Mass of spinor genus, positive integral quadratic forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:33:40Z http://mathoverflow.net/feeds/question/54027 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54027/mass-of-spinor-genus-positive-integral-quadratic-forms Mass of spinor genus, positive integral quadratic forms Will Jagy 2011-02-01T21:40:53Z 2011-02-02T02:54:06Z <p>There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some number, that is different, indeed some recent authors write of representation mass of numbers and it throws me off). Authors R. Scharlau and R. Schulze-Pillot have been loosely mentioned in this regard.</p> <p>Does anyone know for sure, and in particular know a simple (published) reference for positive forms? Evidently there is an analogous formulation where it is true for indefinite forms, see The Hirzebruch-Mumford volume for the orthogonal group and applications, by Valery Gritsenko, Klaus Hulek, G.K. Sankaran,</p> <p><a href="http://arxiv.org/abs/math/0512595" rel="nofollow">http://arxiv.org/abs/math/0512595</a></p> <p>I should add that Kneser (1961) is sometimes mentioned in this regard, but my take is that, while his methods can be used to reconstruct the result, he is not explicit about the mass.</p> <p>Here is computer output on an example that appears in Benham, Earnest, Hsia, Hung (1990), formula (3.8), positive ternary forms, including one of the 29 known spinor regular forms that are not regular. The sextuple a b c d e f refers to the form \$T(x,y,z) = a x^2 + b y^2 + c z^2 + d y z + e z x + f x y,\$ with discriminant \$\Delta = 4 a b c + d e f - a d^2 - b e^2 - c f^2.\$ </p> <pre><code>=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ===Discriminant 343 ==Genus Size== 3 343 = 7^3 All forms in regular spinor genus represent 1 ----------------------------- Spinor genus misses 1 4 16 64 121 256 484 529 841 343: 2 7 8 7 1 0 auto 4 Level 196 irreg spin candidate --------------------------size 1 Spinor genus misses no exceptions 343: 1 2 49 0 0 1 auto 8 Level 196 343: 1 7 14 7 0 0 auto 8 Level 196 --------------------------size 2 Disc 343 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= </code></pre>