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

```


Mass of spinor genus,  positive integral quadratic forms
Better late than never:


In the adelic setup, the mass or measure of a lattice $L$ (the german word Maß translates to measure in English) is given as 
$\mu(O(V)\backslash O(V)O\_{\mathbb A}(L))$, where $\mu$ is the (Tamagawa-normalized) Haar measure on the adelic orthogonal group $O\_{\mathbb A}(V)$ of the underlying quadratic space $V$ and $O\_{\mathbb A}(L)$ is the adelic isometry group of $L$; it is proportional to $\frac{1}{\vert O(L)\vert}$ in the definite case (here $O(L)$ is the global group of isometries of $L$, which is finite in the definite case).


If we sum that up over a set of representatives of the classes in the spinor genus of $L$ we obtain $\mu(O(V)\backslash O(V)O'\_{\mathbb A}(V)O\_{\mathbb A}(L))$, where $O'\_{\mathbb A}(V)$ is the group of adelic orthogonal transformations of $V$ which have spinor norm $1$ at each place.


If we replace here the lattice $L$ by another lattice $\sigma L$ in the genus of $L$ (with $\sigma \in O\_{\mathbb A}(V)$), we have


$$O(V)O'\_{\mathbb A}(V)O\_{\mathbb A}(\sigma L)=O(V)O'\_{\mathbb A}(V)\sigma O\_{\mathbb A}(L)\sigma^{-1}=O(V)O'\_{\mathbb A}(V)O\_{\mathbb A}(L),$$
since $O'\_{\mathbb A}(V)$ contains all commutators.


So, the measures for the spinor genus of $L$ and for that of $\sigma L$ are the same.


The argument is implicit in Kneser's 1961 article and also in Weil's 1962 article, but unfortunately neither of these authors bothered to write down the statement explicitly. 


