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arXiv · 1807.10082

Computing $\mathcal{L}$-invariants via the Greenberg-Stevens formula

Samuele Anni·Gebhard Boeckle·Peter Mathias Graef·Alvaro Troya
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Abstract

In this article, we describe how to compute slopes of $p$-adic $\mathcal{L}$-invariants of arbitrary weight and level by means of the Greenberg-Stevens formula. Our method is based on work of Lauder and Vonk on computing the reverse characteristic series of the $U_p$ operator on overconvergent modular forms. Using higher derivatives of this characteristic series, we construct a polynomial whose zeros are precisely the $\mathcal{L}$-invariants appearing in the corresponding space of modular forms with fixed sign of the Atkin-Lehner involution at $p$. In addition, we describe how to compute this polynomial efficiently. In the final section, we give computational evidence for relations between slopes of $\mathcal{L}$-invariants for small primes.

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Samuele Anni, Gebhard Boeckle, Peter Mathias Graef, Alvaro Troya. 2018-07-26. Computing $\mathcal{L}$-invariants via the Greenberg-Stevens formula. https://arxiv.org/abs/1807.10082

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