|
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
To compute the L-function, one can use
the intrinsic LFunction, or alternatively one can do it "by hand",
using the tools in this section and the next section
(as illustrated in Example H114E6).
LFunction(E,s) : CrvEll[FldFunRat], <SeqEnum> -> RngUPolElt
Check: BoolElt Default: true
The L-function of the elliptic curve E, which should be
defined over a rational function field over a finite field.
To speed up the computation, the points in s are used to
reduce the unknown part of H2;
s should be a sequence of points on E, or on some base change of E
by an extension of the constant field.
If the parameter Check is false, then
the points in s must form a basis for a subgroup of the geometric
Mordell-Weil group modulo torsion that is mapped to itself by the
Galois group of the extension of the fields of constants.
The L-function for the base change of the elliptic curve E by a
constant field extension of degree e.
The Mordell--Weil rank of E as predicted by Tate's version of the
Birch--Swinnerton-Dyer conjecture (or equivalently, the Artin-Tate
conjecture). This is an unconditional
upperbound, and
is proved to be the actual rank for several classes of curves, such as
those arising from rational elliptic surfaces, and K3 elliptic
surfaces. It is required that E is defined over a function field
over a finite field.
AnalyticInformation(E,L) : CrvEll[FldFunG], RngUPolElt -> Tup
AnalyticInformation(E,e) : CrvEll[FldFunG], RngIntElt -> Tup
This function computes what is predicted by the Birch--Swinnerton-Dyer
(or Artin--Tate) conjecture
about E, which should be an elliptic curve defined over a function
field over a finite field.
The data is given as a tuple consisting of the rank, geometric rank,
and the product of the order of Tate--Shafarevich group and the regulator.
When a polynomial L is given as a second argument, Magma will
assume that L is the L-function of E.
When an integer e is given as a second argument, the program will
return analytic data for the base change of E by the constant field
extension of degree e. This is more efficient than simply calling
AnalyticInformation on the base change.
The curve y2 = x3 + 2t2x + (t3 + 2t2 + 2t) over F3(t)
has rank 0 and its Tate-Shafarevich group contains Z/3 direct-sum Z/3.
We may conclude this (unconditionally) from the analytic information,
since the Tate conjecture is proved for rational elliptic surfaces.
> F<t> := FunctionField(GF(3));
> E := EllipticCurve([ 2*t^2, t^3 + 2*t^2 + 2*t ]);
> AnalyticInformation(E);
<0, 4, 9>
Furthermore, the rank of E(/line(F3)(t)) is 4.
We consider again the curve y2 = x3 + (t4 + 2t2)x + t3 + 2t over F3(t),
which was found to have rank 2 and regulator 12 in Example H114E1.
> F<t> := FunctionField(GF(3));
> E := EllipticCurve([ t^4 + 2*t^2, t^3 + 2*t ]);
> AnalyticInformation(E);
<2, 8, 12>
This confirms that the rank is 2, and tells us that the regulator multiplied by
the order of Sha is 12, in other words the Tate-Shafarevich group is trivial.
Again this is unconditional.
Number of points on the Kodaira-Neron model of the elliptic curve E over
an extension of the constant field of degree e.
If an elliptic curve E is defined over a function field over a finite field
of q elements, then this intrinsic computes a sequence containing the number
of points on the corresponding Kodaira-N'eron model of E over extensions
of GF(q) of degree up to e, and the number of points on the base curve over
these extensions.
The i-th Betti number of the Kodaira-N'eron model of the elliptic
surface corresponding to the elliptic curve E.
Given a sequence traces=[t1, t2, ..., td], this computes a polynomial
of degree d with constant coefficient equal to 1, of which the
j-th powersum of the reciprocal zeroes is equal to tj. This is
typically used when one wants to compute the characteristic polynomial
of Frobenius acting on some piece of cohomology from traces of various
powers of Frobenius.
Given a sequence of field elements [t1, t2, ..., tk],
and integers d,
q and i, this computes a polynomial of degree d with constant
coefficient 1, for which the reciprocals of zeroes have absolute
value qi/2, and the j-th powersum of the reciprocals of zeroes equals
tj. For this function one must have k at least equal to
Floor(d/2). A priori the leading coefficient of the polynomials is
not be determined from the list of traces; in this case two polynomials are returned,
one for each choice of sign.
One usually needs to provide Ceiling(d/2)
traces in order to determine the sign, but in certain cases more
traces are required.
This function is typically used when one wants
to compute the characteristic polynomial of Frobenius acting on a
piece of the i-th cohomology group from traces of various powers of
Frobenius.
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
