|
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
The Hasse--Minkowski principle for quadratic forms implies that a
conic has a point over a number field if and only if it has a point
over its completion at every finite and infinite prime.
The provides an effective set of conditions for determining whether
a conic has a point: only the finite number of primes dividing the
discriminant of the curve need to be checked, and by Hensel's Lemma,
it is only necessary to check this condition to finite precision.
The theory holds over any global field (a number field or the
function field of a curve over a finite field), but the algorithms
implemented at present only treat the field Q.
Subsections
We say that p is a ramified or bad prime for a conic C
if there exists no p-integral model for C with nonsingular
reduction. Every such prime is a divisor of the coefficients of
the Lengendre polynomial. However in general it is not possible
to have a Legendre model whose coefficients are divisible only
by the ramified primes. We use the term ramified or bad prime
for a conic to refer to the local properties of C which are
independent of the models.
Given a conic C over the rationals, returns the sequence of
the finite ramified primes of C, at which C has intrinsic
locally singular reduction. This uses quaternion algebras.
N.B. Although the infinite prime is not included, the data of
whether or not C/Q is ramified at infinity is carried by the
length of this sequence. The number of bad primes, including
infinity, must be even, so the parity of the sequence length
specifies the ramification information at infinity.
Hilbert's norm residue symbol logically pertains to the theory
of quadratic forms and lattices, and to that of quadratic fields
and their norm equations. We express it here for its relevance
to determining conditions for existence of local points on conics
over Q. The norm residue symbol gives a precise condition
for a quadratic form to represent zero over Qp, or more
generally over any local field, which is equivalent to the condition
that a conic has a point.
An explicit treatment of the properties and theory of the norm
residue symbol can be found in Cassels [Cas78]; the
Hilbert symbol is treated by Lam [Lam73].
NormResidueSymbol(a,b,p) : RngIntElt, RngIntElt, RngIntElt -> RngIntElt
Given two rational numbers or integers a and b, and a prime p,
returns 1 if the quadratic form ax2 + by2 - z2 represents zero
over Qp, and returns -1 otherwise.
HilbertSymbol(a, b, p) : FldNumElt, FldNumElt, RngOrdIdl -> RngIntElt
Al: MonStgElt Default: "NormResidueSymbol"
Computes the Hilbert symbol (a, b)p,
where a, b are elements of a number field, and p is either a prime number
(if a, b ∈Q) or a prime ideal.
For a, b ∈Q, by default Magma uses table-lookup to compute the Hilbert symbol
(or the NormResidueSymbol); one may
insist on using the full algorithm by setting the optional argument Al to Evaluate.
In the following example we show how the norm residue symbols can
be used to determine the bad primes of a conic.
> P2<x,y,z> := ProjectiveSpace(RationalField(),2);
> a := 234; b := -33709; c := 127;
> C := Conic(P2,a*x^2 + b*y^2 + c*z^2);
> HasRationalPoint(C);
false
> fac := Factorization(Integers()!Discriminant(C));
> fac;
[ <2, 3>, <3, 2>, <13, 2>, <127, 1>, <2593, 1> ]
So we only need to test the primes 2, 3, 13, 127 and 2593.
By scaling the defining polynomial of the curve ax2 + by2 + cz2 = 0
by -1/c, we obtain the quadratic form -a/c x2 - b/c y2 - z2,
which means we want to check the Hilbert symbols for the coefficient
pair ( - a/c, - b/c).
> [ NormResidueSymbol(-a/c,-b/c,p[1]) : p in fac ];
[ -1, 1, 1, -1, 1 ];
The norm residue symbol indicates that only 2 and 127 have
no local p-adic solutions, which confirms the bad primes as
reported by Magma.
> BadPrimes(C);
[ 2, 127 ]
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
