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).
Click here for a TeX version of the above