Similar sublattices of integral lattices

par Marc Zimmerman (University of Cologne)

jeudi 26 sept. 2024, 16:00 → 17:00 Europe/Paris

Salle Pellos (1R2)

A sublattice $M$ of a lattice $L$ is called a similar sublattice of norm $c$ of $L$ if it can be obtained from $L$ by composing an isometry of space and a dilation by factor $c$. So it is a sublattice of $L$ which essentially is a scaled copy of $L$.

In this talk I will discuss how we can decide for lattices with nice properties (i.e. being integral, maximal, unimodular, and so on) whether they allow for such sublattices and of which norms.

I will put my main focus on an approach which is joint work with Rudolf Scharlau: We use the arithmetic theory of integral lattices to relate similar sublattices of maximal integral lattices and maximal totally isotropic submodules of regular quadratic modules over the residue class rings $\mathbb{Z}/c\mathbb{Z}$.

Using this approach we can count and construct similar sublattices of suitably nice lattices, the most important example being the root lattice $E_8$. For this particularly nice Iattice I will show how we end up writing down a zeta-function for the number of its similar sublattices.