Theta functions and weighted theta functions of Euclidean lattices, with some applications

By “Euclidean space” of dimension n we mean a real vector space of dimension n, equipped with a positive-definite inner product 〈·, ·〉. We usually call such a space “Rn” even when there is no distinguished choice of coordinates. A lattice in Rn is a discrete co-compact subgroup L ⊂ Rn, that is, a discrete subgroup such that the quotient Rn/L is compact (and thus necessarily homeomorphic with the n-torus (R/Z)n). As an abstract group L is thus isomorphic with the free abelian group Zn of rank n. Therefore L is determined by the images, call them v1, . . . , vn, of the standard generators of Zn under a group isomorphism Zn ∼ → L. We say the vi generate, or are generators of, L: each vector in L can be written as ∑n i=1 aivi for some unique integers a1, . . . , an. Vectors v1, . . . , vn ∈ Rn generate a lattice if and only if they constitute an R-linear basis for Rn, and then L is the Z-span of this basis. For instance, the Z-span of the standard orthonormal basis e1, . . . , en of Rn is the lattice Zn. This more concrete definition is better suited for explicit computation, but less canonical because most lattices have no canonical choice of generators even up to isometries of Rn.