Lattices and Their Shadows

Noam D. Elkies (Harvard)

Let L be a unimodular integral lattice in R^n (a.k.a. positive-definite quadratic form over Z with discriminant 1. The "shadow" of L is L + (w/2) for w in L such that (v,w) = (v,v) mod 2 for all v in L. [Such w always exist and constitute a coset of 2L in L.] We show how the theory of theta series and modular forms yields both the classical fact that (w,w) = n mod 8 and the new result (suggested by recent work on 4-manifolds) that Z^n is the lattice with the longest shadow (i.e., the lattice for which min_w (w,w) attains its maximal value n).

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