The genus g function Theta : $\mathbb{C}^g x \mathfrak{H}_g -> \mathbb{C}$ has numerous applications in mathematics, from number theory to non-linear differential equations; in particular, its connection with the Abel-Jacobi map on complex elliptic and hyperelliptic curves has important applications. A connection between some values of this function (the theta-constants) and the arithmetico-geometric mean of Gauss (and its generalization in genus $g$, the Borchardt mean) yields an algorithm to compute any $P$ digits of the theta-constants in roughly $\mathrm{log} P$ multiplication of $P$-bit numbers, which is quasi-optimal. We provide a generalization of this connection using general tau-duplication formulas; with some care, this allows us to devise an algorithm to compute $P$ digits of Theta in the same quasi-linear time in $P$. We also report on an implementation in genus 1 and 2, which beats the naive algorithm for precisions as low as a few thousand digits. This is joint work with Emmanuel Thomé.