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Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory pp 741–763Cite as

Topics on Modular Galois Representations Modulo Prime Powers

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Abstract

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.

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Notes

  1. 1.

    http://lmfdb.org.

  2. 2.

    http://math.uni.lu/~tsaknias/elladicdatabase_2.php.

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Acknowledgements

The authors would like to thank Rajender Adibhatla, Sara Arias-de-Reyna, Gebhard Böckle, Frank Calegari, Imin Chen, Shaunak Deo, Frazer Jarvis, Ian Kiming, Ariel Pacetti, Nadim Ruston and many others for various discussions about topics on modular Galois representations modulo prime powers. They also thank Ken Ribet for having pointed out an inaccuracy in a previous version. Thanks are also due to the referee for a careful reading and useful suggestions. The second author thanks Gabi Nebe for having explained the simple algorithmic idempotent lifting (Eq. (4)) to him a long time ago.

This project was supported by the Luxembourg Research Fund (Fonds National de la Recherche Luxembourg) INTER/DFG/12/10/COMFGREP in the framework of the priority program 1489 of the Deutsche Forschungsgemeinschaft.

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Authors and Affiliations

  1. Université du Luxembourg, Unité de Recherche en Mathématiques, Maison du nombre, 6, avenue de la Fonte, L-4364, Esch-sur-Alzette, Luxembourg

    Panagiotis Tsaknias & Gabor Wiese

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  1. Panagiotis Tsaknias
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  2. Gabor Wiese
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Correspondence to Gabor Wiese .

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Editors and Affiliations

  1. IWR, Heidelberg University, Heidelberg, Germany

    Gebhard Böckle

  2. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Wolfram Decker

  3. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Gunter Malle

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Tsaknias, P., Wiese, G. (2017). Topics on Modular Galois Representations Modulo Prime Powers. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_31

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