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These commands calculate the standard invariants that characterize
a quadratic form over the rationals. Definitions of the invariants
may be found in Conway-Sloane [JC98], Chapter 15, Section 5.1.
pSignature(M,p) : Mtrx, RngIntElt -> RngIntElt
pSignature(L,p) : Lat, RngIntElt -> RngIntElt
The p-signature of the specified quadratic form over the rationals,
where p is a prime number or -1 (designating the real place).
For odd primes p, this is defined by diagonalizing the form,
and adding p-parts of these entries to 4 times the number
of anti-squares (mod p) amongst these entries.
The term "anti-square" modulo p denotes something that
has: odd valuation at p; and the prime-to-p part, called u,
has Kronecker symbol ((u/p))= - 1.
At p=2 it is the sum of the odd parts of the diagonalized
entries plus 4 times the number of anti-squares.
In either case, the final answer is really only defined modulo 8
(this is so that p-signatures are invariant under rational equivalence).
At the real place, it is the difference between the number of
positive and negative eigenvalues (the terminology here can be murky).
Oddity(L) : Lat -> RngIntElt
Oddity(M) : Mtrx -> RngIntElt
This returns the 2-signature of the given quadratic form over the rationals.
pExcess(M, p) : Mtrx, RngIntElt -> RngIntElt
pExcess(L, p) : Lat, RngIntElt -> RngIntElt
The p-excess of the specified quadratic form over the rationals,
where p is a prime number or -1 (designating the real place).
The p-excess is the difference between the p-signature
and dimension for odd primes (including -1),
and is the negation of this for p=2.
The sum of p-excesses over all primes should be 0 modulo 8.
WittInvariant(M, p) : AlgMatElt, RngIntElt -> RngIntElt
Calculates the Witt invariant over Qp of the given quadratic form.
Again the form must be defined
over either the rationals or the integers. The result is returned as
something in the set {-1, + 1}.
This is the class in the Brauer group of the Clifford algebra for even dimension,
of the even Clifford algebra for odd dimension, see Para 3 of Chapter V of Lam [Lam05].
WittInvariant(L, P) : Lat, RngInt -> RngIntElt
WittInvariant(L, P) : LatNF, RngOrdIdl -> RngIntElt
AmbientSpace: BoolElt Default: false
The Witt invariant of the quadratic form at p (or P).
If AmbientSpace is set to true, returns the Witt
invariant of the ambient quadratic space.
HasseInvariant(M, p) : AlgMatElt, RngIntElt -> RngIntElt
HasseMinkowskiInvariant(f, p) : RngMPolElt, RngIntElt -> RngIntElt
HasseMinkowskiInvariant(M, p) : AlgMatElt, RngIntElt -> RngIntElt
Calculates the Hasse (or Hasse-Minkowski) invariant over Qp of the given quadratic form.
Again the form must be defined
over either the rationals or the integers. The result is returned as
something in the set {-1, + 1}. One definition of this invariant is to
diagonalize the form and then take the product (in our multiplicative notation)
of the Hilbert symbols of the (n choose 2)
pairs of distinct nonzero diagonal entries,
as in Para 5.3 of Chapter 15 of Conway-Sloane [JC98],
which is what is implemented here. Another method would be to use
a comparison of p-excesses with the standard form (also in Conway-Sloane).
Starting with a p-adic input could lead to precision problems
at the diagonalization step, and so is not allowed.
Can also be called via HasseMinkowskiInvariant.
HasseMinkowskiInvariant(L, p) : Lat, RngIntElt -> RngIntElt
HasseInvariant(L, P) : Lat, RngInt -> RngIntElt
HasseInvariant(L, P) : LatNF, RngOrdIdl -> RngIntElt
HasseMinkowskiInvariant(L, P) : Lat, RngInt -> RngIntElt
AmbientSpace: BoolElt Default: false
The Hasse invariant of the quadratic form at p (or at P).
If AmbientSpace is set to true, returns the Hasse
invariant of the ambient quadratic space.
WittInvariants(M) : AlgMatElt -> SeqEnum
WittInvariants(L) : Lat -> SeqEnum
HasseInvariants(f) : RngMPolElt -> SeqEnum
HasseInvariants(M) : AlgMatElt -> SeqEnum
HasseInvariants(L) : Lat -> SeqEnum
HasseMinkowskiInvariants(f) : RngMPolElt -> SeqEnum
HasseMinkowskiInvariants(M) : AlgMatElt -> SeqEnum
HasseMinkowskiInvariants(L) : Lat -> SeqEnum
Minimize: BoolElt Default: false
AA: BoolElt Default: false
Compute WittInvariant(f,p) or HasseInvariant(f,p), repsectively,
for all bad primes p, and return the result of a sequence of tuples, each entry
given by < p, Wp(f) >.
The set of bad primes includes the real place, the prime p=2,
and all primes that divide either the numerator or the denominator
of the determinant or symmetric matrix associated to f.
If Minimize is set, the primes greater than 2 must have odd
valuation in the determinant to appear. If AA is set, the result
is return as an associative array.
Can also be called via HasseMinkowskiInvariants.
QuadraticFormInvariants(M) : AlgMatElt -> FldElt, SetEnum, SeqEnum[RngIntElt]
QuadraticFormInvariants(L) : LatNF -> FldElt, SetEnum, SeqEnum[RngIntElt]
Minimize: BoolElt Default: false
AA: BoolElt Default: false
The determinant of the diagonalization, the set of Hasse-Minkowski invariants,
and the number of negative eigenvalues (in each real place)
of the quadratic form M. If AA is set, the Hasse-Minkowski invariants are returned as an associative array.
If Minimize is set, the set of Hasse-Minkowski invariants is minimized, returning only the prime ideals
for which the Hasse-Minkowski invariant is -1.
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