Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Checking the Brumer-Stark conjecture using
PARI/GP
Xavier-François Roblot
IGD, Université Claude Bernard – Lyon 1

September 16th, 2004

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

1

Statement of the conjecture
Definitions
The Brumer element
The Brumer-Stark Conjecture

2

Current status of the conjecture
Some reductions and special cases
Further results

3

Checking the conjecture on an example
The example
The strategy
The verification

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

k is a number field of degree n
K is a finite abelian extension over k
G := Gal(K /k)wK is the number of roots of unity in K
ClK is the class group of K
S is the set of the infinite primes of k and of the finite prime
ideals in k that ramify in K
For each σ ∈ G , the partial zeta-function isζS (s, σ) :=X(a,S)=1
σa =σXavier-François Roblot

1
N as

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

k is a number field of degree n
K is a finite abelian extension over k
G := Gal(K /k)wK is the number of roots of unity in K
ClK is the class group of K
S is the set of the infinite primes of k and of the finite prime
ideals in k that ramify in K
For each σ ∈ G , the partial zeta-function isζS (s, σ) :=X(a,S)=1
σa =σXavier-François Roblot

1
N as

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

k is a number field of degree n
K is a finite abelian extension over k
G := Gal(K /k)wK is the number of roots of unity in K
ClK is the class group of K
S is the set of the infinite primes of k and of the finite prime
ideals in k that ramify in K
For each σ ∈ G , the partial zeta-function isζS (s, σ) :=X(a,S)=1
σa =σXavier-François Roblot

1
N as

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

k is a number field of degree n
K is a finite abelian extension over k
G := Gal(K /k)wK is the number of roots of unity in K
ClK is the class group of K
S is the set of the infinite primes of k and of the finite prime
ideals in k that ramify in K
For each σ ∈ G , the partial zeta-function isζS (s, σ) :=X(a,S)=1
σa =σXavier-François Roblot

1
N as

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

k is a number field of degree n
K is a finite abelian extension over k
G := Gal(K /k)wK is the number of roots of unity in K
ClK is the class group of K
S is the set of the infinite primes of k and of the finite prime
ideals in k that ramify in K
For each σ ∈ G , the partial zeta-function isζS (s, σ) :=X(a,S)=1
σa =σXavier-François Roblot

1
N as

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

Theorem (Deligne and Ribet, Barsky, and Pi. Cassou-Noguès)
For every σ ∈ G
wK ζS (0, σ) ∈ Z

The Brumer element is the element of the group ring Z[G ] definedbyX
γ := wKζS (0, σ)σ −1
σ∈G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

Theorem (Deligne and Ribet, Barsky, and Pi. Cassou-Noguès)
For every σ ∈ G
wK ζS (0, σ) ∈ Z

The Brumer element is the element of the group ring Z[G ] definedbyX
γ := wKζS (0, σ)σ −1
σ∈G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

The Brumer-Stark Conjecture
Conjecture (The Brumer part)
The element γ kills ClK .
That is, for every fractional ideal A of K , the ideal Aγ is principal.
Let K ◦ be the set of anti-units of K
K ◦ := {x ∈ K : |x|P∞ = 1, ∀P∞ | ∞}Conjecture (The Stark part)For every fractional ideal A of K , there exists a generator αA of Aγ
K
that is an anti-unit. Furthermore, define λA ∈ K by λW
A = αA ,then K (λA )/k is an abelian extension.
Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

The Brumer-Stark Conjecture
Conjecture (The Brumer part)
The element γ kills ClK .
That is, for every fractional ideal A of K , the ideal Aγ is principal.
Let K ◦ be the set of anti-units of K
K ◦ := {x ∈ K : |x|P∞ = 1, ∀P∞ | ∞}Conjecture (The Stark part)For every fractional ideal A of K , there exists a generator αA of Aγ
K
that is an anti-unit. Furthermore, define λA ∈ K by λW
A = αA ,then K (λA )/k is an abelian extension.
Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

The Brumer-Stark Conjecture
Conjecture (The Brumer part)
The element γ kills ClK .
That is, for every fractional ideal A of K , the ideal Aγ is principal.
Let K ◦ be the set of anti-units of K
K ◦ := {x ∈ K : |x|P∞ = 1, ∀P∞ | ∞}Conjecture (The Stark part)For every fractional ideal A of K , there exists a generator αA of Aγ
K
that is an anti-unit. Furthermore, define λA ∈ K by λW
A = αA ,then K (λA )/k is an abelian extension.
Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Definitions
The Brumer element
The Brumer-Stark Conjecture

The Brumer-Stark Conjecture
Conjecture (The Brumer part)
The element γ kills ClK .
That is, for every fractional ideal A of K , the ideal Aγ is principal.
Let K ◦ be the set of anti-units of K
K ◦ := {x ∈ K : |x|P∞ = 1, ∀P∞ | ∞}Conjecture (The Stark part)For every fractional ideal A of K , there exists a generator αA of Aγ
K
that is an anti-unit. Furthermore, define λA ∈ K by λW
A = αA ,then K (λA )/k is an abelian extension.
Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true if k = Q (Stickelberger’s Theorem)
The conjecture is true if k is not totally real or K is nottotally complex
The conjecture is satisfied for A if it is a principal ideal
The conjecture is true if K is principal
The set of ideals satisfying the conjecture forms a group,
stable under the action of G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true if k = Q (Stickelberger’s Theorem)
The conjecture is true if k is not totally real or K is nottotally complex
The conjecture is satisfied for A if it is a principal ideal
The conjecture is true if K is principal
The set of ideals satisfying the conjecture forms a group,
stable under the action of G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true if k = Q (Stickelberger’s Theorem)
The conjecture is true if k is not totally real or K is nottotally complex
The conjecture is satisfied for A if it is a principal ideal
The conjecture is true if K is principal
The set of ideals satisfying the conjecture forms a group,
stable under the action of G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true if k = Q (Stickelberger’s Theorem)
The conjecture is true if k is not totally real or K is nottotally complex
The conjecture is satisfied for A if it is a principal ideal
The conjecture is true if K is principal
The set of ideals satisfying the conjecture forms a group,
stable under the action of G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true if k = Q (Stickelberger’s Theorem)
The conjecture is true if k is not totally real or K is nottotally complex
The conjecture is satisfied for A if it is a principal ideal
The conjecture is true if K is principal
The set of ideals satisfying the conjecture forms a group,
stable under the action of G

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true in the following cases
if K /k is quadratic [Tate]
if G ' Z/2Z × Z/2Z in general, and when G is of exponent 2
and has order > 4, assuming K /k is a tame extension [Sands]
if |G | = 4 and K /k is a sub-extension of a non-abelian Galois
extension K /k0 of degree 8 [Tate]
if K /k is a sub-extension of an abelian Galois extension K /k0
for which the conjecture is true [Sands, Hayes]
if G ' Z/4Z and k is real quadratic [Greither]
if [K : k] = 6, and [k : Q] = 2, or 3 and the discriminant of kis coprime with 6 (except for some very special cases)[Greither-Roblot-Tangedal]

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true in the following cases
if K /k is quadratic [Tate]
if G ' Z/2Z × Z/2Z in general, and when G is of exponent 2
and has order > 4, assuming K /k is a tame extension [Sands]
if |G | = 4 and K /k is a sub-extension of a non-abelian Galois
extension K /k0 of degree 8 [Tate]
if K /k is a sub-extension of an abelian Galois extension K /k0
for which the conjecture is true [Sands, Hayes]
if G ' Z/4Z and k is real quadratic [Greither]
if [K : k] = 6, and [k : Q] = 2, or 3 and the discriminant of kis coprime with 6 (except for some very special cases)[Greither-Roblot-Tangedal]

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true in the following cases
if K /k is quadratic [Tate]
if G ' Z/2Z × Z/2Z in general, and when G is of exponent 2
and has order > 4, assuming K /k is a tame extension [Sands]
if |G | = 4 and K /k is a sub-extension of a non-abelian Galois
extension K /k0 of degree 8 [Tate]
if K /k is a sub-extension of an abelian Galois extension K /k0
for which the conjecture is true [Sands, Hayes]
if G ' Z/4Z and k is real quadratic [Greither]
if [K : k] = 6, and [k : Q] = 2, or 3 and the discriminant of kis coprime with 6 (except for some very special cases)[Greither-Roblot-Tangedal]

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true in the following cases
if K /k is quadratic [Tate]
if G ' Z/2Z × Z/2Z in general, and when G is of exponent 2
and has order > 4, assuming K /k is a tame extension [Sands]
if |G | = 4 and K /k is a sub-extension of a non-abelian Galois
extension K /k0 of degree 8 [Tate]
if K /k is a sub-extension of an abelian Galois extension K /k0
for which the conjecture is true [Sands, Hayes]
if G ' Z/4Z and k is real quadratic [Greither]
if [K : k] = 6, and [k : Q] = 2, or 3 and the discriminant of kis coprime with 6 (except for some very special cases)[Greither-Roblot-Tangedal]

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true in the following cases
if K /k is quadratic [Tate]
if G ' Z/2Z × Z/2Z in general, and when G is of exponent 2
and has order > 4, assuming K /k is a tame extension [Sands]
if |G | = 4 and K /k is a sub-extension of a non-abelian Galois
extension K /k0 of degree 8 [Tate]
if K /k is a sub-extension of an abelian Galois extension K /k0
for which the conjecture is true [Sands, Hayes]
if G ' Z/4Z and k is real quadratic [Greither]
if [K : k] = 6, and [k : Q] = 2, or 3 and the discriminant of kis coprime with 6 (except for some very special cases)[Greither-Roblot-Tangedal]

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

Some reductions and special cases
Further results

The conjecture is true in the following cases
if K /k is quadratic [Tate]
if G ' Z/2Z × Z/2Z in general, and when G is of exponent 2
and has order > 4, assuming K /k is a tame extension [Sands]
if |G | = 4 and K /k is a sub-extension of a non-abelian Galois
extension K /k0 of degree 8 [Tate]
if K /k is a sub-extension of an abelian Galois extension K /k0
for which the conjecture is true [Sands, Hayes]
if G ' Z/4Z and k is real quadratic [Greither]
if [K : k] = 6, and [k : Q] = 2, or 3 and the discriminant of kis coprime with 6 (except for some very special cases)[Greither-Roblot-Tangedal]

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification√
Let k = Q( 69), and K = K + (j) where K + is the ray class fieldof k of conductor 3 and j is a primitive third root of unity.
This example is one of the exceptions not covered by [GRT].

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Compute the Brumer element using L-functions
Find a minimal set {A1 , . . . , As } of Z[G ]-generators of ClK
For each A
Compute Aγ and check if it is principal
Call β a generator of Aγ , find a unit u such that α := uβ is ananti-unit
Check if K (α1/wK ) is an abelian extension of k

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Compute the Brumer element using L-functions
Find a minimal set {A1 , . . . , As } of Z[G ]-generators of ClK
For each A
Compute Aγ and check if it is principal
Call β a generator of Aγ , find a unit u such that α := uβ is ananti-unit
Check if K (α1/wK ) is an abelian extension of k

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Compute the Brumer element using L-functions
Find a minimal set {A1 , . . . , As } of Z[G ]-generators of ClK
For each A
Compute Aγ and check if it is principal
Call β a generator of Aγ , find a unit u such that α := uβ is ananti-unit
Check if K (α1/wK ) is an abelian extension of k

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Compute the Brumer element using L-functions
Find a minimal set {A1 , . . . , As } of Z[G ]-generators of ClK
For each A
Compute Aγ and check if it is principal
Call β a generator of Aγ , find a unit u such that α := uβ is ananti-unit
Check if K (α1/wK ) is an abelian extension of k

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Compute the Brumer element using L-functions
Find a minimal set {A1 , . . . , As } of Z[G ]-generators of ClK
For each A
Compute Aγ and check if it is principal
Call β a generator of Aγ , find a unit u such that α := uβ is ananti-unit
Check if K (α1/wK ) is an abelian extension of k

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Compute the Brumer element using L-functions
Find a minimal set {A1 , . . . , As } of Z[G ]-generators of ClK
For each A
Compute Aγ and check if it is principal
Call β a generator of Aγ , find a unit u such that α := uβ is ananti-unit
Check if K (α1/wK ) is an abelian extension of k

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let’s start GP!

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

γ = wKX

LS (0, χ)eχThe example
The strategy
The verification

where eχ :=1 X
χ̄(σ)σ −1
|G |
σ∈G

χ∈Ĝ

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let g be a generator of Clk (3∞1 ∞2 ).
Let σ := σg . Thus G = hσi.
Let ζ6 := exp(2iπ/6).
The character χa represented by [a] is the one defined by
χa (σ) := ζ6a .
An element a0 + a1 σ + · · · + a5 σ 5 ∈ Z[G ] is represented by thevector [a0 , a1 , . . . , a5 ].

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let g be a generator of Clk (3∞1 ∞2 ).
Let σ := σg . Thus G = hσi.
Let ζ6 := exp(2iπ/6).
The character χa represented by [a] is the one defined by
χa (σ) := ζ6a .
An element a0 + a1 σ + · · · + a5 σ 5 ∈ Z[G ] is represented by thevector [a0 , a1 , . . . , a5 ].

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let g be a generator of Clk (3∞1 ∞2 ).
Let σ := σg . Thus G = hσi.
Let ζ6 := exp(2iπ/6).
The character χa represented by [a] is the one defined by
χa (σ) := ζ6a .
An element a0 + a1 σ + · · · + a5 σ 5 ∈ Z[G ] is represented by thevector [a0 , a1 , . . . , a5 ].

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let p be a prime ideal of k, P a prime ideal of K such that
P is above p is above p.
Let θ ∈ K such that K = Q(θ) and assume thatp - (ZK : Z[θ]) .
Then the Frobenius of p is the unique element σ ∈ G such that
σ(θ) ≡ θN p

Xavier-François Roblot(mod P).

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let p be a prime ideal of k, P a prime ideal of K such that
P is above p is above p.
Let θ ∈ K such that K = Q(θ) and assume thatp - (ZK : Z[θ]) .
Then the Frobenius of p is the unique element σ ∈ G such that
σ(θ) ≡ θN p

Xavier-François Roblot(mod P).

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Let p be a prime ideal of k, P a prime ideal of K such that
P is above p is above p.
Let θ ∈ K such that K = Q(θ) and assume thatp - (ZK : Z[θ]) .
Then the Frobenius of p is the unique element σ ∈ G such that
σ(θ) ≡ θN p

Xavier-François Roblot(mod P).

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Recall that wK = 6 so the torsion group of K is generated by ζ6 .
Let N ∈ Z be such thatσ(ζ6 ) = ζ6N .
Then an element α ∈ K is such that K (α1/6 )/k is an abelian
extension iff
αN
αN−σ =σ(α)
is a 6-th power in K .

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP

Statement of the conjecture
Current status of the conjecture
Checking the conjecture on an example

The example
The strategy
The verification

Recall that wK = 6 so the torsion group of K is generated by ζ6 .
Let N ∈ Z be such thatσ(ζ6 ) = ζ6N .
Then an element α ∈ K is such that K (α1/6 )/k is an abelian
extension iff
αN
αN−σ =σ(α)
is a 6-th power in K .

Xavier-François Roblot

Checking the Brumer-Stark conjecture using PARI/GP