Local Invariants

These commands calculate the standard invariants that characterize a quadratic form over the rationals. Definitions of the invariants may be found in Conway-Sloane [J.H. Conway, 1998], Chapter 15, Section 5.1.

pSignature(f, p): RngMPolElt, RngIntElt -> RngIntElt

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 \(({u\over p})=-1\).

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(f): RngMPolElt -> RngIntElt

Oddity(L): Lat -> RngIntElt

Oddity(M): Mtrx -> RngIntElt

This returns the \(2\)-signature of the given quadratic form over the rationals.

pExcess(f, p): RngMPolElt, RngIntElt -> RngIntElt

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(f, p): RngMPolElt, RngIntElt -> RngIntElt

WittInvariant(M, p): AlgMatElt, RngIntElt -> RngIntElt

Calculates the Witt invariant over \({\mathbb{Q}}_p\) 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 §3 of Chapter V of Lam [Lam, 2005].

WittInvariant(L, p): Lat, RngIntElt -> RngIntElt

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(f, p): RngMPolElt, RngIntElt -> RngIntElt

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 \({\mathbb{Q}}_p\) 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 §5.3 of Chapter 15 of Conway-Sloane [J.H. Conway, 1998], 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.

HasseInvariant(L, p): Lat, RngIntElt -> RngIntElt

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(f): RngMPolElt -> SeqEnum

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 \(\langle p,W_p(f)\rangle\). 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(f): RngMPolElt -> FldElt, SetEnum, SeqEnum[RngIntElt]

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 \(-1\).