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 ...

Abstract Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling financial returns, with their extremal properties being of interest market risk management. For GARCH( $$p,q$$ p , q ) $$\max (p,q) = 1$$ <...