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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.

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,

http://arxiv.org/abs/math/0512595

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.

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 $\Delta = 4 a b c + d e f - a d^2 - b e^2 - c f^2.$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

   ===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
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

edited Feb 2 2011 at 2:54

asked Feb 1 2011 at 21:40

Will Jagy
12.5k●1●26●45

	What do you mean when you say "mass"? Could you give us a definition of the mass? Thanks. – emiliocba Dec 28 2011 at 18:14
	Yes, any positive form has a finite number of integral automorphs, those being integral matrices of determinant $\pm 1$ that preserve the quadratic form, so the collection is also called the orthogonal group of the form. For a genus, or, here, a spinor genus, the mass is the sum of the reciprocals of the automorph counts over all classes in the genus (spinor genus). In the example above, the mass of the genus is 1/4 + 1/8 + 1/8 = 1/2. See en.wikipedia.org/wiki/… – Will Jagy Dec 28 2011 at 20:09
	Isn't the mass a genus invariant? – emiliocba Jan 3 2012 at 13:37
	@emiliocba yes. In the same way, Siegel's weighted representation measure of a number can be calculated from purely local information. The mass is sometimes referred to as the measure of representing the 0 form. Meanwhile, for positive forms it is just a finite sum, and one may ask for the value of that finite sum on a subset of the genus, in particular one of the spinor genera when there are more than one of those. – Will Jagy Jan 3 2012 at 21:15

Mass of spinor genus, positive integral quadratic forms

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