Serre in the 1970s was the first to formalize such a question on the way to constructing p-adic L-functions, by way of developing the notion of a p-adic modular form to be the p-adic limit of some compatible family of q-expansions of classical modular forms. Katz came along fairly soon afterwards and generalized the theory to a much more geometric context, and showed that Serre’s p-adic forms e...

The p-adic upper half plane X is a rigid analytic variety over a p-adic field K, on which the group GL2(K) acts, that Mumford introduced (as a formal scheme) as part of his efforts to generalize Tate’s p-adic uniformization of elliptic curves to curves of higher genus. The Cp–valued points of X are just P(Cp)−P(K), with GL2(K) acting by linear fractional transformations. Mumford showed that the...

The field $Q_{p}$ of $p$-adic numbers is defined as the completion of the field of the rational numbers $Q$ with respect to the  $p$-adic norm $|.|_{p}$. In this paper, we study the continuous and discrete $p-$adic shearlet systems on $L^{2}(Q_{p}^{2})$. We also suggest discrete $p-$adic shearlet frames. Several examples are provided.

We study the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete limit can be regarded as a classical limit of a quantum object, it implies that a correspondence between classical and qua...