$p$-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 $m\ge 2$, $K(S,m)$ is defined to be the finite subgroup of $K^{\ast }/(K^{\ast }{)}^m$ consisting of elements represented by $a\in K^{\ast }$ whose valuation at all primes not in $S$ is a multiple of $m$. It fits in the short exact sequence

$$1\to O_{K,S}^{\ast }/(O_{K,S}^{\ast }{)}^m\to K(S,m)\to C{l}_{K,S}[m]\to 1$$

where $O_{K,S}^{\ast }$ is the group of $S$-units of $K$ and $C{l}_{K,S}$ 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 $p$-th 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 (mod $p$-th-powers) whose class in the class group C is a $p$-th power, together with a function which takes the S-exponents of such an ideal and returns its coordinates on this basis.

INPUT:

• S – list of prime ideals in a number field K

• C – (class group) the ideal class group of K

• p – prime number

OUTPUT:

(tuple) (b, f) where

• b is a list of ideals which is a basis for the group of ideals supported on S (modulo $p$-th powers) whose ideal class is a $p$-th power;

• f is a function which takes such an ideal and returns its coordinates with respect to this basis.

EXAMPLES:

Sage

sage: from sage.rings.number_field.selmer_group import basis_for_p_cokernel
sage: x = polygen(ZZ, 'x')
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)]

Python

>>> from sage.all import *
>>> from sage.rings.number_field.selmer_group import basis_for_p_cokernel
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - x + Integer(58), names=('a',)); (a,) = K._first_ngens(1)
>>> S = K.ideal(Integer(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)]
>>> C = K.class_group()
>>> C.gens_orders()
(6, 2)
>>> [C(P).exponents() for P in S]
[(5, 0), (1, 0), (3, 1), (1, 1), (5, 1)]
>>> b, f = basis_for_p_cokernel(S, C, Integer(2)); b
[Fractional ideal (2), Fractional ideal (15, a + 13), Fractional ideal (5)]
>>> b, f = basis_for_p_cokernel(S, C, Integer(3)); b
[Fractional ideal (50, a + 18),
Fractional ideal (10, a + 3),
Fractional ideal (3, a + 1),
Fractional ideal (5)]
>>> b, f = basis_for_p_cokernel(S, C, Integer(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 Live

from sage.rings.number_field.selmer_group import basis_for_p_cokernel
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 - x + 58)
S = K.ideal(30).support(); S
C = K.class_group()
C.gens_orders()
[C(P).exponents() for P in S]
b, f = basis_for_p_cokernel(S, C, 2); b
b, f = basis_for_p_cokernel(S, C, 3); b
b, f = basis_for_p_cokernel(S, C, 5); b

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 the $p$-cotorsion $U/U^p$ of the unit group U.

INPUT:

• u – (algebraic unit) a unit in a number field K

• U – (unit group) the unit group of K

• p – 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

sage: from sage.rings.number_field.selmer_group import coords_in_U_mod_p
sage: x = polygen(ZZ, 'x')
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]

Python

>>> from sage.all import *
>>> from sage.rings.number_field.selmer_group import coords_in_U_mod_p
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(5)*x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> U = K.unit_group()
>>> 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
>>> u0, u1, u2, u3 = U.gens_values()
>>> u = u1*u2**Integer(2)*u3**Integer(3)
>>> coords_in_U_mod_p(u,U,Integer(2))
[0, 1, 0, 1]
>>> coords_in_U_mod_p(u,U,Integer(3))
[1, 2, 0]
>>> u*=u0
>>> coords_in_U_mod_p(u,U,Integer(2))
[1, 1, 0, 1]
>>> coords_in_U_mod_p(u,U,Integer(3))
[1, 2, 0]

Sage Live

from sage.rings.number_field.selmer_group import coords_in_U_mod_p
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^4 - 5*x^2 + 1)
U = K.unit_group()
U
u0, u1, u2, u3 = U.gens_values()
u = u1*u2^2*u3^3
coords_in_U_mod_p(u,U,2)
coords_in_U_mod_p(u,U,3)
u*=u0
coords_in_U_mod_p(u,U,2)
coords_in_U_mod_p(u,U,3)

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 field $K$ with respect to the prime ideals in S.

INPUT:

• K – a number field or $\mathbf{Q}$

• S – list of prime ideals in $K$, or prime numbers when $K$ is $\mathbf{Q}$

• p – a prime number

• proof – if True, compute the class group provably correctly. Default is True. Call proof.number_field() to change this default globally.

• debug – boolean (default: False); debug flag

OUTPUT:

(tuple) KSp, KSp_gens, from_KSp, to_KSp where

• KSp is an abstract vector space over $GF(p)$ isomorphic to $K(S,p)$;

• KSp_gens is a list of elements of $K^{\ast }$ generating $K(S,p)$;

• from_KSp is a function from KSp to $K^{\ast }$ implementing the isomorphism from the abstract $K(S,p)$ to $K(S,p)$ as a subgroup of $K^{\ast }/(K^{\ast }{)}^p$;

• to_KSP is a partial function from $K^{\ast }$ to KSp, defined on elements $a$ whose image in $K^{\ast }/(K^{\ast }{)}^p$ lies in $K(S,p)$, mapping them via the inverse isomorphism to the abstract vector space KSp.

ALGORITHM:

The list of generators of $K(S,p)$ is the concatenation of three sublists, called alphalist, betalist and ulist in the code. Only alphalist 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 then alphalist is a list of their generators.

The map from the abstract space to $K^{\ast }$ 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 $a\in K^{\ast }$:

• write the principal ideal $(a)$ in the form $A{B}^p$ 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 $\mathbf{Q}$ the unit contribution is trivial unless $p=2$ and the class group is trivial:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.number_field.selmer_group import pSelmerGroup
>>> QS2, gens, fromQS2, toQS2 = pSelmerGroup(QQ, [Integer(2),Integer(3)], Integer(2))
>>> QS2
Vector space of dimension 3 over Finite Field of size 2
>>> gens
[2, 3, -1]
>>> a = fromQS2([Integer(1),Integer(1),Integer(1)]); a.factor()
-1 * 2 * 3
>>> toQS2(-Integer(6))
(1, 1, 1)

>>> QS3, gens, fromQS3, toQS3 = pSelmerGroup(QQ, [Integer(2),Integer(13)], Integer(3))
>>> QS3
Vector space of dimension 2 over Finite Field of size 3
>>> gens
[2, 13]
>>> a = fromQS3([Integer(5),Integer(4)]); a.factor()
2^5 * 13^4
>>> toQS3(a)
(2, 1)
>>> toQS3(a) == QS3([Integer(5),Integer(4)])
True

Sage Live

from sage.rings.number_field.selmer_group import pSelmerGroup
QS2, gens, fromQS2, toQS2 = pSelmerGroup(QQ, [2,3], 2)
QS2
gens
a = fromQS2([1,1,1]); a.factor()
toQS2(-6)
QS3, gens, fromQS3, toQS3 = pSelmerGroup(QQ, [2,13], 3)
QS3
gens
a = fromQS3([5,4]); a.factor()
toQS3(a)
toQS3(a) == QS3([5,4])

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

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]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.class_number()
2
>>> P2 = K.ideal(Integer(2), -a+Integer(1))
>>> P3 = K.ideal(Integer(3), a+Integer(1))
>>> P5 = K.ideal(a)
>>> KS2, gens, fromKS2, toKS2 = pSelmerGroup(K, [P2, P3, P5], Integer(2))
>>> KS2
Vector space of dimension 4 over Finite Field of size 2
>>> gens
[a + 1, a, 2, -1]

Sage Live

K.<a> = QuadraticField(-5)
K.class_number()
P2 = K.ideal(2, -a+1)
P3 = K.ideal(3, a+1)
P5 = K.ideal(a)
KS2, gens, fromKS2, toKS2 = pSelmerGroup(K, [P2, P3, P5], 2)
KS2
gens

Each generator must have even valuation at primes not in $S$:

Sage

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]

Python

>>> from sage.all import *
>>> [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]

>>> toKS2(Integer(10))
(0, 0, 1, 1)
>>> fromKS2([Integer(0),Integer(0),Integer(1),Integer(1)])
-2
>>> K(Integer(10)/(-Integer(2))).is_square()
True

>>> KS3, gens, fromKS3, toKS3 = pSelmerGroup(K, [P2, P3, P5], Integer(3))
>>> KS3
Vector space of dimension 3 over Finite Field of size 3
>>> gens
[1/2, 1/4*a + 1/4, a]

Sage Live

[K.ideal(g).factor() for g in gens]
toKS2(10)
fromKS2([0,0,1,1])
K(10/(-2)).is_square()
KS3, gens, fromKS3, toKS3 = pSelmerGroup(K, [P2, P3, P5], 3)
KS3
gens

The to and from maps are inverses of each other:

Sage

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)

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> S = K.ideal(Integer(30)).support()
>>> KS2, gens, fromKS2, toKS2 = pSelmerGroup(K, S, Integer(2))
>>> KS2
Vector space of dimension 5 over Finite Field of size 2
>>> assert all(toKS2(fromKS2(v))==v for v in KS2)
>>> KS3, gens, fromKS3, toKS3 = pSelmerGroup(K, S, Integer(3))
>>> KS3
Vector space of dimension 4 over Finite Field of size 3
>>> assert all(toKS3(fromKS3(v))==v for v in KS3)

Sage Live

K.<a> = QuadraticField(-5)
S = K.ideal(30).support()
KS2, gens, fromKS2, toKS2 = pSelmerGroup(K, S, 2)
KS2
assert all(toKS2(fromKS2(v))==v for v in KS2)
KS3, gens, fromKS3, toKS3 = pSelmerGroup(K, S, 3)
KS3
assert all(toKS3(fromKS3(v))==v for v in KS3)

Make live