The André-Oort conjecture for 𝒜_g

Tsimerman, Jacob

doi:10.4007/annals.2018.187.2.2

Cited by 71 publications

(70 citation statements)

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“…In view of the recent results of Andreatta et al, Tsimerman and Yuan and Zhang (cf. [1,34,40]), the André-Oort conjecture is now proven for Abelian type Shimura varieties. In particular, Theorem 2.10 and Proposition 2.12 are unconditional.…”

Section: Zariski Densitymentioning

confidence: 89%

“…[1,34,40]), the André-Oort conjecture now holds for Abelian type Shimura varieties. As an arbitrary self-product of a Hilbert modular Shimura variety is of Abelian type, Theorem A is now unconditional.…”

Section: Introductionmentioning

confidence: 87%

“…The result of Tsimerman builds on a strategy of Pila-Zanier and previous work of Tsimerman-Pila. We refer to the introduction of [34] for an overview.…”

Section: Introductionmentioning

confidence: 99%

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André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

Burungale

Hida²

2016

Forum of Mathematics, Sigma

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Let F/Q be a totally real field and K /F a complex multiplication (CM) quadratic extension. Let f be a cuspidal Hilbert modular new form over F. Let λ be a Hecke character over K such that the Rankin-Selberg convolution f with the θ -series associated with λ is self-dual with root number 1. We consider the nonvanishing of the family of, f ⊗ λχ ), as χ varies over the class group characters of K . Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters χ such that L(, f ⊗ λχ) = 0 increases with the absolute value of the discriminant of K . We crucially rely on the André-Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan-Zhang and Andreatta-Goren-Howard-Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin-Selberg L-values. The Waldspurger formula plays an underlying role.

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Section: Zariski Densitymentioning

confidence: 89%

Section: Introductionmentioning

confidence: 87%

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André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

Burungale

Hida²

2016

Forum of Mathematics, Sigma

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“…These identities yield the following averaged version of the Colmez conjecture. [47], who proved that it implies the André-Oort conjecture for the moduli space of principally polarized abelian varieties of dimension g.…”

Section: The Averaged Colmez Conjecturementioning

confidence: 99%

On the Colmez conjecture for non-abelian CM fields

Sánchez

Masri

2018

Res Math Sci

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The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at s = 0. In this paper, we prove that if F is any fixed totally real number field of degree [F : Q] ≥ 3, then there are infinitely many effective, "positive density" sets of CM extensions E/F such that E/Q is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla-Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by

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“…The Colmez conjecture gives a formula for the Faltings height of a CM abelian variety in terms of log derivatives of L-functions arising from the CM type. This conjecture has proven useful in giving bounds for the Faltings height of CM abelian varieties (see [Col98] for the case of elliptic curves and [Tsi18] where a weaker form of the Colmez conjecture is used in the proof of the André-Oort conjecture for the moduli space of principally polarized abelian varieties).…”

Section: Set Up and Theoremmentioning

confidence: 99%

Signature (n − 2,2) CM types and the unitary Colmez conjecture

Parenti

2018

Comptes Rendus. Mathématique

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Colmez conjectured a formula relating the Faltings height of CM abelian varieties to a certain linear combination of log derivatives of L-functions. In this paper, we study the case of unitary CM fields and by studying the class functions that arise, we reduce the conjecture to a special case. Using the Galois action, we prove more cases of the Colmez Conjecture.2010 Mathematics Subject Classification. Primary: 11G15.

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The André-Oort conjecture for 𝒜_g

Cited by 71 publications

References 20 publications

André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

On the Colmez conjecture for non-abelian CM fields

Signature (n − 2,2) CM types and the unitary Colmez conjecture

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The André-Oort conjecture for 𝒜_g

Cited by 71 publications

References 20 publications

André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

On the Colmez conjecture for non-abelian CM fields

Signature (n − 2,2) CM types and the unitary Colmez conjecture

Contact Info

Product

Resources

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Signature (n − 2,2) CM types and the unitary Colmez conjecture