We study finite dimensional vector spaces over fields of elliptic functions equipped with two commuting aotomorphisms \sigma and \tau induced by isogenies relatively prime orders. give a structure theorem for such objects, that reveals connection to the classification bundles on curves Atiyah. As an application we prove analogue conjecture Loxton van der Poorten which has been recently proved A...

We study the p-Laplace equation with Potentials − div(|∇u|p−2∇u) + λV (x)|u|p−2u = |u|q−2u, u ∈ W 1,p(RN ), x ∈ RN where 2 ≤ p, p < q < p∗. Using a concentrationcompactness principle from critical point theory, we obtain existence, multiplicity solutions, and concentration of solutions.

In this paper, we prove that a semimartingale local time is a rough path of roughness p for any p ∈ [2, 3) and establish the integral ∫∞ −∞ g(x)dLt(x) for any finite q-variation function g (2 ≤ q < 3) using Lyons’ rough path integration theory. We therefore obtain the Meyer-Tanaka’s formula for continuous function f if ∇−f exists and is of finite q-variation when 2 ≤ q < 3. The case when 1 ≤ q ...

Brief introduction to cyclotomic theory over Q using adeles. Discussion of the definitions of modular forms and automorphic forms. Introducing the adelic automorphic forms via strong approximation theorem. Discussion of the connected components of Shimura varieties (modular curves). Smooth/admissible representations of locally finite groups. Definition and admissibility of (cuspidal) automorphi...

Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, let $H_K$ its Hilbert class field. For each prime $p$ we examine two $p$-adic abelian objects in view of theory. First, investigate the image a Galois representation $\mathrm{Gal}(\overline{\mathbb{Q}}/H_K)$ attached to elliptic curve with complex multiplication. Second, construct form group ...

Let p/q be a rational number. Numeration in base is defined by function that evaluates each finite word over A_p={0,1,...,p-1} to some We let N_p/q denote the image of this evaluation function. In particular, contains all nonnegative integers and literature on usually focuses set words are evaluated integers; it rather chaotic language which not context-free. On contrary, we study here subsets ...