Abstract : (joint work with G. Wüstholz) Roughly, $1$-dimensional periods are the complex numbers obtained by integrating a differential form on an algebraic curve over $\overline{\mathbf{Q}}$ over a suitable domain of integration. One of the alternative characterisations is as periods of Deligne $1$-motives.
We clear up the linear relations between these numbers, proving Kontsevich's version of the period conjecture for $1$-dimensional periods. In particular, a $1$-dimensional period is shown to be algebraic if and only if it is of the form ${\int }_{\gamma }(\varphi +df)$ with ${\int }_{\gamma }\varphi =0$. We also get formulas for the spaces of periods of a given $1$-motive, generalising Baker's theorem on logarithms of algebraic numbers.
The proof is based on a version of Wüstholz's analytic subgroup theorem for $1$-motives.

Keywords : $1$-dimensional period; $1$-motive; Deligne motives; Wüstholz theorem