Published by De Gruyter March 18, 2015

Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points

Ziyang Gao

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2014-0127

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Abstract

We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of 𝒜6n and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

A Appendix

We prove here Theorem 7.6 when E=T is an algebraic torus over ℂ (which corresponds to the case W=U) and when E=A is a complex abelian variety (which corresponds to the case W=V). The proof is a rearrangement of existing proofs (combine the point counting of Pila–Zannier [52] and volume calculation of Ullmo–Yafaev [64]). Use notation from Section 11.

Case (i): E=A

In this case, W=V and ΓV=⊕i=12⁢nℤ⁢ei⊂Lie⁡(A)=ℂn=ℝ2⁢n is a lattice. Denote univ:Lie⁡(A)→A. Let ℱV:=∑i=12⁢n(-1,1)⁢ei. Then ℱV is a fundamental set for the action of ΓV on Lie⁡(A) such that univ|ℱV is definable. Define the norm of z=(x1,y1,…,xn,yn)∈Lie⁡(A)=ℝ2⁢n to be

∥z∥:=Max⁡(|x1|,|y1|,…,|xn|,|yn|).

It is clear that for all z∈Lie⁡(A) and all γV∈ΓV such that γV⁢z∈ℱV,

(A.1)H⁢(γV)≪∥xV∥.

Let ωV:=d⁢z1∧d⁢z¯1+…+d⁢zn∧d⁢z¯n be the canonical (1,1)-form of Lie⁡(A)=ℂn. Let pi (i=1,…,n) be the n natural projections of Lie⁡(A)=ℂn to ℂ. Let C be an algebraic curve of Z~ and define CM:={z∈C:∥z∥⩽M}. We have

(A.2)∫C∩ℱVωV⩽d⁢∑i=1n∫pi⁢(C∩ℱV)𝑑zi∧d⁢z¯i

⩽d⁢∑i=1n∫pi⁢(ℱV)𝑑zi∧d⁢z¯i=d⋅O⁢(1)

and

(A.3)∫CMωV⩾O⁢(M2)

with d=deg⁡(C) by [33, Theorem 0.1].

By (A.1),

CM⊂⋃γV∈Θ⁢(Z~,M)(C∩γ-1⁢ℱ).

Integrating both side with respect to ωV we have

M2≪#⁢Θ⁢(Z~,M)

by (A.2) and (A.3).

Let now

StabV⁡(Z~):=ΓV∩StabV⁢(ℝ)⁡(Z~)¯Zar.

By Pila–Wilkie [64, Theorem 3.4], there exists an semi-algebraic block B⊂Σ⁢(Z~) of positive dimension containing arbitrarily many points γV∈ΓV. We have B⁢Z~⊂univ-1⁡(Y) since Σ⁢(Z~)⁢Z~⊂univ-1⁡(Y) by definition. Hence for any γV∈ΓV∩B, Z~⊂γV-1⁢B⁢Z~⊂univ-1⁡(Y), and therefore Z~=γV-1⁢B⁢Z~ by the maximality of Z~. So γV-1⁢(B∩ΓV)⊂StabV⁡(Z~)⁢(ℚ), and hence dim⁡(StabV⁡(Z~))>0. For any point z~∈Z~, StabV⁡(Z~)⁢(ℝ)+z~⊂Z~. By [52, Lemma 2.3], StabV⁡(Z~)⁢(ℝ) is full and complex. Define

V′:=V/StabV⁡(Z~) and ΓV′:=ΓV/(ΓV∩StabV⁡(Z~)⁢(ℚ)),

and then A′:=V′⁢(ℝ)/ΓV′ is a quotient abelian variety of A. Let Y′ (resp. Z~′) be the Zariski closure of the projection of Y (resp. Z~) in A′ (resp. V′⁢(ℝ)). We prove that the image of Z~′ is a point. If not, then proceeding as before for the triple (A′,Y′,Z~′) we can prove that dim⁡(StabV′⁡(Z~′))>0. This contradicts the definition (maximality) of StabV⁡(Z~). Hence Z~ is a translate of StabV⁡(Z~)⁢(ℝ). So Z~ is weakly special.

Case (ii): E=T

Define the norm of xU=(xU,1,xU,2,…,xU,m)∈U⁢(ℂ) to be

∥xU∥:=Max⁡(∥xU,1∥,∥xU,2∥,…,∥xU,m∥).

It is clear that for all xU∈U⁢(ℂ) and all γU∈ΓU such that γU⁢xU∈ℱU,

(A.4)H⁢(γU)≪∥xU∥.

Let ω|T=d⁢z1∧d⁢z¯1+…+d⁢zm∧d⁢z¯m be the canonical (1,1)-form of U⁢(ℂ)≃ℂm. Let pi (i=1,…,m) be the m natural projections of U⁢(ℂ)≃ℂm to ℂ. Let C be an algebraic curve of Z~ and define CM:={x∈C:∥x∥⩽M}. We have

(A.5)∫CM∩ℱUω|T⩽d⁢∑i=1m∫pi⁢(CM∩ℱU)𝑑zi∧d⁢z¯i

⩽d⁢∑i=1m∫{s∈ℂ:-1<Re⁡(s)<1,∥s∥⩽M}𝑑zi∧d⁢z¯i=d⋅O⁢(M),

where d:=deg⁡(C). On the other hand by [33, Theorem 0.1],

(A.6)∫CMω|T⩾O⁢(M2).

By (A.4),

CM⊂⋃γ∈Θ⁢(Z~,M)(CM∩γ-1⁢ℱ).

Integrating both side with respect to ω|T and taking into account that

γ⋅CM⊂(γ⁢C)2⁢M if ⁢H⁢(γ)⩽M,

we have

M2≪#⁢Θ⁢(Z~,M)⋅M

by (A.5) and (A.6). Hence #⁢Θ⁢(Z~,M)≫M.

Let now

StabU⁡(Z~):=ΓU∩StabU⁢(ℂ)⁡(Z~)¯Zar.

By Pila–Wilkie [47, Theorem 3.6], there exists an semi-algebraic subset B⊂Σ⁢(Z~) of positive dimension containing arbitrarily many points γU∈ΓU. We have B⁢Z~⊂univ-1⁡(Y) since Σ⁢(Z~)⁢Z~⊂univ-1⁡(Y) by definition. Hence for any γU∈ΓU∩B, Z~⊂γU-1⁢B⁢Z~⊂univ-1⁡(Y), and therefore Z~=γU-1⁢B⁢Z~ by the maximality of Z~. So γU-1⁢(B∩ΓU)⊂StabU⁡(Z~)⁢(ℚ), and hence dim⁡(StabU⁡(Z~))>0. Let

U′:=U/StabU⁡(Z~) and ΓU′:=ΓU/(ΓU∩StabU⁡(Z~)⁢(ℚ)),

and then T′:=U′⁢(ℂ)/ΓU′ is an algebraic torus over ℂ. Let Y′ (resp. Z~′) be the Zariski closure of the projection of Y (resp. Z~) in T′ (resp. U′⁢(ℂ)). We prove that Z~′ is a point. If not, then proceeding as before for the triple (T′,Y′,Z~′) we can prove dim⁡(StabU′⁡(Z~′))>0. This contradicts the definition (maximality) of StabU⁡(Z~). Hence Z~ is a translate of StabU⁡(Z~)⁢(ℂ). So Z~ is weakly special.

Acknowledgements

This topic was introduced to me by Emmanuel Ullmo. This first part of this paper (Sections 2–8) was done in Leiden University, while the two main theorems were proved when I was in Université Paris-Sud. I would like to express my gratitude to my supervisors Emmanuel Ullmo and Bas Edixhoven for weekly discussions and their valuable suggestions for the writing. I would like to thank Martin Orr for having pointed out a serious gap in Section 9 in a previous version as well as his several valuable remarks, especially for the last part of Section 10. Ya’acov Peterzil pointed out to me that the proof of the definability in Section 10.1 in a previous version was wrong. I have benefited a lot from the discussion with him and Sergei Starchenko for this definability problem. I also had some interesting discussion with Daniel Bertrand and Chao Zhang. Yves André, Daniel Bertrand, Bruno Klingler and Martin Orr have read a previous version of the manuscript and gave me some suggestions to improve the writing of both math and language. I would also like to thank them here. Finally, I thank the referee for his/her careful reading and helpful suggestions thanks to which this article has been improved.

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Received: 2013-12-8

Revised: 2014-11-4

Published Online: 2015-3-18

Published in Print: 2017-11-1