Singular moduli for real quadratic fields: A rigid analytic approach

Darmon, Henri; Vonk, Jan

doi:10.1215/00127094-2020-0035

Cited by 24 publications

(25 citation statements)

References 17 publications

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“…To make such a connection concrete, we require a real quadratic analogue of j(τ 1 ) − j(τ 2 ), and not just the exponents. This connection is the goal of Darmon-Vonk in [DV20].…”

Section: Definementioning

confidence: 96%

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Counting intersection numbers of closed geodesics on Shimura curves

Rickards¹

2021

Preprint

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Let Γ\H be a Shimura curve, where Γ corresponds to the group of units of norm 1 in an Eichler order O of an indefinite quaternion algebra over Q. Closed geodesics on Γ\H correspond to optimal embeddings of real quadratic orders into O. The intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon and Vonk on a real quadratic analogue to j(τ 1 ) − j(τ 2 ). In this paper, we study the total intersection number over all embeddings of a given pair of discriminants D 1 , D 2 . We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection of D 1 , D 2 . This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing nrd(j(τ 1 ) − j(τ 2 )). The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper. Contents 1. Introduction 2. Quaternionic background 3. Basic results on intersection numbers 4.

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“…To make such a connection concrete, we require a real quadratic analogue of j(τ 1 ) − j(τ 2 ), and not just the exponents. This connection is the goal of Darmon-Vonk in [DV20].…”

Section: Definementioning

confidence: 96%

“…Conjecture 1.12 (Conjecture 4.26 of [DV20]). Let q lie above the integer prime q = p. If q is split in…”

Section: Definementioning

confidence: 99%

Counting intersection numbers of closed geodesics on Shimura curves

Rickards¹

2021

Preprint

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“…The aim of this section is to apply our main result to the study of lifting obstructions of theta cocycles. In [8] Darmon and Vonk initiated the theory of rigid meromorphic cocycles, i.e., elements in the cohomology group H 1 (SL 2 (Z[1/p]), M × ), where M × is the group of invertible meromorphic functions on Drinfeld's p-adic upper half plane . They provide a large supply of classes in the space of theta cocycles, i.e., elements of H 1 (SL 2 (Z[1/p]), M × /C × p ) (see also [16] and [19] for a generalization of the theory to other number fields and congruence subgroups).…”

Section: Lifting Obstructions Of Theta Cocyclesmentioning

confidence: 99%

“…This project started as an attempt to understand the relation between lifting obstructions of rigid analytic theta cocycles in the sense of Darmon-Vonk (cf. [8], [7]) and automorphic L-invariants as introduced by Spieß in [22]. As a first application of our main theorem, we show how lifting obstructions of cuspidal theta cocycles for Hilbert modular groups can be computed in terms of L-invariants of the associated Galois representations.…”

Section: Introductionmentioning

confidence: 96%

Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

Gehrmann

2022

Res Math Sci

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Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld’s upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note, we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon–Vonk can be computed in terms of $$\mathcal {L}$$ L -invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups.

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“…In [Ric21a] and [Ric21b], this intersection number was studied, and formulas for the total intersection of a pair of discriminants were derived. The second paper also discussed connections to the work of Gross-Zagier on factorizing Nm(j(τ 1 )−j(τ 2 )) ( [GZ85]), and Darmon-Vonk on a real quadratic analogue of the difference of j−values ( [DV20]). Here, we connect intersection numbers to the Kudla programme by constructing formal power series that turn out to be modular forms.…”

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confidence: 99%

Hecke operators acting on optimal embeddings in indefinite quaternion algebras

Rickards

2021

Preprint

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Let Emb + Z (O) denote Z−linear formal sums of equivalence classes of optimal embeddings of real quadratic orders into O, an Eichler order of level M in an indefinite quaternion algebra over Q. For all positive integers n coprime to M, we define an operator Tn :, which we refer to as a Hecke operator. We introduce the "prime power Hecke graph", and show how it can be used to determine the effect of Tp on a single optimal embedding. To justify the name of the operator, we associate an optimal embedding equivalence class to a closed geodesic on the corresponding Shimura curve. By considering the signed intersection number of pairs of geodesics, we use the operators Tn to define formal power series. In showing that the action of Hecke operators on the homology of the curve commutes with our definition, we prove that the formal power series agree with weight two cusp forms for Γ 0 (N ) on all coefficients of exponents coprime to M.

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Singular moduli for real quadratic fields: A rigid analytic approach

Cited by 24 publications

References 17 publications

Counting intersection numbers of closed geodesics on Shimura curves

Counting intersection numbers of closed geodesics on Shimura curves

Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

Hecke operators acting on optimal embeddings in indefinite quaternion algebras

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