Nonexistence of extremal lattices in certain genera of modular lattices (Q1126376)

The authors use an analytic (as opposed to the geometric) notion of extremality which goes back to Mallows, Odlyzko and Sloane in the selfdual case. Let \(L\) be an even lattice isometric to its \(N\)-scaled dual in \(2k\)-dimensional Euclidean space. If its theta series attains the maximal order to which a modular form in the relevant Fricke eigenspace takes on the value 1 at infinity, then \(L\) is called extremal. What this maximum for general \(k\) is seems to be clear only when \(N+1\) divides 24. For these seven levels there also exists a legion of more or less prominent extremal lattices, and one would like to know all dimensions \(n\) in which such lattices occur. These dimensions are known to be finite in number, and the first open case \(n=12\), \(N=11\) is decided here in the negative. To prove their result the authors make essential use of Siegel modular forms of degree 2.