The hyperbolic Ax-Lindemann-Weierstraß conjecture

Klingler, Bruno; Ullmo, Emmanuel; Yafaev, Andrei

doi:10.1007/s10240-015-0078-9

Cited by 68 publications

(97 citation statements)

References 35 publications

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“…This was first proved for products of modular curves by Pila in his seminal work [Pil11], then for compact Shimura varieties by Ullmo and Yafaev in [UY14b] and for A g by Pila and Tsimerman in [PT14]. The case of a general Shimura variety has recently been demonstrated by Klingler, Ullmo and Yafaev (see [KUY16]). …”

Section: Conjecture 13 Let S Be a Shimura Variety And Let σ Be A Sementioning

confidence: 89%

Heights of pre-special points of Shimura varieties

Daw

Orr

2015

Math. Ann.

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Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding Z-Hodge structure. Our bound is the final step needed to complete a proof of the André-Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.

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Section: Conjecture 13 Let S Be a Shimura Variety And Let σ Be A Sementioning

confidence: 89%

Heights of pre-special points of Shimura varieties

Daw

Orr

2015

Math. Ann.

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“…We point out that, although our result is independent of the realisation of the symmetric domain D uniformising S, 3 we use in a crucial way that there is a bounded realisation. Indeed this allows us to reduce the proof to the definable sets V i and is again used in a fundamental way in the proof of the above cited Lemma 3.3.…”

Section: Theorem 11 (Bloch-ochiai) Let a Be An Abelian Variety And mentioning

confidence: 97%

“…In the proof of Theorem 3.2 for algebraic curves given in [3], the only part that relies on the curve being algebraic is the analogue of the following lemma. On D F we have the Poincaré metric defined by…”

Section: Holomorphic Curves and Fundamental Domainsmentioning

confidence: 99%

Holomorphic curves in Shimura varieties

Giacomini

2018

Arch. Math.

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Abstract. We prove a hyperbolic analogue of the Bloch-Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura varieties. The statement which we consider here was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.Mathematics Subject Classification. 14G35, 03C64, 14P10, 32H02, 32M15.

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“…Given an algebraic subsetỸ of C N , the set Y :=Ỹ ∩ X + is a complex analytic set having finitely many complex analytic components which are semi-algebraic (in particular definable in any o-minimal structure over R). All these facts can be found in [3,Appendix B;11].…”

Section: Introductionmentioning

confidence: 82%

“…In [6], Pila proved Theorem 1.3 for products of modular curves and used o-minimality to prove the André-Oort conjecture for such products. In [7], Theorem 1.3 (and the André-Oort conjecture) is proved for the moduli space of principally polarized abelian surfaces and in [8], Theorem 1.3 is proved for moduli spaces of principally polarized abelian varieties of arbitrary dimension g. In [13], Theorem 1.3 is proved for compact Shimura varieties and in [3] it is proved for all Shimura varieties.…”

Section: Introductionmentioning

confidence: 99%