Solutions to arithmetic differential equations in algebraically closed fields
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.