Arithmetic series

Arithmetic hypergeometric series

The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems. Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general. Bibliography: 183 titles.

Estimation of Arithmetic Linear Series

In the paper [5], Lazarsfeld and Mustaţă propose general and systematic usage of Okounkov’s idea in order to study asymptotic behavior of linear series on an algebraic variety. It is a very simple way, but it yields a lot of consequences, like Fujita’s approximation theorem. Yuan [8] generalized this way to the arithmetic situation, and he established the arithmetic Fujita’s approximation theor...

Arithmetic Linear Series with Base Conditions

In this note, we study the volume of arithmetic linear series with base conditions. As an application, we consider the problemof Zariski decompositions on arithmetic varieties. Introduction LetXbeaprojective andflat integral schemeoverZ. Weassume thatX is normal and thegeneric fiber ofX → Spec(Z) is a d-dimensional smoothvariety overQ. Let Div(X) be the groupofCartier divisors onX and letDiv(X)...

Arithmetic Gevrey Series and Transcendence. a Survey

Effective Bound of Linear Series on Arithmetic Surfaces