Department of Mathematics
Colloquium
by
Prof. Noam Elkies

A characteristic vector of a unimodular lattice L is a lattice vector v such that (v,w) = (w,w) mod 2 for all w in L. These vectors constitute a coset of 2L in L. The "shadow" of L is its translate L+(v/2) for v in the characteristic coset. For instance if L=Z^n then the shadow consists of all vectors with coordinates in Z + 1/2. Note that in this case L is positive definite of rank n and its shadow has minimal norm n/4. It turns out that these three properties characterize L. In this talk we give the background for the interest in this result, some of which comes from 4-dimensional differential topology a la Donaldson. We also outline a proof using modular forms, and consider related problems.

Additional references:

(1) Chapters 1 and 16 in SPLAG, i.e. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. New York: Springer 1993. (General stuff about lattices)

(2) Serre, A Course in Arithmetic, Chapter V (About unimodular lattices), and Chapter VII (About modular forms. Note especially the concluding remarks, i.e. paragraph 6.7, "Complements")

(3) Elkies, Math. Research Letters 2 (1995): "A Characterization of the Z^n Lattice" (321-326) and "Lattices and Codes with Long Shadows" (643-651).