Faltings heights of abelian varieties with complex multiplication

Andreatta, Fabrizio; Goren, Eyal Z.; Howard, Benjamin; Pera, Keerthi Madapusi

doi:10.4007/annals.2018.187.2.3

Cited by 63 publications

(123 citation statements)

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“…Yang [51][52][53] proved the Colmez conjecture for a large class of CM fields of degree 4, including the first known cases when E/Q is non-abelian. Colmez [16] also stated an averaged version of his conjecture, in which the Faltings heights are averaged over the different CM types for E. The averaged Colmez conjecture was recently proved independently by Andreatta et al [1] and Yuan and Zhang [55]. The averaged Colmez conjecture will play a crucial role in the proofs of the results in this paper (see, e.g., the discussion in Sect.…”

Section: Statement Of the Main Resultsmentioning

confidence: 74%

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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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Section: Statement Of the Main Resultsmentioning

confidence: 74%

“…, 1 The shaded parallelogram in Fig. 2 is the subset of the Shintani cone C(ε) determined by the inequalities 0 < t 1 ≤ 1 and 0 ≤ t 2 < 1, which correspond to the inequalities appearing in the definition of R ε, …”

Section: −1mentioning

confidence: 99%

On the Colmez conjecture for non-abelian CM fields

Sánchez

Masri

2018

Res Math Sci

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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: 88%

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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“…The second is that the divisor of these modular forms are known to be a linear combination of special divisors dictated by the principal part of the input weakly holomorphic forms. The second feature has been extended to produce so-called automorphic green functions for special divisors using harmonic Maass forms via regularized theta lifting by Bruinier [1] and BruinierFunke [2], which turned out to be very useful to generalization of the well-known GrossZagier formula [3] and the beautiful Gross-Zagier factorization formula of singular moduli [4] to Shimura varieties of orthogonal type (n, 2) and unitary type (n, 1) (see for example [5][6][7][8][9][10][11]). On the other hand, the Borcherds product expansion and in particular its integral structure is essential to prove modularity of some generating functions of arithmetic divisors on these Shimura varieties [12,13].…”

Section: Introductionmentioning

confidence: 99%