local-global problems laura...
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Local-global problems
Local-global problems
Definition.A global field is a finite extension of Q or a finite extension of Fp(t).
Definition.A number field k is a finite extension of Q.
Definition.An absolute value of a number field k is a function | | : k −→ Rsatisfying the following properties, for all x , y ∈ k .
(i) |x | ≥ 0, e |x | = 0 if and only if x = 0(ii) |xy | = |x ||y |(iii) |x + y | ≤ |x |+ |y |
Local-global problems
Definition.A global field is a finite extension of Q or a finite extension of Fp(t).
Definition.A number field k is a finite extension of Q.
Definition.An absolute value of a number field k is a function | | : k −→ Rsatisfying the following properties, for all x , y ∈ k .
(i) |x | ≥ 0, e |x | = 0 if and only if x = 0(ii) |xy | = |x ||y |(iii) |x + y | ≤ |x |+ |y |
Local-global problems
Definition.A global field is a finite extension of Q or a finite extension of Fp(t).
Definition.A number field k is a finite extension of Q.
Definition.An absolute value of a number field k is a function | | : k −→ Rsatisfying the following properties, for all x , y ∈ k .
(i) |x | ≥ 0, e |x | = 0 if and only if x = 0(ii) |xy | = |x ||y |(iii) |x + y | ≤ |x |+ |y |
Local-global problems
Examples.
• | |∞ the usual absolute value over Q• Let a =b
c , with b, c ∈ Z, coprime. Let p be a prime number. Assume
a = pl b′
c ′, (b′c ′, p) = 1
The function | |p, defined by |a|p := 1pl , is an absolute value of Q,
named the p-adic absolute value.
Local-global problems
Examples.
• | |∞ the usual absolute value over Q• Let a =b
c , with b, c ∈ Z, coprime. Let p be a prime number. Assume
a = pl b′
c ′, (b′c ′, p) = 1
The function | |p, defined by |a|p := 1pl , is an absolute value of Q,
named the p-adic absolute value.
Local-global problems
Examples.
• | |∞ the usual absolute value over Q• Let a =b
c , with b, c ∈ Z, coprime. Let p be a prime number. Assume
a = pl b′
c ′, (b′c ′, p) = 1
The function | |p, defined by |a|p := 1pl , is an absolute value of Q,
named the p-adic absolute value.
Local-global problems
Definition.We say that two absolute values of k are equivalent if they induce thesame topology over k .
Ostrowski’s Theorem.Every absolute value of Q is equivalent to one of the absolute values | |∞or | |p.
Definition.The field obtained as a completion of Q by the absolute value | |p iscalled p-adic field and it is denoted by Qp. The elements of Qp are calledp-adic numbers.
Local-global problems
Definition.We say that two absolute values of k are equivalent if they induce thesame topology over k .
Ostrowski’s Theorem.Every absolute value of Q is equivalent to one of the absolute values | |∞or | |p.
Definition.The field obtained as a completion of Q by the absolute value | |p iscalled p-adic field and it is denoted by Qp. The elements of Qp are calledp-adic numbers.
Local-global problems
Definition.We say that two absolute values of k are equivalent if they induce thesame topology over k .
Ostrowski’s Theorem.Every absolute value of Q is equivalent to one of the absolute values | |∞or | |p.
Definition.The field obtained as a completion of Q by the absolute value | |p iscalled p-adic field and it is denoted by Qp. The elements of Qp are calledp-adic numbers.
Local-global problems
Definition.We say that two absolute values of k are equivalent if they induce thesame topology over k .
Ostrowski’s Theorem.Every absolute value of Q is equivalent to one of the absolute values | |∞or | |p.
Definition.The field obtained as a completion of Q by the absolute value | |p iscalled p-adic field and it is denoted by Qp. The elements of Qp are calledp-adic numbers.
Local-global problems
Definition.A local field is a field obtained as a completion of a global field by one ofits absolute values.
In particular the fields Qp are local fields.
Local-global problems
Definition.A local field is a field obtained as a completion of a global field by one ofits absolute values.
In particular the fields Qp are local fields.
Local-global problems
Definition.A local field is a field obtained as a completion of a global field by one ofits absolute values.
In particular the fields Qp are local fields.
Local-global problems
Hasse Principle, 1923-1924.
Let k be a number field and let F (X1, ...,Xn) ∈ k[X1, ...,Xn] be aquadratic form. If F = 0 has a non-trivial solution in kv , for allcompletions kv of k , where v is a place of k , then F = 0 has a non-trivialsolution in k .
The assumption that F is isotropic in kv for all but finitely manycompletions implies the same conclusion.
Since then, many mathematicians have been concerned with similarso-called local-global problems, i.e. they have been questioning if, given aglobal field k , the validity of some properties for all but finitely many localfields kv could ensure the validity of the same properties for k .
Local-global problems
Hasse Principle, 1923-1924.
Let k be a number field and let F (X1, ...,Xn) ∈ k[X1, ...,Xn] be aquadratic form. If F = 0 has a non-trivial solution in kv , for allcompletions kv of k , where v is a place of k , then F = 0 has a non-trivialsolution in k .
The assumption that F is isotropic in kv for all but finitely manycompletions implies the same conclusion.
Since then, many mathematicians have been concerned with similarso-called local-global problems, i.e. they have been questioning if, given aglobal field k , the validity of some properties for all but finitely many localfields kv could ensure the validity of the same properties for k .
Local-global problems
Hasse Principle, 1923-1924.
Let k be a number field and let F (X1, ...,Xn) ∈ k[X1, ...,Xn] be aquadratic form. If F = 0 has a non-trivial solution in kv , for allcompletions kv of k , where v is a place of k , then F = 0 has a non-trivialsolution in k .
The assumption that F is isotropic in kv for all but finitely manycompletions implies the same conclusion.
Since then, many mathematicians have been concerned with similarso-called local-global problems, i.e. they have been questioning if, given aglobal field k , the validity of some properties for all but finitely many localfields kv could ensure the validity of the same properties for k .
Notation
Notation
k̄ the algebraic closure of k
Gk the absolute Galois group Gal(k̄/k)
Gk = {σ ∈ Aut(k̄)|σ(x) = x , for every x ∈ k}
Mk the set of places v ∈ k
Notation
k̄ the algebraic closure of k
Gk the absolute Galois group Gal(k̄/k)
Gk = {σ ∈ Aut(k̄)|σ(x) = x , for every x ∈ k}
Mk the set of places v ∈ k
Notation
k̄ the algebraic closure of k
Gk the absolute Galois group Gal(k̄/k)
Gk = {σ ∈ Aut(k̄)|σ(x) = x , for every x ∈ k}
Mk the set of places v ∈ k
Notation
A a commutative algebraic group defined over k
A[pl ] the pl -torsion subgroup of A
A[pl ] = {P ∈ A|plP = 0}
k(A[pl ]) the number field obtained by adding tok the coordinates of the points in A[pl ]
Notation
A a commutative algebraic group defined over k
A[pl ] the pl -torsion subgroup of A
A[pl ] = {P ∈ A|plP = 0}
k(A[pl ]) the number field obtained by adding tok the coordinates of the points in A[pl ]
Notation
A a commutative algebraic group defined over k
A[pl ] the pl -torsion subgroup of A
A[pl ] = {P ∈ A|plP = 0}
k(A[pl ]) the number field obtained by adding tok the coordinates of the points in A[pl ]
The Local-Global Divisibility Problem
The local-global divisibility problem
Local-Global Divisibility Problem. (Dvornicich, Zannier,2001)
Let P ∈ A(k). Suppose for all but finitely many v ∈ Mk , there existsDv ∈ A(kv ) such that P = plDv . Is it possible to conclude that thereexists D ∈ A(k) such that P = plD?
This problem has a cohomological interpretation.
The local-global divisibility problem
Local-Global Divisibility Problem. (Dvornicich, Zannier,2001)
Let P ∈ A(k). Suppose for all but finitely many v ∈ Mk , there existsDv ∈ A(kv ) such that P = plDv . Is it possible to conclude that thereexists D ∈ A(k) such that P = plD?
This problem has a cohomological interpretation.
The local-global divisibility problem
Local-Global Divisibility Problem. (Dvornicich, Zannier,2001)
Let P ∈ A(k). Suppose for all but finitely many v ∈ Mk , there existsDv ∈ A(kv ) such that P = plDv . Is it possible to conclude that thereexists D ∈ A(k) such that P = plD?
This problem has a cohomological interpretation.
The local-global divisibility problem
Local-Global Divisibility Problem. (Dvornicich, Zannier,2001)
Let P ∈ A(k). Suppose for all but finitely many v ∈ Mk , there existsDv ∈ A(kv ) such that P = plDv . Is it possible to conclude that thereexists D ∈ A(k) such that P = plD?
This problem has a cohomological interpretation.
The local-global divisibility problem
Local-Global Divisibility Problem. (Dvornicich, Zannier,2001)
Let P ∈ A(k). Suppose for all but finitely many v ∈ Mk , there existsDv ∈ A(kv ) such that P = plDv . Is it possible to conclude that thereexists D ∈ A(k) such that P = plD?
This problem has a cohomological interpretation.
The local-global divisibility problem
Definition.Let G be a group and let M be a G -module. A cocycle of G with valuesin M (or a crossed homomorphism of G in M) is a map
Z :G −→ Mσ 7→ Zσ
such thatZστ = Zσ + σ(Zτ ),
for every σ, τ ∈ G .
The cocycles of G with values in M form a group denoted by Z (G ,M).
The local-global divisibility problem
Definition.Let G be a group and let M be a G -module. A cocycle of G with valuesin M (or a crossed homomorphism of G in M) is a map
Z :G −→ Mσ 7→ Zσ
such thatZστ = Zσ + σ(Zτ ),
for every σ, τ ∈ G .
The cocycles of G with values in M form a group denoted by Z (G ,M).
The local-global divisibility problem
Definition.Let G be a group and let M be a G -module. A coboundary of G withvalue in M is a cocycle Z of G with value in M such that
Zσ = (σ − 1)A,
for some A ∈ M.
The coboundaries of G with values in M form a group denoted byB(G ,M).
The local-global divisibility problem
Definition.Let G be a group and let M be a G -module. A coboundary of G withvalue in M is a cocycle Z of G with value in M such that
Zσ = (σ − 1)A,
for some A ∈ M.
The coboundaries of G with values in M form a group denoted byB(G ,M).
The local-global divisibility problem
Definition.Let G be a group and let M be a G -module. The first cohomology groupof G with values in M is defined as the quotient Z (G ,M)/B(G ,M) andit is denoted by H1(G ,M).
The local-global divisibility problem
Let D ∈ A(k̄) such that plD = P . We can define a cocycle of Gk withvalues in A[pl ] by setting
Zσ := σ(D)− D, σ ∈ Gk .
Proposition.The class of Z is 0 in H1(Gk ,A[pl ]), if and only if there exists D ′ ∈ A(k)such that plD ′ = P .
CorollaryIf H1(Gk ,A[pl ]) = 0, then the local-global divisibility by pl holds in Aover k .
Let Σ ⊆ Mk , containing all the places v , for which the hypotheses of theproblem hold. Then Z vanishes in H1(Gkv ,A[pl ]), for every v ∈ Σ.
The local-global divisibility problem
Let D ∈ A(k̄) such that plD = P . We can define a cocycle of Gk withvalues in A[pl ] by setting
Zσ := σ(D)− D, σ ∈ Gk .
Proposition.The class of Z is 0 in H1(Gk ,A[pl ]), if and only if there exists D ′ ∈ A(k)such that plD ′ = P .
CorollaryIf H1(Gk ,A[pl ]) = 0, then the local-global divisibility by pl holds in Aover k .
Let Σ ⊆ Mk , containing all the places v , for which the hypotheses of theproblem hold. Then Z vanishes in H1(Gkv ,A[pl ]), for every v ∈ Σ.
The local-global divisibility problem
Let D ∈ A(k̄) such that plD = P . We can define a cocycle of Gk withvalues in A[pl ] by setting
Zσ := σ(D)− D, σ ∈ Gk .
Proposition.The class of Z is 0 in H1(Gk ,A[pl ]), if and only if there exists D ′ ∈ A(k)such that plD ′ = P .
CorollaryIf H1(Gk ,A[pl ]) = 0, then the local-global divisibility by pl holds in Aover k .
Let Σ ⊆ Mk , containing all the places v , for which the hypotheses of theproblem hold. Then Z vanishes in H1(Gkv ,A[pl ]), for every v ∈ Σ.
The local-global divisibility problem
Let D ∈ A(k̄) such that plD = P . We can define a cocycle of Gk withvalues in A[pl ] by setting
Zσ := σ(D)− D, σ ∈ Gk .
Proposition.The class of Z is 0 in H1(Gk ,A[pl ]), if and only if there exists D ′ ∈ A(k)such that plD ′ = P .
CorollaryIf H1(Gk ,A[pl ]) = 0, then the local-global divisibility by pl holds in Aover k .
Let Σ ⊆ Mk , containing all the places v , for which the hypotheses of theproblem hold. Then Z vanishes in H1(Gkv ,A[pl ]), for every v ∈ Σ.
The local-global divisibility problem
Let D ∈ A(k̄) such that plD = P . We can define a cocycle of Gk withvalues in A[pl ] by setting
Zσ := σ(D)− D, σ ∈ Gk .
Proposition.The class of Z is 0 in H1(Gk ,A[pl ]), if and only if there exists D ′ ∈ A(k)such that plD ′ = P .
CorollaryIf H1(Gk ,A[pl ]) = 0, then the local-global divisibility by pl holds in Aover k .
Let Σ ⊆ Mk , containing all the places v , for which the hypotheses of theproblem hold. Then Z vanishes in H1(Gkv ,A[pl ]), for every v ∈ Σ.
The local-global divisibility problem
The first local cohomology group of A over k is defined as
H1loc(G ,A[pl ]) :=
⋂v∈Σ ker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
where resv is the usual restriction map and G = Gal(k(A[pl ])/k).
Proposition. (Dvornicich, Zannier, 2001)
If H1loc(G ,A[pl ]) = 0, then the local-global divisibility by pl holds in A
over k .
The local-global divisibility problem
The first local cohomology group of A over k is defined as
H1loc(G ,A[pl ]) :=
⋂v∈Σ ker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
where resv is the usual restriction map and G = Gal(k(A[pl ])/k).
Proposition. (Dvornicich, Zannier, 2001)
If H1loc(G ,A[pl ]) = 0, then the local-global divisibility by pl holds in A
over k .
The local-global divisibility problem
The first local cohomology group of A over k is defined as
H1loc(G ,A[pl ]) :=
⋂v∈Σ ker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
where resv is the usual restriction map and G = Gal(k(A[pl ])/k).
Proposition. (Dvornicich, Zannier, 2001)
If H1loc(G ,A[pl ]) = 0, then the local-global divisibility by pl holds in A
over k .
The local-global divisibility problem
H1loc(G ,A[pl ]) =
⋂v∈Σ ker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
This definition is very similar to the one of the Tate-Shafarevich group
X(k ,A[pl ]) :=⋂
v∈Mkker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
The local-global divisibility problem
H1loc(G ,A[pl ]) =
⋂v∈Σ ker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
This definition is very similar to the one of the Tate-Shafarevich group
X(k ,A[pl ]) :=⋂
v∈Mkker{H1(Gk ,A[pl ])
resv−−−−→ H1(Gkv ,A[pl ])}.
Cassels’ question
Cassels’ question
Cassels’ Question, 1962.
Let k be a number field and E : y2 = x3 + bx + c an elliptic curve definedover k . Are the elements of X(k , E) infinitely divisible by a prime p whenconsidered as elements of the group H1(Gk , E)?
Proposition.If X(k , E [pl ]) = 0, for every l , then Cassels’ question has an affirmativeanswer for p.
Cassels’ question
Cassels’ Question, 1962.
Let k be a number field and E : y2 = x3 + bx + c an elliptic curve definedover k . Are the elements of X(k , E) infinitely divisible by a prime p whenconsidered as elements of the group H1(Gk , E)?
Proposition.If X(k , E [pl ]) = 0, for every l , then Cassels’ question has an affirmativeanswer for p.
Solutions
Solutions
(Tate, 1962)
Cassels’ question has an affirmative answer for the divisibility by p (onetime).
The question for the divisibility by powers of p remained open fordecades, for every p.
Solutions
(Tate, 1962)
Cassels’ question has an affirmative answer for the divisibility by p (onetime).
The question for the divisibility by powers of p remained open fordecades, for every p.
Solutions
Theorem. (P., Ranieri, Viada, 2012)
The local-global divisibility by pl holds in E over k , for allp > (3[k:Q]/2 + 1)2 and l ≥ 1.
Corollary. (P., Ranieri, Viada, 2012)
Cassels’ question has an affirmative answer over k for allp > (3[k:Q]/2 + 1)2.
Solutions
Theorem. (P., Ranieri, Viada, 2012)
The local-global divisibility by pl holds in E over k , for allp > (3[k:Q]/2 + 1)2 and l ≥ 1.
Corollary. (P., Ranieri, Viada, 2012)
Cassels’ question has an affirmative answer over k for allp > (3[k:Q]/2 + 1)2.
Solutions
Theorem. (P., Ranieri, Viada, 2012-2014)
The local-global divisibility by pl holds in E over Q, for all p ≥ 5 andl ≥ 1.
Corollary. (P., Ranieri, Viada, 2012-2014)
Cassels’ question has an affirmative answer over Q for all p ≥ 5.
A second proof.
Theorem. (Çiperiani, Stix, 2015)
Cassels’ question has an affirmative answer in elliptic curves defined overQ, for all p ≥ 11.
Solutions
Theorem. (P., Ranieri, Viada, 2012-2014)
The local-global divisibility by pl holds in E over Q, for all p ≥ 5 andl ≥ 1.
Corollary. (P., Ranieri, Viada, 2012-2014)
Cassels’ question has an affirmative answer over Q for all p ≥ 5.
A second proof.
Theorem. (Çiperiani, Stix, 2015)
Cassels’ question has an affirmative answer in elliptic curves defined overQ, for all p ≥ 11.
Solutions
Theorem. (P., Ranieri, Viada, 2012-2014)
The local-global divisibility by pl holds in E over Q, for all p ≥ 5 andl ≥ 1.
Corollary. (P., Ranieri, Viada, 2012-2014)
Cassels’ question has an affirmative answer over Q for all p ≥ 5.
A second proof.
Theorem. (Çiperiani, Stix, 2015)
Cassels’ question has an affirmative answer in elliptic curves defined overQ, for all p ≥ 11.
Solutions
Counterexamples
in elliptic curves over Q for all 2n, with n ≥ 2 (P., 2011);
in elliptic curves over Q for all 3n, with n ≥ 2 (Creutz, 2016).
Solutions
Counterexamples
in elliptic curves over Q for all 2n, with n ≥ 2 (P., 2011);
in elliptic curves over Q for all 3n, with n ≥ 2 (Creutz, 2016).
Solutions
Counterexamples
in elliptic curves over Q for all 2n, with n ≥ 2 (P., 2011);
in elliptic curves over Q for all 3n, with n ≥ 2 (Creutz, 2016).
Solutions
Theorem. (P., 2019)
Let p be a prime number. Let k be a number field and let A be acommutative algebraic group defined over k , with A[p] ' (Z/pZ)n.Assume that A[p] is an irreducible N-module or a direct sum of irreducibleN-modules, for every subnormal subgroup N of Gal(k(A[pl ])/k).If p > n
2 + 1, then the local-global divisibility by p holds in A over k andX(k ,A[p]) = 0.
Solutions
Theorem. (P., 2019)
Let p be a prime number. Let k be a number field and let A be acommutative algebraic group defined over k , with A[p] ' (Z/pZ)n.Assume that A[p] is an irreducible N-module or a direct sum of irreducibleN-modules, for every subnormal subgroup N of Gal(k(A[pl ])/k).If p > n
2 + 1, then the local-global divisibility by p holds in A over k andX(k ,A[p]) = 0.
Solutions
Theorem. (P., 2019)
Let p be a prime number. Let k be a number field and let A be acommutative algebraic group defined over k , with A[p] ' (Z/pZ)n.Assume that A[p] is an irreducible N-module or a direct sum of irreducibleN-modules, for every subnormal subgroup N of Gal(k(A[pl ])/k).If p > n
2 + 1, then the local-global divisibility by p holds in A over k andX(k ,A[p]) = 0.
Solutions
Theorem. (P., 2019)
Let p be a prime number. Let G be a group and let M = (Z/pZ)n aG -module.Assume that M is an irreducible N-module or a direct sum of irreducibleN-modules, for every subnormal subgroup N of G .
If p >(n2 + 1
)2, then H1(G ,M) = 0.
Solutions
Theorem. (P., 2019)
Let p be a prime number. Let G be a group and let M = (Z/pZ)n aG -module.Assume that M is an irreducible N-module or a direct sum of irreducibleN-modules, for every subnormal subgroup N of G .
If p >(n2 + 1
)2, then H1(G ,M) = 0.
Solutions
Theorem. (P., 2019)
Let p be a prime number. Let G be a group and let M = (Z/pZ)n aG -module.Assume that M is an irreducible N-module or a direct sum of irreducibleN-modules, for every subnormal subgroup N of G .
If p >(n2 + 1
)2, then H1(G ,M) = 0.
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