We consider integral lattices $L$ in an Euclidean space $V = \mathbb{R}^m$, i.e. $\mathbb{Z}$-submodules of full rank in $V$ such that all vectors in $L$ have integral length. It is impossible to classify such lattices up to isometry, there are just too many of them in general, even if we ﬁx additional invariants such as the discriminant. Therefore one looks for interesting subclasses of lattices, in particular “extremal lattices”, characterized by the property that the smallest length of a non-zero vector in $L$ is “as large as possible”. There are several ways to make this more precise, we will focus on analytic extremality, where modular forms come in. In particular, we will consider extremality for maximal lattices.

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Last updated: 24 Jan 2020