A control theorem for p-adic automorphic forms and Teitelbaum’s L-invariant - Gräf, Peter Mathias (The Ramanujan journal, 2019)
In this article, we describe an efficient method for computing Teitelbaum’s p-adic LL\mathcal {L}-invariant. These invariants are realized as the eigenvalues of the LL\mathcal {L}-operator acting on a space of harmonic cocycles on the Bruhat-Tits tree TT{\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 LL\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 LL\mathcal {L}-invariants of different levels and weights for p=2p=2p=2.