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A control theorem for p-adic automorphic forms and Teitelbaum’s \(\mathcal {L}\)-invariant

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Abstract

In this article, we describe an efficient method for computing Teitelbaum’s p-adic \(\mathcal {L}\)-invariant. These invariants are realized as the eigenvalues of the \(\mathcal {L}\)-operator acting on a space of harmonic cocycles on the Bruhat–Tits tree \({\mathcal {T}}\), which is computable by the methods of Franc and Masdeu described in (LMS J Comput Math 17:1–23, 2014). The main difficulty in computing the \(\mathcal {L}\)-operator is the efficient computation of the p-adic Coleman integrals in its definition. To solve this problem, we use overconvergent methods, first developed by Darmon, Greenberg, Pollack and Stevens. In order to make these methods applicable to our setting, we prove a control theorem for p-adic automorphic forms of arbitrary even weight. Moreover, we give computational evidence for relations between slopes of \(\mathcal {L}\)-invariants of different levels and weights for \(p=2\).

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Acknowledgements

The author wishes to thank Gebhard Böckle for suggesting this interesting topic and for his support and encouragement. He is grateful to Marc Masdeu, Tommaso Centeleghe and Samuele Anni for many enlightening discussions and their comments on this article. The author expresses his gratitude to Adrian Iovita and the Mathematics department of Concordia University for their hospitality.

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  1. IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Peter Mathias Gräf

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Correspondence to Peter Mathias Gräf.

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The author was supported by the DFG via the Forschergruppe 1920 and the SPP 1489. Moreover, part of this work was done during a stay at Concordia University, Montréal funded by a DAAD-Doktorandenstipendium.

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Gräf, P.M. A control theorem for p-adic automorphic forms and Teitelbaum’s \(\mathcal {L}\)-invariant. Ramanujan J 50, 13–43 (2019). https://doi.org/10.1007/s11139-019-00160-1

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