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A Robust Implementation for Solving the S-Unit Equation and Several Applications Read chapter Read first chapter

2021 | OriginalPaper | Chapter

Visualizing Modular Forms

Author: David Lowry-Duda

Published in: Arithmetic Geometry, Number Theory, and Computation

Publisher: Springer International Publishing

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Abstract

We examine several currently used techniques for visualizing complex-valued functions applied to modular forms. We plot several examples and study the benefits and limitations of each technique. We then introduce a method of visualization that can take advantage of colormaps in Python’s matplotlib library, describe an implementation, and give more examples. Much of this discussion applies to general visualizations of complex-valued functions in the plane.
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Footnotes
1
Adjusting the map from magnitude to brightness for SageMath’s complex_plot is possible, but nontrivial; the map is defined internally and the current plotting interface doesn’t give any option to choose or alter this map. In order to adjust this map, it is necessary to modify the source for SageMath’s plotting routines directly.
 
2
In the period between writing and publishing this paper, the LMFDB changed its method for plotting modular forms to incorporate methods described in this paper. Thus in the rest of this paper, LMFDB-style plots refer to the style of plots in the LMFDB on October 1, 2020.
 
3
The behavior in Matlab and Maple appears similar, though it is possible to define a color scheme. On the other hand Mathematica has an extensive library of complex plotting color schemes.
 
Literature
[DS05]
Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228. Springer Verlag, 2005. MATH
[Far98]
Frank A Farris. Visual complex analysis by Tristan Needham. The American Mathematical Monthly, 105(6):570–576, 1998.
[Far15]
Frank A Farris. Creating Symmetry: The artful mathematics of wallpaper patterns. Princeton University Press, 2015.
[Hun07]
J. D. Hunter. Matplotlib: A 2d graphics environment. Computing in Science & Engineering, 9(3):90–95, 2007. CrossRef
[Kov15]
Peter Kovesi. Good colour maps: How to design them. arXiv preprint arXiv:1509.03700, 2015.
[LD20]
[LMF19]
[NAR18]
Jamie R Nuñez, Christopher R Anderton, and Ryan S Renslow. Optimizing colormaps with consideration for color vision deficiency to enable accurate interpretation of scientific data. PloS one, 13(7):e0199239, 2018.
[Oli06]
Travis E Oliphant. A guide to NumPy, volume 1. Trelgol Publishing USA, 2006.
[Sag20]
[WS10]
Elias Wegert and Gunter Semmler. Phase plots of complex functions: a journey in illustration. Notices AMS, 58:768–780, 2010. MathSciNetMATH
Metadata
Title
Visualizing Modular Forms
Author
David Lowry-Duda
Copyright Year
2021
Book
Arithmetic Geometry, Number Theory, and Computation

Print ISBN: 978-3-030-80913-3

Electronic ISBN: 978-3-030-80914-0

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-80914-0_19

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