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We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We introduce perfect and imperfect overconvergent period rings, and we use the Tate--Sen method to construct overconvergent $(\varphi, \Gamma)$-modules for Galois representations over pseudorigid spaces., Comment: Final version. To appear in J. Theor. Nombres Bordeaux

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer [math] as the sum of [math] triangular numbers is asymptotically equivalent to the modified divisor function [math] .

We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also formulated a version of the epsilon-conjecture for affinoid families of (phi,Gamma)-modules over the Robba ring, and proved his conjecture in the rank-1 case. He used this case to construct an epsilon-isomorphism for families of trianguline (phi,Gamma)-modules, depending on a fixed triangulation. Our results imply that this epsilon-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [Ber04] and following the approach of Kisin to the construction of potentially semi-stable deformation rings [Kisa]., Comment: Revised. To appear in IMRN