Species and Dimension Formulas for Period Spaces

(joint work with Martin Kalck) Periods are complex numbers obtained by integrating algebraic differential forms over $\mathbf{Q}$ over a domain of algebraic nature. This includes numbers like $\pi$, $\log(2)$ or the values of the Riemann zeta function at integers. More conceptually, periods are the entries of the period pairing between the singular and de Rham realisation of a mixed motive over $\mathbf{Q}$. The period conjecture makes a sweeping qualitative prediction on the relations between period numbers, e.g., that $\pi$ is transcendental.

We develop and apply the abstract theory of a very special class of Noetherian rings, namely finite dimensional $\mathbf{Q}$-algebras, in order to deduce formulas for the expected dimension of the space of periods of a single motive. This is of particular interest in the case of $1$-motives (or periods of curves), where the linear version of the period conjecture is a theorem due to Huber and Wüstholz.