  We consider the minimization of theta functions
$\theta\_\Lambda(\alpha)=\sum\_{p\in\Lambda}e^{-\pi\alpha|p|^2}$ amongst
lattices $\Lambda\subset \mathbb R^d$, by reducing the dimension of the
problem, following as a motivation the case $d=3$, where minimizers are
supposed to be either the BCC or the FCC lattices. A first way to reduce
dimension is by considering layered lattices, and minimize either among
competitors presenting different sequences of repetitions of the layers, or
among competitors presenting different shifts of the layers with respect to
each other. The second case presents the problem of minimizing theta functions
also on translated lattices, namely minimizing $(L,u)\mapsto
\theta\_{L+u}(\alpha)$. Another way to reduce dimension is by considering
lattices with a product structure or by successively minimizing over concentric
layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics
questions, which we study in detail in two dimensions.
