Local-global problems Laura...

Local-global problems

Laura [email protected]

4th Number Theory MeetingTorino, October 25 2019



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 |












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.


Examples.



a = pl b′

c ′, (b′c ′, p) = 1




Examples.



a = pl b′

c ′, (b′c ′, p) = 1




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.














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.








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





Notation





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[pl ] = {P ∈ A|plP = 0}


Notation



A[pl ] = {P ∈ A|plP = 0}


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.


















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).


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).


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).


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).


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).


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.



Zσ := σ(D)− D, σ ∈ Gk .






Zσ := σ(D)− D, σ ∈ Gk .






Zσ := σ(D)− D, σ ∈ Gk .






Zσ := σ(D)− D, σ ∈ Gk .





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 .



H1loc(G ,A[pl ]) :=






over k .



H1loc(G ,A[pl ]) :=






over k .


H1loc(G ,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 ])



H1loc(G ,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 ])


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





A second proof.



Solutions





A second proof.



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



Solutions

Counterexamples



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)



Solutions

Theorem. (P., 2019)



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)


If p >(n2 + 1

)2, then H1(G ,M) = 0.

Solutions

Theorem. (P., 2019)


If p >(n2 + 1

)2, then H1(G ,M) = 0.

Thank you for your attention!

Date post:	26-Oct-2020
Category:	Documents
Upload:	others
View:	9 times
Download:	0 times

Local-global problems Laura...

Documents