We construct approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory, a nonlocal theory with derivatives of all orders. Novel features include the existence of slow-roll solutions even when the slow-roll parameters, as usually defined, are much greater than unity, as well as the need for the Hubble parameter to exceed the string mass scale ms. We show t...

Herrlich showed that a Mumford curve of genus g > 1 over the p-adic complex field Cp has at most 48(g− 1), 24(g− 1), 30(g− 1) or 12(g− 1) automorphisms as p = 2, 3, 5 or p > 5. The Mumford curves attaining these bounds are uniformised by normal subgroups of finite index in certain p-adic triangle groups ∆p for p ≤ 5, or in a p-adic quadrangle group p for p > 5. The finite groups attaining these...

A new incremental algorithm for data compression is presented. For a sequence of input symbols algorithm incrementally constructs a p-adic integer number as an output. Decoding process starts with less significant part of a p-adic integer and incrementally reconstructs a sequence of input symbols. Algorithm is based on certain features of p-adic numbers and p-adic norm. p-adic coding algorithm ...

We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

Although N is discrete in R, it is not discrete in Qp, and in fact has closure Zp. This raises the possibility of p-adically interpolating a sequence an, which is really a function n 7→ an on N, to a continuous function x 7→ ax with x ∈ Zp. Our basic question is this: when does a function f : N→ Qp extend to a continuous function Zp → Qp? We will look at some concrete examples, then see what a ...

We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the met...

2 Derived p-adic heights 2.1 Derived heights for cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Comparison of pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Compatibility of the derived heights . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Derived p-adic heights . . . . . . . . . . . . . . . . . . . . . . . . . ....

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a p...

The purpose of this paper is to survey some recent results in the theory of “p-adic cohomology”, by which we will mean several different (but related) things: the de Rham or p-adic étale cohomology of varieties over p-adic fields, or the rigid cohomology of varieties over fields of characteristic p > 0. Our goal is to update Illusie’s beautiful 1994 survey [I] by reporting on some of the many i...