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In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field. In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields, and in 3 dimensions some partial results were given by Ferdinand Eisenstein. The mass formula in higher dimensions was first given by , though his results were forgotten for many years. It was rediscovered by , and an error in Minkowski's paper was found and corrected by . Many published versions of the mass formula have errors; in particular the 2-adic densities are difficult to get right, and it is sometimes forgotten that the trivial cases of dimensions 0 and 1 are different from the cases of dimension at least 2. give an expository account and precise statement of the mass formula for integral quadratic forms, which is reliable because they check it on a large number of explicit cases. For recent proofs of the mass formula see and . The Smith–Minkowski–Siegel mass formula is essentially the constant term of the Weil–Siegel formula. ==Statement of the mass formula== If ''f'' is an ''n''-dimensional positive definite integral quadratic form (or lattice) then the mass of its genus is defined to be : where the sum is over all integrally inequivalent forms in the same genus as ''f'', and Aut(Λ) is the automorphism group of Λ. The form of the mass formula given by states that for ''n'' ≥ 2 the mass is given by : for sufficiently large ''r'', where ''p''''s'' is the highest power of ''p'' dividing the determinant of ''f''. The number ''N''(''p''''r'') is the number of ''n'' by ''n'' matrices ''X'' with coefficients that are integers mod ''p'' ''r'' such that : where ''A'' is the Gram matrix of ''f'', or in other words the order of the automorphism group of the form reduced mod ''p'' ''r''. Some authors state the mass formula in terms of the ''p''-adic density : instead of the ''p''-mass. The ''p''-mass is invariant under rescaling ''f'' but the ''p''-density is not. In the (trivial) cases of dimension 0 or 1 the mass formula needs some modifications. The factor of 2 in front represents the Tamagawa number of the special orthogonal group, which is only 1 in dimensions 0 and 1. Also the factor of 2 in front of ''m''''p''(''f'') represents the index of the special orthogonal group in the orthogonal group, which is only 1 in 0 dimensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Smith–Minkowski–Siegel mass formula」の詳細全文を読む スポンサード リンク
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