Transcendence of 1-periods

(joint work with G. Wüstholz) 1-periods are complex numbers obtained as path intervals of algebraic 1-forms on algebraic varieties over the field of algebraic numbers. The set contains famous numbers like 2pi i or values of log in algebraic numbers. They are a long-standing object of transcendence theory.

We will explain a sharp transcendence criterion and describe more generally all linear relations between 1-periods. The proof uses the theory of 1-motives in an essential way, allowing us to reduce the question to the seminal Analytic Subgroup Theorem of Wüstholz.

In the second half of the talk, we will discuss 1-motives in more detail. This leads to quantitive  versions of the theorem, i.e., formulas for the dimension of the space of periods of a given 1-motive.