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The functions in this section are for elliptic curves defined over
number fields, including Q. For curves over Q,
they provide complementary method to those
described already (with not much overlap).
In particular, the algorithms related to
ranks and Mordell--Weil groups use the "algebraic"
method of first determining some Selmer group,
in contrast to the search for homogeneous
spaces carried out in MordellWeilGroup or Rank over Q.
The most important pieces of machinery implemented here are 2-descent
and descent by 2-isogenies, and standard height machinery.
Subsections
Given an elliptic curve E defined over a number field K, returns
the places of K of bad reduction for E. i.e., the places dividing
the discriminant of E.
Given an elliptic curve E defined over a number field K and a
number field L, such that K is a subfield of L, returns the
places of L of bad reduction for E.
The conductor of the elliptic curve E defined over a number field.
UseGeneratorAsUniformiser: BoolElt Default: false
Implements Tate's algorithm for the elliptic curve E over a number field.
This intrinsic computes
local reduction data at the prime ideal P, and a local minimal model.
The model is not required to be integral on input.
Output is < P, vp(d), fp, cp, K, s > and Emin
where P is the prime ideal, vp(d) is the valuation of the
local minimal discriminant, fp is the valuation of the conductor,
cp is the Tamagawa number, K is the Kodaira Symbol, and s is false
if the curve has non-split multiplicative reduction at P and true otherwise.
Emin is a model of E (integral and) minimal at P.
When the optional parameter UseGeneratorAsUniformiser is set true,
the computation checks whether P is principal,
and if so, uses generator of P as the uniformiser.
This means that at primes other that P,
the returned model will still be integral or minimal
if the given curve was.
Return a sequence of the tuples returned by LocalInformation(E, P)
for all prime ideals P in the decomposition of the discriminant of E
in the maximal order of the number field the elliptic curve E is over.
Given an elliptic curve E over a number field given by a model which
is integral
at p and has good reduction at p, returns an elliptic curve over the
residue field of p which represents the reduction of E at p. The
reduction map is also returned.
The local root number of the elliptic curve E
(defined over a number field) at the prime ideal P.
Calculates the global root number of an elliptic curve E
defined over a number field K.
This is the product of local root numbers over all places of K
(-1 from each infinite place), and is the (conjectural) sign
in the functional equation relating L(E/K,s) to L(E/K,2-s).
Given an elliptic curve E over a number field,
the function determines whether the curve has complex multiplication and,
if so, also returns the discriminant of the CM quadratic order.
The algorithm uses
fairly straightforward analytic methods, which are not suited to very high
degree j-invariants with CM by orders with discriminants more than
a few thousand.
Given an elliptic curve E defined over a number field, returns a bound
on the size of the torsion subgroup of E by looking at the first n
non-inert primes of sufficiently low ramification degree and
good reduction.
Bound: RngIntElt Default: -1
Given an elliptic curve E defined over a number field or Q, returns
the p-power torsion of E over its base field as an abelian group,
together with the map from the group into the curve.
One can specify a bound on the size of the p-power
torsion, which may help the computation.
Given an elliptic curve E defined over a number field, returns the
torsion subgroup of E.
Given a point P on an elliptic curve over a number field K, the function
returns the naive height of P; i.e., the absolute logarithmic height
of the x-coordinate.
Precision: RngIntElt Default: 27
Extra: RngIntElt Default: 8
Given a point P on an elliptic curve defined over a number field K,
returns the N{'e}ron-Tate height of P .
(This is based on the absolute logarithmic height of the x-coordinate.)
The parameter Precision may be used to specify the desired
precision of the output.
The parameter Extra is used in the cases when one of the points
2n P gets too close to the point at infinity, in which case
there is a huge loss in precision in the archimedean height and
increasing Extra remedies that problem.
Compute the height pairing matrix for an array of points.
Same varargs as above.
Precision: RngIntElt Default: 0
Extra: RngIntElt Default: 8
Given a point P on an elliptic curve defined over a number
field K, and a place Pl (finite or infinite) of K,
returns the local height λPl(P) of P
at (Pl). The parameter Precision sets the precision of the
output. A value of 0 takes the default precision.
When Pl is an infinite place, the parameter Extra can be used
to remedy the huge loss in precision when one of the points 2n P
gets too close to the point at infinity.
First we give a short overview of the theory of Selmer groups. This enables us
to fix the notation that is used in the naming of the Magma functions. For a
more complete account, see [Sil86].
The actual algorithms to compute the Selmer groups
are closer to the description in [Cas66].
Let E', E be elliptic curves over a number field K and let
φ:E' to E be an isogeny. The only cases currently implemented are where
φis a 2-isogeny, i.e., an isogeny of degree 2, and where E'=E and
φis multiplication by 2.
Let E'[φ] be the kernel subscheme of φ.
By taking Galois cohomology of the short exact sequence of
schemes over K,
0to E'[φ]to E'to E to 0,
we obtain
E'(K)to E(K)to H1(K, E'[φ]) to H1(K, E').
For the middle map, we write μ:E(K)to H1(K, E'[φ]).
Thus, E(K)/φ(E'(K)) injects in H1(K, E'[φ]) and the image
consists exactly of
the cocycles that vanish in H1(K, E').
As an approximation to this image,
we define the φ-Selmer group of E over
K to consist of those cocycles H1(K, E'[φ]) that vanish in all
restrictions H1(Kp, E'), where p runs though all places of K.
It fits in the exact sequence
0to S(φ)(E/K)to H1(K, E'[φ])to ∏p H1(Kp, E')
The main application of Selmer groups is that they provide the bound:
#E(K)/φE'(K)≤# S(φ)(E/K). If φ^ * :E to E' is the
isogeny dual to φ, then φ φ^ * :E to E is multiplication by
d=( Deg)(φ). Thus, one can use the φand φ^ * Selmer groups to
provide a bound on #E(K)/dE(K) and thus on the Mordell--Weil rank of E(K).
Representation of H1(K, E'[φ]):
If φis a 2-isogeny, then H1(K, E'[φ])~K^ * /K * 2. Thus, we can
represent elements by δ∈K^ *. In Magma, the map μcorresponds
to a representation of the map E(K)to K^ *. The map μalso accepts just
x-coordinates.
If φis multiplication-by-2 and E : y2=f(x), we write
A=K[x]/(f(x)) then
H1(K, E[2])~{δ∈A^ * /A * 2| NA/K(δ)∈K * 2}.
In fact, the full set A^ * /A * 2 corresponds to H1(K, E[2] x {+- 1}).
The set H1(K, E'[φ]) also corresponds to the set of
covers τδ:Tδto
E over K, modulo isomorphy over K, that are isomorphic to φ:E'to E
over the algebraic closure of K (see [Sil86], Theorem X.2.2).
In this section, we write tor for the map
δ|-> τδ.
CasselsMap(phi) : Map -> Map, Map
Fields: SetEnum Default: { }
Given an isogeny φ: E to E1 of elliptic curves over a number field K
(or Q), the function returns the connecting homomorphism
μ: E1(K) to H1(K, E[φ]),
and a map τsending an element of H1(K, E[φ]) to the corresponding homogeneous space.
Here elements of H1(K, E[φ]) are represented as elements of A^ * /(A^ * )2
(as described above).
The maps are actually given as maps to,
and from, A (rather than A^ * /(A^ * )2).
The isogeny φmust be either a 2-isogeny or multiplication--by--2.
When φis multiplication--by--two, then the computation of μinvolves a call to AbsoluteAlgebra.
The optional parameter Fields is passed on to that.
If the fields mentioned in this set are found to be of any use,
then these will be used and whatever class group and unit data stored on the
fields will be used in subsequent computations.
When φis multiplication--by--two,
the second map τaccepts all elements δ∈A^ *.
An element δ∈A^ *
represents an element of H1(K, E[2]) if and only if
δmust have square norm. In general, δ∈A^ * represents
an element of H1(K, E[2] x {+- 1}), in which case τ(δ) is the
corresponding covering Tδto( P)1.
(This covering is a twist of the covering E to E to ( P)1
given by P |-> 2P |-> x(2P).)
Hints: SetEnum Default: { }
Raw: BoolElt Default: false
Bound: RngIntElt Default: -1
SetVerbose("Selmer", n): Maximum: 2
Given an isogeny φ: E to E1 defined over a number field K,
computes the associated Selmer group Sel(φ) := Selφ(E/K).
The Selmer group is returned as a finite abelian group S,
together with a map AtoS : A to S, where A is as in the introduction.
This is a map only in the Magma sense; it is defined only on a finite
subset of A. Its "inverse" S to A provides the mathematically
meaningful injection S -> A^ * /(A^ * )2.
The standard map
E1(K) to E1(K)/φE(K) to Sel(φ)
is given by the composition of μwith AtoS,
where μ: E1(K) to H1(K, E[φ])
is the first map returned by DescentMaps.
If the optional parameter Hints is given, it is used as
a list of x-coordinates to try first when determining local images.
Supplying Hints does not change the outcome, but may speed up
the computation.
The calculation of the Selmer group involves possibly expensive class group
and unit group computations. If no such data has been precomputed,
SelmerGroup
will attempt to obtain this information unconditionally, unless Bound is
positive. This bound is then passed on to any called ClassGroup. However,
if such data is already stored, it will be used and subsequent results will be
conditional on whatever assumptions were made while computing this information.
If conditional results are desired (for instance, assuming GRH),
one should precompute class group information on the codomain of
CasselsMap(phi) prior to calling SelmerGroup(phi).
If Raw is true, then three technical items are also returned.
The first two of these, toVec and FB,
enable one to represent elements of the Selmer
group in terms of a "factor base" FB consisting of elements of A
that generate a relevant subgroup of A^ * /(A^ * )2.
The map toVec sends an element of S to an exponent vector
relative to FB.
The map from S to A obtained by multiplying
out the results is inverse to AtoS.
The final returned value (when Raw is true) is a set of Hints
(just as in the optional parameter).
Hints: SetEnum Default: { }
Raw: BoolElt Default: false
Bound: RngIntElt Default: -1
SetVerbose("EllSelmer", n): Maximum: 2
The 2-Selmer group of an elliptic curve defined over Q or a number field.
The function simply calls SelmerGroup for the
multiplication--by--two isogeny.
The given model for E should be integral.
The options and return values are the same as for SelmerGroup.
Given a nonzero element e in an univariate quotient polynomial algebra A
with A cubic over Q, return a hyper-elliptic curve associated
to it by the standard quadric intersection construction.
Also returns a map to model given by y2=f(x) where f is the minimal
polynomial of the given element e.
Isogeny: Map Default:
The upper bound on the rank of E(K), for an elliptic curve
over a number field K (or Q), obtained by computing
the Selmer group(s) associated to some isogeny on E.
The isogeny is either multiplication by 2, or a 2-isogeny if any exist,
and may be specified by the optional parameter Isogeny.
In this example, we determine the rank of y2 = x3 + 9x2 - 10x + 1
by computing the 2-Selmer group.
> E := EllipticCurve([0,9,0,-10,1]);
> two := MultiplicationByMMap(E,2);
> mu, tor := DescentMaps(two);
The hard work: computing the Selmer group.
> S, AtoS := SelmerGroup(two);
> #S;
8
So the Selmer rank is 3. We deduce the following upper bound on the rank
of E(Q), taking into account any 2-torsion.
> RankBound(E : Isogeny := two);
3
In fact, there are 3 points: (0, 1), (1, 1) and (2, 5).
> g1 := E![ 0, 1 ];
> g2 := E![ 1, 1 ];
> g3 := E![ 2, 5 ];
We now map these points into the Selmer group and check that they generate it.
It will follow that the points generate E(K)/2E(K), which means they are
independent nontorsion points in E(K)
(since we know there is no 2-torsion).
> s1 := AtoS(mu(g1));
> s2 := AtoS(mu(g2));
> s3 := AtoS(mu(g3));
> sub<S | [s1, s2, s3]> eq S;
true
Next we compute the homogeneous space associated to the point g1 + g2.
It must have a rational point mapping to g1 + g2.
Note that @@ always denotes "preimage" in Magma.
> coveringmap := tor((s1+s2) @@ AtoS);
> Domain(coveringmap); // Print the homogeneous space but not the map to E
Curve over Rational Field defined by
-u0^2 - 2*u0*u1 - 18*u1*u2 + 82*u2^2,
-10*u0^2 - 18*u0*u1 - 18*u0*u2 - 9*u1^2 + 2*u1*u2 - 90*u2^2 + u3^2
The next command finds all Q-rational points in the preimage of
the point g1 + g2 on E.
(In Magma the preimage is constructed as a scheme.)
> RationalPoints( (g1+g2) @@ coveringmap);
{@ (10 : -9 : -1 : 1) @}
We consider the elliptic curve E : y2 = dx(x + 1)(x + 3)
where d is the product of the primes less than 50.
Since E has full 2-torsion, we can carry out 2-isogeny descent
in three non-equivalent ways, resulting in three different rank bounds.
> P<x> := PolynomialRing(Integers());
> d := &*[ p : p in [1..50] | IsPrime(p) ];
> E := EllipticCurve(HyperellipticCurve(d*x*(x + 1)*(x + 3)));
The nontrivial 2-torsion points in E(Q):
> A, mp := TorsionSubgroup(E);
> T := [ t : a in A | t ne E!0 where t := mp(a) ];
The corresponding 2-isogenies:
> phis := [ TwoIsogeny(t) : t in T ];
The rank bounds obtained from these isogenies:
> [ RankBound(E : Isogeny := phi) : phi in phis ];
[ 9, 5, 7 ]
We find [9, 5, 7]! Each descent gives a different rank bound.
However, a full 2-descent gives:
> two := MultiplicationByMMap(E,2);
> RankBound(E : Isogeny := two);
1
Now doing full 2-descent on the three isogenous curves,
we see where the obstacle for sharp rank bounds comes from.
> OtherTwos := [ MultiplicationByMMap(Codomain(phi), 2) : phi in phis ];
> [ RankBound(Domain(two) : Isogeny := two) : two in OtherTwos ];
[ 9, 5, 7 ]
We find [9, 5, 7] again.
The next example is a classic one from [Kra81].
> E := EllipticCurve([0, 977, 0, 976, 0]);
> twoE := MultiplicationByMMap(E, 2);
> muE := CasselsMap(twoE);
> SE, toSE := SelmerGroup(twoE);
> rkEQ := RankBound(E);
> ptE := [ E | [-1, 0], [0, 0], [-4, 108], [4, 140] ];
> sub<SE | [ toSE(muE(p)) : p in ptE ]> eq SE;
true
> // So E is really of rank 2
> d := 109;
> Ed := QuadraticTwist(E, d);
> Ed![ -976, -298656, 1];
(-976 : -298656 : 1)
> // So Ed is of rank at least 1
> rkEdQ := RankBound(Ed);
> _<x> := PolynomialRing(Rationals());
> K := NumberField(x^2 - d);
> EK := BaseChange(E, K);
> rkEK := RankBound(EK);
> rkEQ;
2
> rkEdQ;
3
> rkEK;
3
We know that the rank of Ed(Q) must be the rank of E(K) minus the
rank of E(Q); thus Ed(Q) has rank 1.
This is smaller than the bound of 3 we get from a 2-descent on
Ed alone.
Here we give some examples of using TwoDescent,
TwoSelmerGroup, and TwoCover.
> E := EllipticCurve( [ 0, 0, 1, -7, 6] ); // rank 3 curve
> T := TwoDescent(E);
> T[6];
Hyperelliptic Curve defined by y^2 = 3*x^4 - 10*x^3 + 10*x + 1 over
Rational Field
> G, m := TwoSelmerGroup(E);
> G.1 @@ m;
theta^2 - 12*theta + 33
> Parent($1); // Modulus has y^2 = modulus isomorphic to E
Univariate Quotient Polynomial Algebra in theta over Rational Field
with modulus theta^3 - 112*theta + 400
> TwoCover( (G.1 + G.2) @@ m);
Hyperelliptic Curve defined by y^2 = 3*x^4 - 10*x^3 + 10*x + 1 over
Rational Field
> TwoCover( Domain(m) ! 1 );
Hyperelliptic Curve defined by y^2 = 2*x^3 - 12*x^2 - 32*x + 196 over
Rational Field
The pairing between elements of the 2-Selmer group is implemented,
in the same way as for curves over the rationals.
This is described in Section The Cassels-Tate Pairing.
PseudoMordellWeilGroup(E) : CrvEll -> BoolElt, GrpAb, Map
Isogeny: Map Default:
SearchBound: RngIntElt Default:
UseHomogeneousSpaces: BoolElt Default:
Given an elliptic curve defined over Q or a number field K,
this function combines several functions from this chapter, in an attempt
find a subgroup of odd, finite index in E(K).
It chooses an isogeny and computes the appropriate Selmer
groups (either multiplication by 2, or a 2-isogeny if any exist),
which the user may specify with the optional parameter Isogeny.
The program then searches on E up to the SearchBound
to find rational points, using RationalPoints.
When UseHomogeneousSpaces is true
(or by default when it is efficient to do so),
it will also search for points on the relevant homogeneous spaces.
The returned values are bool, A and AtoE where A is an
abstract abelian group representing a subgroup of the Mordell--Weil group,
and the map AtoE sends elements of the abstract group A to points on E.
When bool is true, the computation has proved that the rank of A
equals the rank of E(K), in which case A has odd index in E(K)
(in any case, A is 2-saturated).
This refers to a method for finding the rational points on a curve, if the
curve admits a suitable map to an elliptic curve over some extension field.
The method was developed principally by Nils Bruin
(see [Bru03] or [Bru04]).
The first intrinsic follows a Mordell-Weil sieve based strategy, similar to
that used in Chabauty for genus 2 curves (see [BS09]).
InertiaDegreeBound: RngIntElt Default: 20
SmoothBound: RngIntElt Default: 50
PrimeBound: RngIntElt Default: 30
IndexBound: RngIntElt Default: -1
InitialPrimes: RngIntElt Default: 50
SetVerbose("EllChab", n): Maximum: 3
Let E be an elliptic curve defined over a number field K.
This function attempts to determine the subset of E(K) consisting
of those points that have a Q-rational image under a given map E to ( P)1.
The arguments are as follows:
 - a map MWmap: A to E from an abstract abelian group into E(K)
(for instance, the map returned by PseudoMordellWeilGroup),
- A map Ecov : E to( P)1 defined over K.
The returned values are a finite subset V of A and an integer R. The set
V consists of all the points in the image of A in E(K) that have a Q-rational
image under the given map Ecov. If the index of A in E(K) is finite and coprime to R
then V consists of all points in E(K) with Q-rational image. Changing the various optional
parameters can change the performance of this routine considerably. In general, it is not
proven that this routine will return at all.
The parameter InertiaDegreeBound determines the maximum inertia degree of primes at which
local information is obtained.
Only places at with the group order of the rational points on the reduction of E are
SmoothBound-smooth are used.
The Mordell-Weil sieving process only uses information from E(K)/ B E(K), where B is
PrimeBound-smooth.
If IndexBound is set to a value different from -1, then the returned value R will only
contain prime divisors from IndexBound. Setting this parameter may make it impossible for the routine
to return.
Cosets: Tup Default:
Aux: SetEnum Default:
Precision: RngIntElt Default:
Bound: RngIntElt Default:
SetVerbose("EllChab", n): Maximum: 3
Let E be an elliptic curve defined over a number field K.
This function bounds the set of points in E(K)
whose images under a given map E to ( P)1 are Q-rational.
The arguments are as follows:
- a map MWmap : A to E from an abstract group into E(K)
(for instance, the map returned by PseudoMordellWeilGroup),
- a map of varieties Ecov : E to( P)1 defined over K, and
- a rational prime p, such that E and the map Ecov have good reduction
at primes above p.
The returned values are N, V, R and L
as follows. Let G denote the image of A in E(K).
- N is an upper bound for the number of points P ∈G such that
( Ecov)(P) ∈( P)1(Q) (note that N can be ∞).
- V is a set of elements of A found by the program that have images in
( P)1(Q),
- R is a number with the following property: if
[E(K) : G] is finite and coprime to R, then
N is also an upper bound for the number of points P ∈E(K)
such that ( Ecov)(P) ∈( P)1(Q), and
- L is a collection of cosets in A such that (bigcup
L)∪V contains all elements of A that have images in
( P)1(Q).
If Cosets (<Cj>) are supplied (a coset collection of A),
then the bounds and results are computed for A ∩(bigcup Cj).
If Aux is supplied (a set of rational primes),
then the information at p is
combined with the information at the other primes supplied.
If Precision is supplied, this is the p-adic precision used in the
computations. If not, a generally safe default is used.
If Bound is supplied, this determines a search bound for finding V.
The algorithm used is based on [Bru03] and [Bru02].
For examples and a more elaborate discussion, see [Bru04].
This example is motivated and explained in great detail in
[Bru04] (Section 7).
Let E be the elliptic curve
y2 = x3 +
( - 3ζ3 - ζ+ 1)dx2
+ ( - ζ2 - ζ- 1)d2x,
d = 2ζ3 - 2ζ2 - 2
over K := Q(ζ) where ζis a primitive 10th root of unity.
> _<z> := PolynomialRing(Rationals());
> K<zeta> := NumberField(z^4-z^3+z^2-z+1);
> OK := IntegerRing(K);
> d := 2*zeta^3-2*zeta^2-2;
> E<X,Y,Z> := EllipticCurve(
> [ 0, (-3*zeta^3-zeta+1)*d, 0, (-zeta^2-zeta-1)*d^2, 0 ]);
Next we determine as much as possible of E(K),
allowing Magma to choose the method.
> success, G, GtoEK := PseudoMordellWeilGroup(E);
> G;
Abelian Group isomorphic to Z/2 + Z/2 + Z + Z
Defined on 4 generators
Relations:
2*G.1 = 0
2*G.2 = 0
> success;
true
Here G is an abstract group and GtoEK injects it into E(K).
Since the flag is true, this is a subgroup of finite, odd index
in E(K). Next, we determine the points (X:Y:Z) in this subgroup
for which the function
u : E to ( P)1 : (X:Y:Z) to d ( - X + (ζ3 - 1) Z : X + d ( - ζ3 - ζ) Z)
takes values in ( P)1(Q).
> P1 := ProjectiveSpace(Rationals(),1);
> u := map< E->P1 | [-X + (zeta^3 - 1)*d*Z, X+(-zeta^3-zeta)*d*Z] >;
> V, R := Chabauty( GtoEK, u);
> V;
{
0,
G.3 - G.4,
-G.3 + G.4,
G.3 + G.4,
-G.3 - G.4
}
> R;
320
We see that the routine assumed that the image of GtoEK is
2- and 5-saturated in E(K). We can ask it to not assume anything
outside 2.
> V2, R := Chabauty( GtoEK, u: IndexBound:= 2);
> V eq V2;
true
> R;
16
This means that we have found all points in E(K) that have a rational
image under u. If one wants to find the images of these points then
it is necessary in this example to first extend u
(by finding alternative defining equations for it)
so that it is defined on all the points.
> u := Extend(u);
> [ u( GtoEK(P) ) : P in V ];
[ (-1 : 1), (-1/3 : 1), (-1/3 : 1), (-3 : 1), (-3 : 1) ]
Alternatively, we can apply Chabauty's method without Mordell-Weil sieving.
> N, V, R, C := Chabauty( GtoEK, u, 3);
> N;
5
> V;
{
0,
G.3 - G.4,
-G.3 + G.4,
G.3 + G.4,
-G.3 - G.4
}
> R;
4
The Chabauty calculations prove that there
are at most N elements in G whose image under u is Q-rational.
Also, V is a set of elements with this property.
Since here N = 5 = #V, these are the only such elements.
Moreover, the calculations prove
that if [E(K) : G] is coprime to R,
then N is actually an upper bound on the number of elements in E(K)
whose image under u is Q-rational.
We know from the PseudoMordellWeilGroup computation
that [E(K) : G] is odd. Therefore we have solved our problem for E(K),
not just for G.
This section contains machinery for number fields (and more generally
"etale algebras", i.e. algebras of the form Q[x]/p(x)), intended to
perform the number field calculations that are involved in computing
Selmer groups of geometric objects.
It has become conventional to refer to "the p-Selmer group" of
a number field K (or, more generally, of an etale algebra)
relative to a finite set S
of K-primes. It means the finite subgroup K(S, p) of K^ * /(K^ * )p,
consisting of those elements whose valuation at every prime
outside S is a multiple of p.
Fields: SetEnum Default: { }
SetVerbose("EtaleAlg", n): Maximum: 1
Given a separable commutative algebra over Q, an absolute number field
or a finite field,
the function returns the isomorphic direct sum of absolute fields as a
cartesian product and the isomorphisms to and from this product. The
optional parameter Fields enables the user to suggest representations
of the absolute fields. If this function finds it needs a field isomorphic
to one occurring in the supplied list, then it will use the given field.
Otherwise it will construct a field itself. The isomorphism is returned
as a second argument. If called twice on the same algebra, it will recompute
if the Fields argument is given. Otherwise it will return the result
computed the first time.
SetVerbose("EtaleAlg", n): Maximum: 1
Returns the p-Selmer group of a semi-simple algebra A.
The set S of ideals
should be prime ideals of the underlying number field. The group
returned is the direct sum of the p-Selmer groups of the irreducible
summands of A and the map is the obvious embedding in the multiplicative
group of A.
An implied restriction is that BaseRing(A) be of type FldNum.
See also pSelmerGroup.
If an algebra over the rationals is required, create a degree 1
extension by
QasaNumberField := NumberField(Rationals());
Let K be the number field associated with the prime ideal P.
The map returned is
K^ * to KP^ * / KP * 2, where KP is the completion of K at P.
The codomain is represented as a finite abelian group.
Let K be the base field of the commutative algebra A
and let P be a prime ideal in K.
Then this function returns a map A^ * to A^ * /A * 2 tensor KP,
where KP is the completion of K at P.
The codomain is represented as a finite abelian group.
This map is computed by using LocalTwoSelmerMap(Q) for the various
extensions Q of P to the fields making up AbsoluteAlgebra(A).
The map returned is essentially the direct sum of all these maps.
For technical purposes, one may also wish to use the components of the
map coming from each number field; these are given by the second return value,
which is a sequence of records (ordered in the same way as the results are
concatenated in the returned map).
Each record contains items i, p, map and vmap.
Here i is an index indicating which number field
in AbsoluteAlgebra(A) the record corresponds to,
p is an extension of P to a prime in that number field,
map is LocalTwoSelmerMap(p),
and vmap is the valuation map at p on that number field.
> P<x> := PolynomialRing(Rationals());
> A := quo<P | x^3 - 1>;
> AA := AbsoluteAlgebra(A);
> AA;
Cartesian Product
<Number Field with defining polynomial x - 1 over the Rational Field,
Number Field with defining polynomial x^2 + x + 1 over the Rational Field>
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