q). The Andre–Oort´ conjec- ∈ This statement is analogous to (but much easier than) the Andre–Oort´ conjecture for A1 A1: ture for A1 A1 is equivalent to the statement × × torsion points correspond to special points, and the exponential map corresponds to the given in the previous section. quotient map H H A1 A1. The most interesting case of the Andre–´ ± × ± → × Oort conjecture is for the moduli space Ag of (principally polarized) complex abelian vari- We can now state an explicit case of Andre’s´ Rutger Noot [10] on a conjecture of Robert eties of dimension g: question. Let Z C2 be the set of zeros Coleman on curves with complex multiplica- ⊂ of an irreducible complex polynomial in two tions. Ag := GSp2g(Z) Hg±, variables. Assume that Z contains inﬁnitely Since the general theory of Shimura vari- \

many points (λ1, λ2) such that both λ1 and eties is rather technical, we will restrict our-

λ2 are special, and that Z is not a fiber of selves to examples. where Hg± is the space of symmetric complex one of the two coordinate projections. In this Lattices 1 and 2 give isomorphic ellip- g by g matrices whose imaginary part is defi- case, Andre´ asked if for all (λ1, λ2) in Z, there tic curves C/ 1 and C/ 2 if and only if they nite. The group GSp2g(R) of symplectic simil- Λ Λ is a non-zero complex number α such that are homothetic, that is, if there is a complex itudes acts transitively on Hg± by Λ Λ the pair of lattices ( 1, 2) corresponding to number α such that α 1 = 2. Every lattice (λ1, λ2) satisfies α 1 2? (The answer is is homothetic to a lattice of the form ⊂ a b 1 yes, as was shown, independently,Λ Λ in [4] (un- Λ Λ τ := (aτ + b)(cτ + d)− c d ! · Λ Λ τ := Z 1 + Z τ der GRH) and in [2].) The relation between λi · · and is not algebraic, which makes it dif- i Λ ficult to use the polynomial relation between for a τ H± := C R. The group GL2(R) acts where now a, b, c, d are real g by g matri- Λ ∈ − λ1 and λ2. transitively on H± by ces. The special points of Ag correspond to abelian varieties A with many endomor- Statement of the conjecture phisms (H1(A, Q) is generated over Q End(A) a b aτ + b ⊗ The Andre–Oort´ conjecture is the following τ := c d ! · cτ + d by one element). statement. A general Shimura variety A is of the form X, where is a discrete subgroup of G(R) \ Conjecture. Let A be a Shimura variety, and and homothety classes of lattices correspond for some matrix group G over Q, and where Γ Γ Z A an irreducible algebraic subvariety. to orbits under the discrete subgroup GL2(Z). G(R) acts transitively on X. The Shimura sub- ⊂ Assume that Z contains a subset of spe- The quotient varieties of A are images of orbits H(R) x for · cial points that is not contained in a strict certain algebraic subgroups H of G over Q and Σ A1 := GL2(Z) H± X subvariety of Z. Then Z is a Shimura sub- \ certain x in . The zero-dimensional Shimura variety. subvarieties are precisely the special points. is an example of a Shimura variety. It is the The set of special points is dense in A. We will say more about Shimura varieties moduli space of elliptic curves: the points of Shimura and Deligne have shown that

and special points below. A1 are in bijection with isomorphism class- each Shimura variety has a natural structure This conjecture was formulated (as a ques- es of complex elliptic curves. As before a of algebraic variety, deﬁned over a number

tion) by Andre´ for Z of dimension 1. Inde- point x A1 is special if its corresponding field K. It is the subspace of a complex projec- ∈ pendently, this was also formulated by Frans lattice has complex multiplications. These tive space defined by a finite system of poly-

Oort, for the Shimura variety Ag (see be- are precisely the images of the points in nomial equalities and inequalities (=, not <) 6 low). Both Andre´ and Oort were inspired H ofΛ the form a + b√ d with a, b and d with coefficients in K. The special points are ± − by the analogy with the Manin–Mumford rational. defined over finite extensions of K, and all

conjecture (see box). Oort’s motivation al- The only Shimura subvarieties of A1 are Galois conjugates of special points are spe- so came from work of Johan de Jong and the special points and A1 itself, so that the cial points.

2 2 Bas Edixhoven, Lenny Taelman The Andr´e–Oortconjecture NAW 5/15 nr. 4 december 2014 281

n 2n Strategies and results the Galois orbits GalK z, and sufficient con- open subset of C = R in which one again · In his thesis Ben Moonen gave a thorough trol on the complexity of the g that can be has the notion of algebraic subsets (defined treatment of Shimura subvarieties, including used. Lower bounds for Galois orbits depend by polynomial equations) and even the notion two characterisations. Using one of these he on lower bounds for class numbers of number of algebraic subsets defined over Q. This is proved the Andre–Oort´ conjecture for subva- fields. For the choice of g one needs suffi- relevant to the problem: the inverse images

rieties of Ag under an additional hypothesis ciently many small primes in number fields. of special subvarieties are of that kind. Pila on the set [13, 3.7 and 4.5]. Both are hard problems in analytic number imported the tool of O-minimal structures [3] As remarked above, Andre´ proved the con- theory. The best known bounds depend on to deal with this mixed algebraic and analytic Σ jecture for A1 A1 in [2]. the Generalised Riemann Hypothesis (GRH) context. Here, Pila could apply (a generali- × Let A = X be a Shimura variety, and for number fields. sation of) his result with Alex Wilkie [15] (see \ Z A an irreducible algebraic subvariety. Using this strategy, the Andre–Oort´ con- box). This result is used to show that large ⊂ Γ Assume that Z contains an infinite subset jecture was proved, under GRH, for A1 A1 Galois orbits of special points z in Z A × ⊂ of special points that is not contained in in [4], for Hilbert modular surfaces in [6], for give rise to positive dimensional special sub-

a strict subvariety of Z. A general strate- curves in general Shimura varieties (Andre’s´ varieties Yz Z, as in the previous strate- ⊂ Σgy for proving the conjecture is to ﬁrst show question) by Andrei Yafaev in [21] (building gy, but now without having to take intersec-

that for almost all z there is a positive- on [7, 20]), for powers of A1 in [8]. Finally, tions. An important intermediate result is the ∈ dimensional Shimura subvariety Yz with z Bruno Klingler, Emmanuel Ullmo and Yafaev Ax–Lindeman theorem, which is also proved Σ ∈ Yz Z, and to deduce from this that Z it- treated the general case (under GRH) in [12, using [15]; it says that maximal irreducible al- ⊂ self is a Shimura subvariety. The ﬁrst step is 19]. To make the strategy work in this case is gebraic subsets of the inverse image of Z are the hardest. a real tour de force, which took the authors already very close to being the inverse image One method, introduced in [4], for produc- (and the referees) quite a few years (the ﬁrst of a special subvariety.

ing such Yz is to exploit the Galois action on versions are from 2006). Pila and Jacob Tsimerman generalised Pi- the set of special points in A combined with Another strategy was introduced more re- la’s strategy to Ag in [17]. They proved the the action of G(Q) on X. The idea is to inter- cently by Jonathan Pila in [16], where he conjecture in that case, under GRH. In the n sect Z with a Z obtained from the action by proved the conjecture for powers A = A . The case of Ag for g 6 (and products of those) ′ 1 ≤ a carefully chosen g in G(Q), such that Z Z main idea is to work in X = (H )n instead of in they give an unconditional proof. In this case ∩ ′ ± contains GalK z and such that the Galois the quotient A = X. Of course, everything GRH is not needed, as there are sufficiently · \ orbit has so many elements that Z Z′ can- that takes place in A can be seen in X, but, strong unconditional lower bounds for Galois ∩ Γ not be finite. Then Yz is obtained as an irre- because the quotient map is not algebraic, orbits. ducible component of a repeated such inter- the inverse images of algebraic subvarieties section. This method works if one has suf- of A are then genuine complex analytic sub- Epilogue ficiently good lower bounds for the sizes of varieties of X. On the other hand, X is an We have seen that the Andre–Oort´ conjecture has been proved, under GRH, but many ques- tions remain open. The Pila–Wilkie theorem At this moment, Klingler, Ullmo, Yafaev n A subset of R is called definable in Ran,exp if it can be defined using finitely many formulas and Chris Daw are making Pila’s strategy work involving the logical symbols , , , , , addition and multiplication, inequalities, for general Shimura varieties [11]. It is not in- ∃ ∀ ¬ ∧ ∨ real numbers (occurring as ‘constants’), the real exponential function ex, and functions conceivable that number theorists can make [0, 1]m R that can be extended to a real analytic function on an open neighbourhood of the proof unconditional, by providing suffi- → [0, 1]m. For example, semi-algebraic sets (defined by polynomial inequalities) are definable. cient lower bounds for Galois orbits (see [5, The Pila–Wilkie theorem roughly states that if a definable X contains many points with Problem 14] for what is needed). Ziyang Gao rational coordinates, then these must accumulate on semi-algebraic subsets of X. More has announced a proof, under GRH, of the precisely, for a subset X of Rn we define the counting function conjecture generalised to mixed Shimura varieties [22]. p p Each Shimura variety A has a natural prob- N(X, t) := 1 ,..., n X p , q Z [ t, t] .  i i  ability measure. One expects that for a se- q1 qn ∈ ∈ ∩ −   quence zn of special points in A such that no

  subsequence is contained in a strict Shimu- For X = Rn we see that N(Rn, t) ct2n for some c. Now let X be a deﬁnable subset of Rn, ra subvariety, the Galois orbits of the z are ∼ n and let Xalg be the union of all positive-dimensional semi-algebraic subsets of X. equidistributed. This would imply Andre–´ Oort immediately, but this is known only in Theorem (Pila–Wilkie [15]). For every ǫ > 0 there is a c such that for all t we have N(X very special cases. − Xalg, t) < ctǫ. In the wider context of ‘unlikely intersec- tions’, Richard Pink has formulated [18] a As an example, let X R2 be the graph of a function f : [0, 1] R. If f is a polynomial conjecture on subvarieties of mixed Shimu- ⊂ → with rational coefﬁcients, then f (x) is rational for every rational x and N(X, t) will grow ra varieties, simultaneously generalising the polynomially in t. But if we take f (x) = sin(πx) then the theorem says that this cannot Andre–Oort,´ Manin–Mumford, Mordell–Lang happen, since Xalg = . In fact by Niven’s theorem N(X, t) 5. and Zilber conjectures. Pink’s conjecture re- ∅ ≤ mains wide open. k

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