Smith–Minkowski–Siegel mass formula
Encyclopedia
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 formula
s 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 .
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
where mp(f) is the p-mass of f, given by
for sufficiently large r, where ps is the highest power of p dividing the determinant of f. The number N(pr) 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 mp(f) represents the index of the special orthogonal group in the orthogonal group, which is only 1 in 0 dimensions.
(for n = dim(ƒ) even)
(for n = dim(ƒ) odd)
where the Legendre symbol in the second line is interpreted as 0 if p divides 2 det(ƒ).
If all the p-masses have their standard value, then the total mass is the
standard mass
(For n odd)
(For n even)
where
The values of the Riemann zeta function for an even integers s are given in terms of Bernoulli number
s by
So the mass of ƒ is given as a finite product of rational numbers as
where q runs through powers of p and fq has determinant prime to p and dimension n(q),
then the p-mass is given by
Here n(II) is the sum of the dimensions of all Jordan consituens of type 2 and p = 2, and n(I,I) is the totla number of pairs of afjacent consituents fq, f2q that are both of type I.
The factor Mp(fq) is called a diagonal factor and is a power of p times the order of a certain orthogonal group over the field with p elements.
For odd p its value is given by
when n is odd, or
when n is even and (−1)n/2dq is a quadratic residue. or
when n is even and (−1)n/2dq is a quadratic nonresidue.
For p = 2 the diagonal factor Mp(fq) is notoriously tricky to calculate. (The notation is misleading as it depends not only on fq but also on f2q and fq/2.)
Then the diagonal factor Mp(fq) is given as follows.
when the form is bound or has octane value +2 or −2 mod 8 or
when the form is free and has octane value −1 or 0 or 1 mod 8 or
when the form is free and has octane value −3 or 3 or 4 mod 8.
with χ(m) given by 0 if m is even, and the Jacobi symbol
is m is odd. We write k for the modulus of this character and k1 for its conductor, and put χ = χ1ψ where χ1 is the principal character mod k and ψ is a primitive character mod k1. Then
The functional equation for the L-series is
where G is the Gauss sum
If s is a positive integer then
where Bs(x) is a Bernoulli polynomial.
s Λ of dimension n > 0 divisible by 8 the mass formula is
where Bk is a Bernoulli number
.
There is exactly one even unimodular lattice of dimension 8, the E8 lattice
, whose automorphism group is the Weyl group of E8 of order 696729600, so this verifies the mass formula in this case.
Smith originally gave a nonconstructive proof of the existence of an even unimodular lattice of dimension 8 using the fact that the mass is non-zero.
There are two even unimodular lattices of dimension 16, one with root system E82
and automorphism group of order 2×6967296002 = 970864271032320000, and one with root system D16 and automorphism group of order 21516! = 685597979049984000.
So the mass formula is
s. The mass formula for them is checked in .
unimodular lattices of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass. By refining this argument, showed that there are more than a billion such lattices. In higher dimensions the mass, and hence the number of lattices, increases very rapidly.
Tamagawa showed that the mass formula was equivalent to the statement that the Tamagawa number of
the orthogonal group is 2, which is equivalent to saying that the Tamagawa number of its simply connected cover the spin group is 1. André Weil
conjectured more generally that the Tamagawa number of any simply connected semisimple group is 1
, and this conjecture was proved by Kottwitz in 1988.
gave a mass formula for unimodular lattice
s without roots (or with given root system).
In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formula
Class number formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function-General statement of the class number formula:...
s for imaginary quadratic fields, and in 3 dimensions some partial results were given by Ferdinand Eisenstein
Ferdinand Eisenstein
Ferdinand Gotthold Max Eisenstein was a German mathematician. He specialized in number theory and analysis, and proved several results that eluded even Gauss. Like Galois and Abel before him, Eisenstein died before the age of 30. He was born and died in Berlin, Germany.-Early life:He was born...
.
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 .
Statement of the mass formula
If f is an n-dimensional positive definite integral quadratic form (or lattice) then the massof 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
where mp(f) is the p-mass of f, given by
for sufficiently large r, where ps is the highest power of p dividing the determinant of f. The number N(pr) 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 mp(f) represents the index of the special orthogonal group in the orthogonal group, which is only 1 in 0 dimensions.
Evaluation of the mass
The mass formula gives the mass as an infinite product over all primes. This can be rewritten as a finite product as follows. For all but a finite number of primes (those not dividing 2 det(ƒ)) the p-mass mp(ƒ) is equal to the standard p-mass stdp(ƒ), given by (for n = dim(ƒ) even) (for n = dim(ƒ) odd)where the Legendre symbol in the second line is interpreted as 0 if p divides 2 det(ƒ).
If all the p-masses have their standard value, then the total mass is the
standard mass
(For n odd) (For n even)where
- D = (−1)n/2 det(ƒ)
The values of the Riemann zeta function for an even integers s are given in terms of Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s by
So the mass of ƒ is given as a finite product of rational numbers as
Evaluation of the p-mass
If the form f has a p-adic Jordan decompositionwhere q runs through powers of p and fq has determinant prime to p and dimension n(q),
then the p-mass is given by
Here n(II) is the sum of the dimensions of all Jordan consituens of type 2 and p = 2, and n(I,I) is the totla number of pairs of afjacent consituents fq, f2q that are both of type I.
The factor Mp(fq) is called a diagonal factor and is a power of p times the order of a certain orthogonal group over the field with p elements.
For odd p its value is given by
when n is odd, or
when n is even and (−1)n/2dq is a quadratic residue. or
when n is even and (−1)n/2dq is a quadratic nonresidue.
For p = 2 the diagonal factor Mp(fq) is notoriously tricky to calculate. (The notation is misleading as it depends not only on fq but also on f2q and fq/2.)
- We say that fq is odd if it represents an odd 2-adic integer, and even otherwise.
- The octane value of fq is an integer mod 8; if fq is even its octane value is 0 if the determinant is +1 or −1 mod 8, and is 4 if the determinant is +3 or −3 mod 8, while if fq is odd it can be diagonalized and its octane value is then the number of diagonal entries that are 1 mod 4 minus the number that are 3 mod 4.
- We say that fq is bound if at least one of f2q and fq/2 is odd, and say it is free otherwise.
- The integer t is defined so that the dimension of for fq is 2t if fq is even, and 2t + 1 or 2t + 2 if fq is odd.
Then the diagonal factor Mp(fq) is given as follows.
when the form is bound or has octane value +2 or −2 mod 8 or
when the form is free and has octane value −1 or 0 or 1 mod 8 or
when the form is free and has octane value −3 or 3 or 4 mod 8.
Evaluation of ζD(s)
The required values of the Dirichlet series ζD(s) can be evaluated as follows. We write χ for the Dirichlet characterDirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
with χ(m) given by 0 if m is even, and the Jacobi symbol
Jacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
is m is odd. We write k for the modulus of this character and k1 for its conductor, and put χ = χ1ψ where χ1 is the principal character mod k and ψ is a primitive character mod k1. ThenThe functional equation for the L-series is
where G is the Gauss sum
Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...
If s is a positive integer then
where Bs(x) is a Bernoulli polynomial.
Examples
For the case of even unimodular latticeUnimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
s Λ of dimension n > 0 divisible by 8 the mass formula is
where Bk is a Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
.
Dimension n = 0
The formula above fails for n = 0, and in general the mass formula needs to be modified in the trivial cases when the dimension is at most 1. For n = 0 there is just one lattice, the zero lattice, of weight 1, so the total mass is 1.Dimension n = 8
The mass formula gives the total mass asThere is exactly one even unimodular lattice of dimension 8, the E8 lattice
E8 lattice
In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...
, whose automorphism group is the Weyl group of E8 of order 696729600, so this verifies the mass formula in this case.
Smith originally gave a nonconstructive proof of the existence of an even unimodular lattice of dimension 8 using the fact that the mass is non-zero.
Dimension n = 16
The mass formula gives the total mass asThere are two even unimodular lattices of dimension 16, one with root system E82
and automorphism group of order 2×6967296002 = 970864271032320000, and one with root system D16 and automorphism group of order 21516! = 685597979049984000.
So the mass formula is
Dimension n = 24
There are 24 even unimodular lattices of dimension 24, called the Niemeier latticeNiemeier lattice
In mathematics, a Niemeier lattice is one of the 24positive definite even unimodular lattices of rank 24,which were classified by . gave a simplified proof of the classification. has a sentence mentioning that he found more than 10 such lattices, but gives no further details...
s. The mass formula for them is checked in .
Dimension n = 32
The mass in this case is large, more than 40 million. This implies that there are more than 80 million evenunimodular lattices of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass. By refining this argument, showed that there are more than a billion such lattices. In higher dimensions the mass, and hence the number of lattices, increases very rapidly.
Generalizations
Siegel gave a more general formula that counts the weighted number of representations of one quadratic form by forms in some genus; the Smith–Minkowski–Siegel mass formula is the special case when one form is the zero form.Tamagawa showed that the mass formula was equivalent to the statement that the Tamagawa number of
the orthogonal group is 2, which is equivalent to saying that the Tamagawa number of its simply connected cover the spin group is 1. André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
conjectured more generally that the Tamagawa number of any simply connected semisimple group is 1
Weil conjecture on Tamagawa numbers
In mathematics, the Weil conjecture on Tamagawa numbers is a result about algebraic groups formulated by André Weil in the late 1950s and proved in 1989...
, and this conjecture was proved by Kottwitz in 1988.
gave a mass formula for unimodular lattice
Unimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
s without roots (or with given root system).