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Abstract

Arithmetic differential equations or δ-geometry exploits analogies between derivations and p-derivations δ arising from lifts of Frobenius to study problems in arithmetic geometry. Along the way, two main classes such functions, describable as series, arose prominently namely δ-characters of abelian schemes and (isogeny covariant) δ-modular forms. However, the theory of these δ-functions is not as straightforward in ramified settings. Overconvergence was introduced in [13] to account for these issues which essentially imposes growth conditions extensions of these series to a fixed level of ramification; necessary as such extensions have non-trivial fractional coefficients. In this article, we introduce a rescaling process which identifies a class of δ-functions we call totally overconvergent, which extend all the way to the algebraic closure of ring of integers of the maximally unramified extension of Q_p . Applications built on these functions allow one to remove boundedness assumptions on ramification. The bulk of the article is devoted to establishing that most δ-functions arising in practice, namely those in the applications described in [5], [7], [8], are totally overconvergent, which essentially extends results in [13] to unbounded ramification.