Based on joint with Adam Keilthy.

In isolated examples that appear in the literature, deformation coboundaries constructed geometrically for modular curves feature non-critical periods. These periods cancel in the associated deformation cocycle, but it was not clear whether non-critical periods are an artifact of a specific construction or genuinely contribute to the theory. In recent work, Bogo used an explicit hypergeometric uniformization to enable a calculation of deformation cocycles. He revealed a connection to quasi-modular forms, and confirmed the appearance of non-critical periods.

Using a formal deformation approach to the topic, we provide unique logarithmic extensions of any first order deformation, canonical and universal deformation families, and establish uniqueness of logarithmic deformation coboundaries. In particular, non-critical periods are not an artifact previous methods.