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MIT Applied Math Colloquium, May 1, 4:15 p.m., Rm 2-105

Professor Noam D.\ Elkies (Harvard)

Lattices and their shadows

Let $L$ be a unimodular integral lattice in $R^n$
(a.k.a. positive-definite
quadratic form over $\Bbb Z$ with discriminant 1).  
The ``shadow" of $L$ is $L + {w\over 2}$  for $w\in L$  
such that  $(v,w) \equiv (v,v) {\rm \ mod\,} 2$  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  $(w,w) \equiv n {\rm \ mod\,} 8$  
and the new result (suggested by recent work on 4-manifolds) 
that  ${\Bbb Z}^n$  is the lattice with the longest shadow 
(i.e., the lattice for which  $\displaystyle\min_w\,(w,w)$  attains
its maximal value $n$).

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