# $$p$$-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

$1 \rightarrow O^*_{K,S}/(O^*_{K,S})^m \rightarrow K(S,m) \rightarrow Cl_{K,S}[m] \rightarrow 1$

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 (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) – a list of prime ideals in a number field K.

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

• p (prime) – a 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: 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 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) – 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 field K with respect to the prime ideals in S

INPUT:

• K (number field) – a number field, or $$\QQ$$.

• S (list) – a list of prime ideals in K, or prime numbers when K is $$\QQ$$.

• p (prime) – a prime number.

• proof - if True then 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 $$K^*$$ generating $$K(S,p)$$;

• from_KSp is a function from KSp 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 $$K^*$$ to KSp, 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 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 $$K^*$$ 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.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 and from maps are inverses of each other: