Computing Tangent Spaces to Eigenvarieties

Abstract

We develop an effective algorithm to compute the derivative of a Bianchi modular form with respect to weight space as it varies in a $p$-adic family. This method is entirely local at the modular form, and does not compute the family anywhere outside an infinitesimal neighbourhood. We numerically verify some conjectures surrouding smoothness of the eigenvariety (equivalently, uniqueness of families) and the ``direction over weight space'' of the family. The methods are also applied to study elliptic modular forms and their $\mathcal{L}$-invariants.

Figures (6)

• Figure 1: Deformation directions for some Bianchi cusp forms of weight (2, 2).
• Figure 2: Derivatives of the $U_p$ operator for a selection of cusp forms
• Figure 3: Derivatives of some Hecke operators for the cusp form with LMFDB label 11.2.a.a
• Figure 4: Derivatives of some Hecke operators for the cusp form with LMFDB label 55.2.a.a
• Figure 5: Derivatives of some Hecke operators for the cusp form with LMFDB label 15.2.a.a

Theorems & Definitions (22)

• Theorem 1
• Definition 1
• Theorem 2: Eichler-Shimura-Harder Isomorphism
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof