Twisted traces of CM values of weak Maass forms

Alfes, Claudia; Ehlen, Stephan

doi:10.1016/j.jnt.2012.10.008

Cited by 27 publications

(60 citation statements)

References 19 publications

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“…In particular, if M * has genus zero, then these cycle integrals are closely related to the Date: November 27, 2017. 1 winding number of the closed geodesics around the poles of η. To see this, we take as an example Γ = SL 2 (Z) and consider (1.2) η := j ′ (z) j(z) − 1728 dz,…”

Section: Introductionmentioning

confidence: 99%

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Modularity of generating series of winding numbers

et al. 2018

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The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

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Section: Introductionmentioning

confidence: 99%

“…(3.34) intertwines the representations ρ L and ρ L ∆[1]. With respect to the pairings on C[L ′ /L] and C[L ′ /L ∆ ], we obtain a linear map ψ * ∆,r : A k (ρ L ∆ ) → A k (ρ L ).…”

mentioning

confidence: 99%

Modularity of generating series of winding numbers

et al. 2018

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“…Fix a class r mod 2N with r 2 ≡ ∆ mod 4N. Following the Bruinier and Ono's work [BO], Alfes and Ehlen constructed a C[L ♯ /L]− valued theta function [AE,Section 4] (2.16)…”

Section: Thus One Hasmentioning

confidence: 99%

“…where Θ ∆,r (τ, z) is the twisted Kudla-Millson theta kernel defined by (2.16), following [AE,Section 4]. Define the normalized Eisenstein series of weight 0 as in [DY] by…”

Section: Introductionmentioning

confidence: 99%

Twisted arithmetic Siegel Weil formula on X0(N)

Yang

2019

Journal of Number Theory

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In this paper, we study twisted arithmetic divisors on the modular curve X 0 (N ) with N square-free. For each pair (∆, r) where ∆ > 0 and ∆ ≡ r 2 mod 4N , we constructed a twisted arithmetic theta function φ ∆,r (τ ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of φ ∆,r (τ ), along the way, we also identify the arithmetic pairing φ ∆,r (τ ), ω N with special value of some Eisenstein series, where ω N is a normalized metric Hodge line bundle.

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“…Z and ϕ ∈ S L , we can define the divisor This result comes out of calculating the effect of the intertwining operator in [AE,Section 3] applied to the generating series A m (τ ) associated to the scaled lattice (∆L,…”

Section: Mmentioning

confidence: 99%

Heegner divisors in generalized Jacobians and traces of singular moduli

Bruinier¹,

Li²

2016

Algebra Number Theory

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Abstract. In the present paper we prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.

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Twisted traces of CM values of weak Maass forms

Cited by 27 publications

References 19 publications

Modularity of generating series of winding numbers

Modularity of generating series of winding numbers

Twisted arithmetic Siegel Weil formula on X0(N)

Heegner divisors in generalized Jacobians and traces of singular moduli

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