The classical unit root splitting of the Hodge filtration on universal p-adic de Rham cohomology over the ordinary locus of Shimura varieties, due to Dwork and Katz, has had many applications in the study of p-adic modular forms, constructions of theta operators and Iwasawa theory. The well-known obstruction to extending this splitting into the supersingular locus is the non-overconvergence of Katz's p-adic weight 2 Eisenstein series $E_2$. In this talk we discuss a new period sheaf over which the Hodge filtration splits canonically, containing periods which can roughly be thought of as analytic continuations of $E_2$. Using this sheaf, we define a new theory of quasi-overconvergent modular forms and p-adic Maass-Shimura operators acting on these forms, which specialize to Katz's p-adic modular forms and theta operators on the ordinary locus. Using these operators, we generalize and unify previous constructions due to the author and Andreatta-Iovita of p-adic L-functions of Katz- and BDP-type for p nonsplit in the CM field K, defining a locally analytic p-adic L-function on the full space of p-adic central critical characters which specializes on certain subdomains to each of these constructions. We will also discuss the arithmetic applications of these constructions, including generalizing the author's previous work on Iwasawa theory over K, Sylvester's conjecture on sums of rational cubes and Goldfeld's conjecture for the congruent number family.