\(p\)-Selmer groups of number fields#
This file contains code to compute \(K(S,p)\) where
\(K\) is a number field
\(S\) is a finite set of primes of \(K\)
\(p\) is a prime number
For \(m\ge2\), \(K(S,m)\) is defined to be the finite subgroup of \(K^*/(K^*)^m\) consisting of elements represented by \(a\in K^*\) whose valuation at all primes not in \(S\) is a multiple of \(m\). It fits in the short exact sequence
where \(O^*_{K,S}\) is the group of \(S\)-units of \(K\) and \(Cl_{K,S}\) the \(S\)-class group. When \(m=p\) is prime, \(K(S,p)\) is a finite-dimensional vector space over \(GF(p)\). Its generators come from three sources: units (modulo \(p\)’th powers); generators of the \(p\)’th powers of ideals which are not principal but whose \(p\)’the powers are principal; and generators coming from the prime ideals in \(S\).
The main function here is pSelmerGroup()
. This will not
normally be used by users, who instead will access it through a method
of the NumberField class.
AUTHORS:
John Cremona (2005-2021)
- sage.rings.number_field.selmer_group.basis_for_p_cokernel(S, C, p)#
Return a basis for the group of ideals supported on
S
(modp
’th-powers) whose class in the class groupC
is ap
’th power, together with a function which takes theS
-exponents of such an ideal and returns its coordinates on this basis.INPUT:
S
(list) – a list of prime ideals in a number fieldK
.C
(class group) – the ideal class group ofK
.p
(prime) – a prime number.
OUTPUT:
(tuple) (
b
,f
) whereb
is a list of ideals which is a basis for the group of ideals supported onS
(modulop
’th powers) whose ideal class is ap
’th power;f
is a function which takes such an ideal and returns its coordinates with respect to this basis.
EXAMPLES:
sage: from sage.rings.number_field.selmer_group import basis_for_p_cokernel sage: K.<a> = NumberField(x^2 - x + 58) sage: S = K.ideal(30).support(); S [Fractional ideal (2, a), Fractional ideal (2, a + 1), Fractional ideal (3, a + 1), Fractional ideal (5, a + 1), Fractional ideal (5, a + 3)] sage: C = K.class_group() sage: C.gens_orders() (6, 2) sage: [C(P).exponents() for P in S] [(5, 0), (1, 0), (3, 1), (1, 1), (5, 1)] sage: b, f = basis_for_p_cokernel(S, C, 2); b [Fractional ideal (2), Fractional ideal (15, a + 13), Fractional ideal (5)] sage: b, f = basis_for_p_cokernel(S, C, 3); b [Fractional ideal (50, a + 18), Fractional ideal (10, a + 3), Fractional ideal (3, a + 1), Fractional ideal (5)] sage: b, f = basis_for_p_cokernel(S, C, 5); b [Fractional ideal (2, a), Fractional ideal (2, a + 1), Fractional ideal (3, a + 1), Fractional ideal (5, a + 1), Fractional ideal (5, a + 3)]
- sage.rings.number_field.selmer_group.coords_in_U_mod_p(u, U, p)#
Return coordinates of a unit
u
with respect to a basis of thep
-cotorsion \(U/U^p\) of the unit groupU
.INPUT:
u
(algebraic unit) – a unit in a number fieldK
.U
(unit group) – the unit group ofK
.p
(prime) – a prime number.
OUTPUT:
The coordinates of the unit \(u\) in the \(p\)-cotorsion group \(U/U^p\).
ALGORITHM:
Take the coordinate vector of \(u\) with respect to the generators of the unit group, drop the coordinate of the roots of unity factor if it is prime to \(p\), and reduce the vector mod \(p\).
EXAMPLES:
sage: from sage.rings.number_field.selmer_group import coords_in_U_mod_p sage: K.<a> = NumberField(x^4 - 5*x^2 + 1) sage: U = K.unit_group() sage: U Unit group with structure C2 x Z x Z x Z of Number Field in a with defining polynomial x^4 - 5*x^2 + 1 sage: u0, u1, u2, u3 = U.gens_values() sage: u = u1*u2^2*u3^3 sage: coords_in_U_mod_p(u,U,2) [0, 1, 0, 1] sage: coords_in_U_mod_p(u,U,3) [1, 2, 0] sage: u*=u0 sage: coords_in_U_mod_p(u,U,2) [1, 1, 0, 1] sage: coords_in_U_mod_p(u,U,3) [1, 2, 0]
- sage.rings.number_field.selmer_group.pSelmerGroup(K, S, p, proof=None, debug=False)#
Return the
p
-Selmer group \(K(S,p)\) of the number fieldK
with respect to the prime ideals inS
INPUT:
K
(number field) – a number field, or \(\QQ\).S
(list) – a list of prime ideals inK
, or prime numbers whenK
is \(\QQ\).p
(prime) – a prime number.proof
- if True then compute the class group provably correctly. Default is True. Callproof.number_field()
to change this default globally.debug
(boolean, defaultFalse
) – debug flag.
OUTPUT:
(tuple)
KSp
,KSp_gens
,from_KSp
,to_KSp
whereKSp
is an abstract vector space over \(GF(p)\) isomorphic to \(K(S,p)\);KSp_gens
is a list of elements of \(K^*\) generating \(K(S,p)\);from_KSp
is a function fromKSp
to \(K^*\) implementing the isomorphism from the abstract \(K(S,p)\) to \(K(S,p)\) as a subgroup of \(K^*/(K^*)^p\);to_KSP
is a partial function from \(K^*\) toKSp
, defined on elements \(a\) whose image in \(K^*/(K^*)^p\) lies in \(K(S,p)\), mapping them via the inverse isomorphism to the abstract vector spaceKSp
.
ALGORITHM:
The list of generators of \(K(S,p)\) is the concatenation of three sublists, called
alphalist
,betalist
andulist
in the code. Onlyalphalist
depends on the primes in \(S\).ulist
is a basis for \(U/U^p\) where \(U\) is the unit group. This is the list of fundamental units, including the generator of the group of roots of unity if its order is divisible by \(p\). These have valuation \(0\) at all primes.betalist
is a list of the generators of the \(p\)’th powers of ideals which generate the \(p\)-torsion in the class group (so is empty if the class number is prime to \(p\)). These have valuation divisible by \(p\) at all primes.alphalist
is a list of generators for each ideal \(A\) in a basis of those ideals supported on \(S\) (modulo \(p\)’th powers of ideals) which are \(p\)’th powers in the class group. We find \(B\) such that \(A/B^p\) is principal and take a generator of it, for each \(A\) in a generating set. As a special case, if all the ideals in \(S\) are principal thenalphalist
is a list of their generators.
The map from the abstract space to \(K^*\) is easy: we just take the product of the generators to powers given by the coefficient vector. No attempt is made to reduce the resulting product modulo \(p\)’th powers.
The reverse map is more complicated. Given \(a\in K^*\):
write the principal ideal \((a)\) in the form \(AB^p\) with \(A\) supported by \(S\) and \(p\)’th power free. If this fails, then \(a\) does not represent an element of \(K(S,p)\) and an error is raised.
set \(I_S\) to be the group of ideals spanned by \(S\) mod \(p\)’th powers, and \(I_{S,p}\) the subgroup of \(I_S\) which maps to \(0\) in \(C/C^p\).
Convert \(A\) to an element of \(I_{S,p}\), hence find the coordinates of \(a\) with respect to the generators in
alphalist
.after dividing out by \(A\), now \((a)=B^p\) (with a different \(a\) and \(B\)). Write the ideal class \([B]\), whose \(p\)’th power is trivial, in terms of the generators of \(C[p]\); then \(B=(b)B_1\), where the coefficients of \(B_1\) with respect to generators of \(C[p]\) give the coordinates of the result with respect to the generators in
betalist
.after dividing out by \(B\), and by \(b^p\), we now have \((a)=(1)\), so \(a\) is a unit, which can be expressed in terms of the unit generators.
EXAMPLES:
Over \(\QQ\) the unit contribution is trivial unless \(p=2\) and the class group is trivial:
sage: from sage.rings.number_field.selmer_group import pSelmerGroup sage: QS2, gens, fromQS2, toQS2 = pSelmerGroup(QQ, [2,3], 2) sage: QS2 Vector space of dimension 3 over Finite Field of size 2 sage: gens [2, 3, -1] sage: a = fromQS2([1,1,1]); a.factor() -1 * 2 * 3 sage: toQS2(-6) (1, 1, 1) sage: QS3, gens, fromQS3, toQS3 = pSelmerGroup(QQ, [2,13], 3) sage: QS3 Vector space of dimension 2 over Finite Field of size 3 sage: gens [2, 13] sage: a = fromQS3([5,4]); a.factor() 2^5 * 13^4 sage: toQS3(a) (2, 1) sage: toQS3(a) == QS3([5,4]) True
A real quadratic field with class number 2, where the fundamental unit is a generator, and the class group provides another generator when \(p=2\):
sage: K.<a> = QuadraticField(-5) sage: K.class_number() 2 sage: P2 = K.ideal(2, -a+1) sage: P3 = K.ideal(3, a+1) sage: P5 = K.ideal(a) sage: KS2, gens, fromKS2, toKS2 = pSelmerGroup(K, [P2, P3, P5], 2) sage: KS2 Vector space of dimension 4 over Finite Field of size 2 sage: gens [a + 1, a, 2, -1]
Each generator must have even valuation at primes not in \(S\):
sage: [K.ideal(g).factor() for g in gens] [(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)), Fractional ideal (-a), (Fractional ideal (2, a + 1))^2, 1] sage: toKS2(10) (0, 0, 1, 1) sage: fromKS2([0,0,1,1]) -2 sage: K(10/(-2)).is_square() True sage: KS3, gens, fromKS3, toKS3 = pSelmerGroup(K, [P2, P3, P5], 3) sage: KS3 Vector space of dimension 3 over Finite Field of size 3 sage: gens [1/2, 1/4*a + 1/4, a]
The
to
andfrom
maps are inverses of each other:sage: K.<a> = QuadraticField(-5) sage: S = K.ideal(30).support() sage: KS2, gens, fromKS2, toKS2 = pSelmerGroup(K, S, 2) sage: KS2 Vector space of dimension 5 over Finite Field of size 2 sage: assert all(toKS2(fromKS2(v))==v for v in KS2) sage: KS3, gens, fromKS3, toKS3 = pSelmerGroup(K, S, 3) sage: KS3 Vector space of dimension 4 over Finite Field of size 3 sage: assert all(toKS3(fromKS3(v))==v for v in KS3)