How to implement a modular form

AbstractWe present a model for Fourier expansions of arbitrary modular forms. This model takes precisions and symmetries of such Fourier expansions into account. The value of this approach is illustrated by studying a series of examples. An implementation of these ideas is provided by the author. We discuss the technical background of this implementation, and we explain how to implement arbitrary Fourier expansions and modular forms. The framework allows us to focus on the considerations of a mathematical nature during this procedure. We conclude with a list of currently available implementations and a discussion of possible computational research

Dual weights in the theory of harmonic Siegel modular forms

We define harmonic Siegel modular forms based on a completely new approach using vector-valued covariant operators. The Fourier expansions of such forms are investigated for two distinct slash actions. Two very different reasons are given why these slash actions are natural. We prove that they are related by xi-operators that generalize the xi-operator for elliptic modular forms. We call them dual slash actions or dual weights, a name which is suggested by the many properties that parallel the elliptic case. Based on Kohnen's limit process for real-analytic Siegel Eisenstein series, we show that, under mild assumptions, Jacobi forms can be obtained from harmonic Siegel modular forms, generalizing the classical Fourier-Jacobi expansion. The resulting Fourier-Jacobi coefficients are harmonic Maass-Jacobi forms, which are defined in full generality in this work. A compatibility between the various xi-operators for Siegel modular forms, Jacobi forms, and elliptic modular forms is deduced, relating all three kinds of modular forms.Duale Gewichte in der Theorie harmonischer Siegelscher Modulformen Fußend auf einem vollständig neuen Ansatz, dem vektorwertige kovariante Operatoren zu Grunde liegen, definieren wir den Begriff der harmonischen Siegelschen Modulform. Dieser Definition schließt sich eine Untersuchung der für zwei verschiedene Strichoperationen auftretenden Fourier-Entwicklungen an. Die besagten Operationen sind natürlich in zweierlei Hinsicht, auf die wir beide näher eingehen. Darüber hinaus besteht eine Verbindung zwischen diesen beide Strichoperatoren, die durch zwei xi-Operatoren, die wiederum den elliptischen xi-Operator verallgemeinern, vermittelt wird. Die bemerkenswerte Ähnlichkeit zum Verhalten von elliptischen Modulformen dual Gewichts legt die Verwendung dieses Begriffs auch für die hier untersuchten Gewichte Siegelscher Modulformen nahe. Eine Verallgemeinerung der klassischen Fourier-Jacobi-Entwicklung kann aufbauend auf Kohnens Grenzwertprozess für reell-analytische Siegelsche Eisensteinreihen für eine große Klasse von harmonischen Siegelschen Modulformen hergele\-tet werden. Die herbei auftretenden Fourier-Jacobi-Entwicklungen stellen sich als Maaß-Jacobiformen heraus, die in voller Allgemeinheit in dieser Arbeit definiert werden. Wir zeigen schließlich, dass die verschiedenen xi-Operatoren für Siegelsche Modulformen, Jacobiformen und elliptische Modulformen miteinander verträglich sind und stellen so einen Zusammenhang zwischen diesen drei Arten von Modulformen her

Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms

We investigate Poincar\'e series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincar\'e series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincar\'e series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel-Poincar\'e series. We surprisingly discover that these Poincar\'e series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls

Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms

The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms

Achieving impact from ecosystem assessment and valuation of urban greenspace: The case of i-Tree Eco in Great Britain

Numerous tools have been developed to assist environmental decision-making, but there has been little examination of whether these tools achieve this aim, particularly for urban environments. This study aimed to evaluate the use of the i-Tree Eco tool in Great Britain, an assessment tool developed to support urban forest management. The study employed a documentary review, an online survey, and interviews in six case study areas to examine five impacts (instrumental, conceptual, capacity-building, enduring connectivity, and culture/attitudes towards knowledge exchange) and to identify which factors inhibited or supported achievement of impact. It revealed that the i-Tree Eco projects had helped to increase knowledge of urban forests and awareness of the benefits they provide. While there was often broad use of i-Tree Eco findings in various internal reports, external forums, and discussions of wider policies and plans, direct changes relating to improved urban forest management, increased funding or new tree policies were less frequent. The barriers we identified which limited impact included a lack of project champions, policy drivers and resources, problems with knowledge transfer and exchange, organisational and staff change, and negative views of trees. Overall, i-Tree Eco, similar to other environmental decision-making tools, can help to improve the management of urban trees when planned as one step in a longer process of engagement with stakeholders and development of new management plans and policies. In this first published impact evaluation of multiple i-Tree Eco projects, we identified eight lessons to enhance the impact of future i-Tree Eco projects, transferable to other environmental decision-making tools

volumetric characterisation and correlation to established classification systems

Objective and sensitive assessment of cartilage repair outcomes lacks suitable methods. This study investigated the feasibility of 3D ultrasound biomicroscopy (UBM) to quantify cartilage repair outcomes volumetrically and their correlation with established classification systems. 32 sheep underwent bilateral treatment of a focal cartilage defect. One or two years post- operatively the repair outcomes were assessed and scored macroscopically (Outerbridge, ICRS-CRA), by magnetic resonance imaging (MRI, MOCART), and histopathology (O'Driscoll, ICRS-I and ICRS-II). The UBM data were acquired after MRI and used to reconstruct the shape of the initial cartilage layer, enabling the estimation of the initial cartilage thickness and defect volume as well as volumetric parameters for defect filling, repair tissue, bone loss and bone overgrowth. The quantification of the repair outcomes revealed high variations in the initial thickness of the cartilage layer, indicating the need for cartilage thickness estimation before creating a defect. Furthermore, highly significant correlations were found for the defect filling estimated from UBM to the established classification systems. 3D visualisation of the repair regions showed highly variable morphology within single samples. This raises the question as to whether macroscopic, MRI and histopathological scoring provide sufficient reliability. The biases of the individual methods will be discussed within this context. UBM was shown to be a feasible tool to evaluate cartilage repair outcomes, whereby the most important objective parameter is the defect filling. Translation of UBM into arthroscopic or transcutaneous ultrasound examinations would allow non-destructive and objective follow-up of individual patients and better comparison between the results of clinical trials