In this paper we give a short proof of the André-Oort conjecture for products of modular curves under the Generalised Riemann Hypothesis using only simple Galois-theoretic and geometric arguments. We believe this method represents a strategy for proving the conjecture for a general Shimura variety under GRH without using ergodic theory. We also demonstrate a short proof of the Manin–Mumford conjecture for Abelian varieties using similar arguments

Daw, Chris

Archiv der Mathematik

Central Archive at the University of Reading

A simplified proof of the Andre´-Oort conjecture for prod-ucts of modular curvesChristopher DawAbstract. In this paper we give a short proof of the Andre´-Oort conjecture for products of modularcurves under the Generalised Riemann Hypothesis using only simple Galois-theoretic and geometricarguments. We believe this method represents a strategy for proving the conjecture for a generalShimura variety under GRH without using ergodic theory. We also demonstrate a short proof ofthe Manin-Mumford conjecture for Abelian varieties using similar arguments.Mathematics Subject Classification (2010). Primary 11G18; Secondary 14G35, 11G10.Keywords. Andre´-Oort, modular curve, Shimura variety, Manin-Mumford.1. IntroductionThis paper is concerned with the following conjecture.Conjecture 1.1. (Andre´-Oort) Let S be a Shimura variety and let Σ be a set of special points in S.Every irreducible component of the Zariski closure of Σ is a special subvariety of S.In this paper we consider the case when S is a product of modular curves. However, since Salways admits a morphism pi to SLn2 (Z)\Hn and an irreducible subvariety Z is special if and only ifpi(Z) is special, we will assume S to be SLn2 (Z)\Hn = Cn. Special subvarieties here have the followingdescription (see [10], Definition 2.1).Definition 1.2. Let I = {1, ..., n}. A closed irreducible subvariety Z of Cn is called special (of typeΩ = ΩZ) if I has a partition Ω = (I1, ..., It), with |Ii| = ni, such that Z is a product of subvarietiesZi of Cni , each of one of the following forms:1. Ii is a one element set and Zi is a CM point.2. Zi is the image of H in Cni under the map sending τ in H to the image of (gs · τ)s∈Ii in Cni forelements gs ∈ SL2(Q).Given a special subvariety Z of type Ω, we define c(Ω) to be the number of CM factors. A specialsubvariety Z is called strongly special if c(Ω) = 0.The strategy of this paper will be to consider an irreducible subvariety X in Cn containing aZariski dense set Σ of special subvarieties. After, if necessary, replacing Σ by a Zariski dense subset,we may assume that c(ΩZ) is constant as Z ranges through Σ. Hence, we denote its value c(Σ). UnderGRH and using Galois-theoretic and geometric arguments, Ullmo and Yafaev show in [10] that, ifc(Σ) > 0, then X contains a Zariski dense set Σ′ with c(Σ′) = 0. Therefore, we consider sets ofstrongly special subvarieties. These are dealt with in [1] via ergodic theory. In this paper, we showthat the irreducible components of the Zariski closure of a set of strongly special subvarieties arespecial using only simple geometric arguments.2 Christopher DawThe result for products of modular curves has also been attained via relatively elementary meth-ods by Edixhoven in [3]. However, his method relies intrinsically on the properties of products ofmodular curves. The motivation for this paper is the development of a strategy that will apply toa general Shimura variety. Using model theory, Pila has proven the Andre´-Oort conjecture uncon-ditionally for Cn (see [7]) and his ideas have led to unconditional proofs of other special cases e.g.[2].Given a set of strongly special subvarieties Σ, we consider an irreducible component X of itsZariski closure. The idea of the proof is to intersect X with its image under a suitable Hecke cor-respondence and reiterate this proceedure, each time with an irreducible component of the previousintersection. By comparing lower bounds for the degrees of strongly special subvarieties with the de-grees of Hecke correspondences, we arrive at a nonproper intersection. For each Z in Σ, we produce aZ ′ strictly containing Z and repeat the argument.2. Degrees of strongly special subvarietiesConsider the image of H in Cn, for some n ∈ N, under the map described in Definition 1.2 (2) for someg1, ..., gn ∈ SL2(Q). For simplicity we may assume that g1 is the identity. This image is the modularcurve Γ′\H embedded in Cn, with Γ′ := Γ ∩ g−12 Γg2 ∩ ... ∩ g−1n Γgn, where Γ := SL2(Z).By [5], Proposition 5.3.2., the projectionpi : Γ′\H→ Γ\H,extends to a morphismpi : Γ′\H→ Γ\Hof their Baily-Borel compactifications, such that the inverse image pi∗LΓ of the Baily-Borel line bundleon Γ\H is LΓ′ , the Baily-Borel line bundle on Γ′\H, which is the restriction of the Baily-Borel linebundle on (P1C)n. Therefore, by the projection formula, we havedegLCn Γ′\H = deg pi · degLΓΓ\H,which is bounded below by the index [Γ : Γ′].Consider the closures Γ = SL2(Zˆ) and Γ′ of Γ and Γ′ in SL2(Zˆ), respectively. We have [Γ : Γ′] ≥[Γ : Γ′], which is equal to∏{p: prime}[SL2(Zp) : (SL2(Zp) ∩ g−12 SL2(Zp)g2 ∩ ... ∩ g−1n SL2(Zp)gn)],noting that, for almost all p, gi ∈ SL2(Zp) for all i = 2, ..., n. Therefore, the above quotients at theseprimes are trivial. Now suppose that gi /∈ SL2(Zp) for some i ∈ {2, ..., n} and a prime p. ConsideringSmith normal forms, it is possible to write gi = γDγ′, where γ, γ′ ∈ SL2(Zp) and D is a diagonalmatrix of the form diag(pn, p−n), for some n ∈ N. ThenSL2(Zp) ∩ g−12 SL2(Zp)g2 ∩ ... ∩ g−1n SL2(Zp)gnis contained in the subgroup of SL2(Zp) whose lower left entry belongs to pZp. The index of thissubgroup can be calculated the same way as the index of Γ0(p) in SL2(Z) (see [6], p81) and is boundedbelow by p.Thus, from the above remarks, we conclude that the degree of Γ′\H with respect to the BailyBorel line bundle on (P1C)n is bounded below by the product of primes p such that gi /∈ SL2(Zp) forsome i ∈ {2, ..., n}.Henceforth, when we refer to the degree of a subvariety of Cn, we will omit reference to the BailyBorel line bundle.A simplified proof of Andre´-Oort for products of modular curves 33. Choosing a suitable Hecke correspondence.Consider the Shimura variety ShK(G,X)C defined by the Shimura datum(G,X) := (GLn2,Q, (H±)n)and K := GLn2 (Zˆ), a compact open subgroup of GLn2 (Af ); this is the connected Shimura variety Cn.Consider GL2,Q emdedded into GLn2,Q via the mapϕ : x 7→ (x, g2xg−12 , ..., gnxg−1n ),for g2, ..., gn ∈ GL2(Q). We denote its image by H and we write X+H for a connected component ofH(R) · (i, g2i, ..., gni) ⊂ (H±)n. Let Z be the image of X+H × {1} in ShK(G,X)C. This is the image ofH described in the previous section.Lemma 3.1. For any α2 ∈ H(Af ), Z is contained in its image under the Hecke correspondence Tα2 .Proof. Let (x, 1) ∈ Z i.e. x ∈ X+H . Let τ ∈ GL2(Af ) be such that τ = ϕ(α). A point (h, g) ∈ShK(G,X)C depends only on g modulo (A∗f )n = (Q∗ · Zˆ∗)n, since these factors are killed in the doublecoset defining the Shimura variety. Hence, we consider the image of τ2 under the standard mappi : GL2,Q → PGL2,Q,which is surjective on adelic points since the kernel Gm,Q is connected. Consider the simply connectedcoveringρ : SL2,Q → PGL2,Q.Its kernel is µ2,Q. Therefore, there exists a υ ∈ SL2(Af ) such that ρ(υ) = pi(τ2). By strong approxi-mation applied to SL2,Q, υ = qk, with q ∈ SL2(Q) and k belonging to the compact open subgroupn⋂i=1g−1i SL2(Zˆ)gi ⊂ SL2(Af ),where g1 = 1. Note that pi restricted to SL2,Q coincides with ρ, since ρ is a universal cover. Now, ϕ(q)belongs to H(Q)+. Hence, ϕ(q) · x belongs to X+H . Therefore, consider (ϕ(q) · x, ϕ(τ2)) ∈ Tα2(Z). Bythe previous discussion, this equals (ϕ(q) · x, ϕ(qk)) = (x, ϕ(k)). However, since ϕ(k) ∈ GLn2 (Zˆ), thispoint is (x, 1). Choose a prime p such that gi ∈ GL2(Zp) for all i = 2, ..., n. Consider the elementP :=(1 00 p−2)∈ GL2(Q)+Let α ∈ H(Qp) be the image of P in GLn2 (Qp) under ϕ. Since the gi belong to GL2(Zp) for alli = 2, ..., n, the double coset KαK equals KβK, where β is P diagonally embedded into GLn2 (Qp).Recall that ([4], Theorem 6.1) the connected components of the correspondence Tβ on Cn arethe Tβi induced by GLn2 (Q)+ acting on Hn such thatGLn2 (Q)+ ∩KβK =∐iΓnβ−1i Γn.Note, however, that GLn2 (Q)+ ∩ KβK = ΓnβΓn. Hence, the correspondence Tβ is irreducible andequal to the standard Hecke correspondence Tp2 on Cn. Combining this observation with lemma 3.1,we obtain the following result:Lemma 3.2. Let Z be the image of H in Cn as defined above. Choose a prime p such that gi ∈ SL2(Zp)for all i = 2, ..., n. Then Z is contained in its image under the Hecke correspondence Tp2 on Cn givenby P−1.4 Christopher Daw4. ProofTheorem 4.1. Assume the GRH for imaginary quadratic fields. Let X be an irreducible component ofthe Zariski closure of a set Σ of special subvarieties in a product S of n modular curves. Then X is aspecial subvariety.By [10] we may assume that c(Σ) = 0 i.e. Σ is a set of strongly special subvarieties. The GRHwill not be used in this case. Using Proposition 2.1 of [4] we may assume that S is the product of ncopies of SL2(Z)\H.Proof. Consider a special subvariety Z ⊂ X ⊂ Cn. Given our description of strongly special sub-varieties we write Z = Z1 × · · · × Zt, where each Zi is the image of H in Cni given by somegi2 , ..., gini ∈ SL2(Q). Therefore, by the arguments in Section 2, the degree of Z is bounded be-low by the product of all primes p such that not all gij ∈ SL2(Zp) for i = 1, ..., t and j = 2, ..., ni. Wedenote this product MZ ∈ N. By Lemma 3.2, for any prime p not dividing MZ , we have Z ⊂ Tp2(Z),By [3], Section 3, we may assume that X is a hypersurface all of whose projections to any n− 1factors of Cn are dominant. Under this assumption, given n ≥ 3, Tp2(X) is irreducible for any p greaterthan degX and 13 by Proposition 3.1 of [10]. If n = 2, Z is either C2 or a modular curve Y0(N), inwhich case X = C2 or X = Y0(N).Consider first the case that X contains a Zariski dense subset Σ′ such that MZ is bounded forall Z in Σ′ i.e. the elements gij defining a given Z must belong to SL2(Zp) for all but a fixed and finiteset of primes whose product we denote M . For any prime p not dividing M , every Z in Σ′ is containedin its image under Tp2 . Hence, Σ′ belongs to X ∩ Tp2(X). This is a closed set and so X ⊂ Tp2(X).However, since Tp2(X) is irreducible (provided p is larger than a uniform constant), we must haveequality. By Lemma 4.4 of [3], for any point z ∈ Cn, the Tp2-orbit ∪i≥0T ip2(z) is dense in Cn for theArchimedean topology. Thus, X is Cn .Therefore, we assume that X contains no such Zariski dense subset. Hence, we may assume thatMZ is larger than any uniform constant for all Z in Σ. We consider an arbitrary Z. By a theorem ofChebyshev, there exist positive absolute constants c1 and c2 such that the number of primes less thana given real number x ≥ 2, pi(x), is bounded below by c1 xlog x and above by c2 xlog x . Therefore, for anyfixed 0 < δ1 < 1 and  > 0,pi(Mδ1Z) Mδ1ZlogMδ1ZM δ1−Z .It is an obvious fact that the number of primes dividing MZ , ω(MZ), satisfiesω(MZ) ≤ logMZlog 2M Z .Therefore, for MZ larger than a constant depending only on X, we can find a prime p1 smallerthan Mδ1Z , not dividing MZ and larger than 3 and the degree of X. The last condition implies thatTp21(X) is irreducible. Therefore, the intersection of X and Tp21(X) is either proper or X = Tp21(X). IfX = Tp21(X) then we are finished, as explained above.Thus, we assume the intersection is proper. For the prime p1, Z ⊂ Tp21(Z), which implies Z ⊂X∩Tp21(X). We relabel X as X1 and let X2 be an irreducible component of the intersection containingZ. Notice that, by Bezout’s theorem ([9], Lemme 3.4), the degree of X2 ⊂ X1 ∩ Tp21(X1) is boundedabove by (degX1)2 · deg Tp21 , wheredeg Tp21 = |Γ : Γ0(p21)|n = pn1 (p1 + 1)n  p2n1 ,i.e. degX2  p2+2n1 M (2+2n)δ1Z .A simplified proof of Andre´-Oort for products of modular curves 5So long as δ1 and  are small enough, we can fix 0 < δ2 < 1 such that (2 + 2n)δ1 + 2 < δ2. Wehavepi(M(2+2n)δ1Z) M(2+2n)δ1ZlogM(2+2n)δ1ZM (2+2n)δ1+Zandpi(M δ2Z) Mδ2ZlogMδ2ZM δ2−Z .Therefore, for MZ larger than a constant depending only on X, we can a find a prime p2smaller than Mδ2Z , not dividing MZ and larger than degX2 and 3. The second condition implies thatZ ⊂ X2∩Tp22(X2). The latter condition implies that if this interesection is not proper i.e.X2 ⊂ Tp22(X2),then X2 is special by [3], Theorem 4.1. It can only be strongly special since it contains Z. It is also ofhigher dimension than Z by comparing degrees. Hence, we replace Z in Σ with X2.Assume, then, that the intersection is proper. We perform the above construction recursively,with suitable δk, assuming MZ is large enough, thus obtaining subvarieties Xk and primes pk. If atsome point we have Xk ⊂ Tp2k(Xk), Xk is a special subvariety containing Z. If this inclusion does notoccur at any of the previous stages, after a finite number of steps bounded by dimX1, we end up inthe following situation:1. dim(Xk) = dim(Z) + 12. deg(Xk)M (2+2n)δk−1Z3. degZ > MZ4. Z ⊂ Xk5. pk < MδkZ6. pk not divinding MZ7. pk > deg(Xk)Therefore, we have Z ⊂ Xk ∩ Tp2k(Xk). By comparing the degrees of Z and Xk ∩ Tp2k(Xk), theintersection cannot be proper. Hence, Xk is contained in Tp2k(Xk) and is, therefore, special by [3],Theorem 4.1.We perform this procedure on all of the Z, replacing them in Σ by strongly special subvarietiesof higher dimension. Reiterating the above argument, we eventually conclude that X must be special.5. Manin-MumfordWe conclude this note by remarking that the techniques exhibited above apply, in a simpler way, tothe Manin-Mumford conjecture for Abelian varieties.Theorem 5.1. (Manin-Mumford) Let K be a number field, A/K an Abelian variety over K and V/Ka geometrically irreducible subvariety of A. If V (K) contains a Zariski dense set of torsion points thenV is the translate of an Abelian subvariety by a torsion point.Many proofs of this theorem exist. Ratazzi and Ullmo have recently given another combiningGalois-theoretic and ergodic methods. We refer to their paper [9] for more details. Here we replacethe ergodic theory with an elementary geometric argument. In this setting, special subvarieties arethe translates of Abelian subvarieties by torsion points.Proof. Let (Σn)n∈N be a sequence of special subvarieties with Zariski closure equal to V . For eachn ∈ N we choose a representationΣn = An + ξn,6 Christopher Dawwhere An ⊂ A is an Abelian subvariety and ξn is a torsion point in the Abelian subvariety A′n suchthat A = An +A′n and An ∩A′n is finite of uniformly bounded order (see [9], Proposition 2.1). Let dndenote the order of the torsion point ξn.Whether or not the sequence (dn)n∈N is bounded is independent of the choice of the ξn ([9],Remarque 3.1). In the case that the sequence (dn)n∈N is unbounded, Section 3.2 of [9] concludes thateach Σn is contained in a special subvariety Σ′n of higher dimension (the arguments here are Galois-theoretic, similar to the Shimura case, but are not dependent on GRH). Therefore, we replace (Σn)n∈Nwith (Σ′n)n∈N and reiterate this argument unless, at some point, we obtain a sequence (Σn)n∈N with(dn)n∈N bounded.In this case, since the set of torsion points of bounded order is finite, we may suppose that eachΣn is of the form An + ξ, for a fixed torsion point ξ. However, since V is special if and only if V − ξis special, we may assume that Σn = An for all n ∈ N.We denote by [m] the multiplication by m ∈ N map on A and choose one of the An. Considerthe intersection V ∩ [m]V . Either it is proper or [m]V = V , in which case V is special by [9], Lemme3.2. Notice that, by Bezout’s theorem, the degree of V ∩ [m]V is bounded above by deg V · deg[m]Vwhere, by [9], Lemme 3.1,deg[m]V ≤ m2 dimV · deg V.It is a classical fact that an Abelian variety contains only finitely many Abelian subvarieties of boundeddegree. Therefore, we may assume that the An have degree exceeding any uniform constant. Therefore,by comparing degrees, if dimV = dimAn+1 and degAn > m2 dimV ·(deg V )2, we must have V = [m]V .If the intersection is proper we choose an irreducible component W of the intersection containingAn. Therefore, An is contained in the intersection W ∩ [m]W . Either this intersection is proper orW = [m]W and W is special, in which case we replace An in (Σn)n∈N by W , a special subvariety ofhigher dimension. Again, comparing degrees, if dimW = dimAn+1 and degAn > m8 dimV · (deg V )4,we must have W = [m]W .Otherwise, if W ∩ [m]W is a proper intersection, we take an irreducible component containingAn and repeat the argument in the previous paragraph. After a finite number of steps, bounded bydimV , we will have found a special subvariety strictly containing An. We perform this procedure onall of the An, replacing them by special subvarieties of V of higher dimension. Reiterating the abovearguments at most dimV − 1 times, we conclude that V must be special.AcknowledgmentThe author would like to express his continuing gratitude to Andrei Yafaev for the abundance of ideasand time that he is willing to share. The author is also grateful to the department of mathematics atUCL for their financial support and to the referee for highlighting a mistake in the first version of thispaper.A simplified proof of Andre´-Oort for products of modular curves 7Bibliography.[1] Clozel, L and Ullmo, E (2005), Equidistribution de sous-varie´te´s spe´ciales, Annals of Maths, 161, 1571-1588[2] Daw, C and Yafaev, A (2011), An unconditional proof of the Andre´-Oort conjecture for Hilbert modularsurfaces, Manuscripta Math. 135, 263-271[3] Edixhoven, B (2005), Special points on products of modular curves, Duke Mathematical Journal. Vol. 126.No. 2[4] Edixhoven, S.J. and Yafaev, A (2003), Subvarieties of Shimura varieties, Annals of Mathematics, Volume157, No. 2, March 2003, 621-645[5] Klingler, B and Yafaev, A (2006), The Andre´-Oort conjecture, Preprint[6] Milne, J.S. (2009), Modular Functions and Modular Forms, available on the author’s web page[7] Pila, J (2011), O-minimality and the Andre-Oort conjecture for Cn, Annals Math. 173, 1779-184[8] Platanov, V. P. and Rapinchuk, A. S. (1991), Algebraic Groups and Number Theory, Academic Press, Inc.[9] Ratazzi, N and Ullmo, E (2009), Galois+Equidistribution=Manin-Mumford, Ecole d’t Arithmetic geom-etry, 419-430, Clay Math. Proc., 8, Amer. Math. Soc., Providence, RI[10] Ullmo, E and Yafaev, A (2009), The Andre´-Oort conjecture for products of modular curves, ArithmeticGeometry, 431-439, Clay Math. Proc., 8, Amer. Math. Soc., Providence, RI[11] Ullmo, E and Yafaev, A (2006), Galois Orbits and equidistribution of special subvarieties: towards theAndre´-Oort conjecture, with an appendix by P. Gille and L. Moret-Bailly, PreprintChristopher DawCollege LondonDepartment of MathematicsGower streetWC1E 6BT LondonUnited Kingdome-mail: c.daw@ucl.ac.uk

A simplified proof of the Andre-Oort conjecture for products of modular curves

Abstract

Similar works

Full text

Available Versions

Central Archive at the University of Reading

Crossref