Niemeier lattices and theta functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:51:47Z http://mathoverflow.net/feeds/question/62070 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62070/niemeier-lattices-and-theta-functions Niemeier lattices and theta functions flor.ian sprung 2011-04-18T01:08:05Z 2011-04-18T21:31:34Z <p>I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice gave you this theta function in the first place?</p> http://mathoverflow.net/questions/62070/niemeier-lattices-and-theta-functions/62071#62071 Answer by Henry Cohn for Niemeier lattices and theta functions Henry Cohn 2011-04-18T01:25:51Z 2011-04-18T01:25:51Z <p>The theta series for a Niemeier lattice determines the lattice in most cases, but there are five ambiguous pairs.</p> <p>The theta series of an even unimodular lattice must be a polynomial in the theta series of \$E_8\$ and \$\Lambda_{24}\$ (this is a modular forms calculation). Thus, for Niemeier lattices, it must be a linear combination of those for \$E_8^3\$ and \$\Lambda_{24}\$. The constant term must be \$1\$, so there is one remaining degree of freedom. This means the theta series for a Niemeier lattice is determined by how many roots (i.e., vectors of norm \$2\$) it has. There are five pairs of Niemeier lattices with the same number of roots, but no triples.</p>