Abelian varieties over function fields and
independence of `-adic representations
Wojciech Gajda
Adam Mickiewicz University
Poznań, POLAND
September 2012
Warwick University
September 2012 Warwick University
1/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Plan
1
Definitions and Notation
2
Results
3
Proof sketch of the corollary
4
Monodromies for abelian varieties
5
An application to arithmetic
6
Proofs
September 2012 Warwick University
2/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Definitions and Notation
` - an odd prime, L = {` : odd primes}
K - a field (later on; finitely generated field)
GK = Gal(K̄ /K ) - absolute Galois group
Assume
for every ` ∈ L there is a representation (= cont. homomorphism)
η` : GK −→ Gln (Zl ), and n is independent of `
Denote by: η : GK −→
Q
`∈L Gln (Z` )
the map induced by η`0 s.
Definition
Q
• family (η` )`∈L is independent (over K ) if η(GK ) = `∈L η` (GK )
Q
• family (η` )`∈L is almost independent if η(H) = `∈L η` (H) for an
open subgroup H ⊂ GK .
September 2012 Warwick University
3/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Definitions and Notation
` - an odd prime, L = {` : odd primes}
K - a field (later on; finitely generated field)
GK = Gal(K̄ /K ) - absolute Galois group
Assume
for every ` ∈ L there is a representation (= cont. homomorphism)
η` : GK −→ Gln (Zl ), and n is independent of `
Denote by: η : GK −→
Q
`∈L Gln (Z` )
the map induced by η`0 s.
Definition
Q
• family (η` )`∈L is independent (over K ) if η(GK ) = `∈L η` (GK )
Q
• family (η` )`∈L is almost independent if η(H) = `∈L η` (H) for an
open subgroup H ⊂ GK .
September 2012 Warwick University
3/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Definitions and Notation
` - an odd prime, L = {` : odd primes}
K - a field (later on; finitely generated field)
GK = Gal(K̄ /K ) - absolute Galois group
Assume
for every ` ∈ L there is a representation (= cont. homomorphism)
η` : GK −→ Gln (Zl ), and n is independent of `
Denote by: η : GK −→
Q
`∈L Gln (Z` )
the map induced by η`0 s.
Definition
Q
• family (η` )`∈L is independent (over K ) if η(GK ) = `∈L η` (GK )
Q
• family (η` )`∈L is almost independent if η(H) = `∈L η` (H) for an
open subgroup H ⊂ GK .
September 2012 Warwick University
3/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Definitions and Notation
` - an odd prime, L = {` : odd primes}
K - a field (later on; finitely generated field)
GK = Gal(K̄ /K ) - absolute Galois group
Assume
for every ` ∈ L there is a representation (= cont. homomorphism)
η` : GK −→ Gln (Zl ), and n is independent of `
Denote by: η : GK −→
Q
`∈L Gln (Z` )
the map induced by η`0 s.
Definition
Q
• family (η` )`∈L is independent (over K ) if η(GK ) = `∈L η` (GK )
Q
• family (η` )`∈L is almost independent if η(H) = `∈L η` (H) for an
open subgroup H ⊂ GK .
September 2012 Warwick University
3/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Define: Kl = K (η` ) := K̄ ker(η` ) ,
then the family (η` )`∈L is independent iff the family of fields (K (η` ))`∈L
is K −linearly disjoint.
EXAMPLES
Let K be a finitely gen. field over Q, i.e., finite ext. of Q(t1 t2 , . . . , ts ).
(1) Let
` : GK −→ Z×
`
be the cyclotomic character. Classically known that (` )`∈L is
independent. Here K (` ) = K (µ`∞ ) is the cyclotomic extension.
(2) Let A/K be an abelian variety of dim. g and let
ρ`,A : GK −→ Gl2g (Z` ) = Aut(T` (A))
be the Tate module representation. Here K (ρ`,A ) = K (A[l ∞ ]) is the
field of `-division points.
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Define: Kl = K (η` ) := K̄ ker(η` ) ,
then the family (η` )`∈L is independent iff the family of fields (K (η` ))`∈L
is K −linearly disjoint.
EXAMPLES
Let K be a finitely gen. field over Q, i.e., finite ext. of Q(t1 t2 , . . . , ts ).
(1) Let
` : GK −→ Z×
`
be the cyclotomic character. Classically known that (` )`∈L is
independent. Here K (` ) = K (µ`∞ ) is the cyclotomic extension.
(2) Let A/K be an abelian variety of dim. g and let
ρ`,A : GK −→ Gl2g (Z` ) = Aut(T` (A))
be the Tate module representation. Here K (ρ`,A ) = K (A[l ∞ ]) is the
field of `-division points.
September 2012 Warwick University
4/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Define: Kl = K (η` ) := K̄ ker(η` ) ,
then the family (η` )`∈L is independent iff the family of fields (K (η` ))`∈L
is K −linearly disjoint.
EXAMPLES
Let K be a finitely gen. field over Q, i.e., finite ext. of Q(t1 t2 , . . . , ts ).
(1) Let
` : GK −→ Z×
`
be the cyclotomic character. Classically known that (` )`∈L is
independent. Here K (` ) = K (µ`∞ ) is the cyclotomic extension.
(2) Let A/K be an abelian variety of dim. g and let
ρ`,A : GK −→ Gl2g (Z` ) = Aut(T` (A))
be the Tate module representation. Here K (ρ`,A ) = K (A[l ∞ ]) is the
field of `-division points.
September 2012 Warwick University
4/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Define: Kl = K (η` ) := K̄ ker(η` ) ,
then the family (η` )`∈L is independent iff the family of fields (K (η` ))`∈L
is K −linearly disjoint.
EXAMPLES
Let K be a finitely gen. field over Q, i.e., finite ext. of Q(t1 t2 , . . . , ts ).
(1) Let
` : GK −→ Z×
`
be the cyclotomic character. Classically known that (` )`∈L is
independent. Here K (` ) = K (µ`∞ ) is the cyclotomic extension.
(2) Let A/K be an abelian variety of dim. g and let
ρ`,A : GK −→ Gl2g (Z` ) = Aut(T` (A))
be the Tate module representation. Here K (ρ`,A ) = K (A[l ∞ ]) is the
field of `-division points.
September 2012 Warwick University
4/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Igusa (1959)
If trdeg K > 0 and g = 1, then (ρ`,A )`∈L is almost independent.
Serre (1972)
If trdeg K = 0 and dim A = 1,then (ρ`,A )`∈L is almost independent.
Serre (1986)
Same true (as for elliptic curves) for dim A > 1.
Question of Serre (1991)
Is the family (ρ`,A )`∈L almost independent, for K finitely gen. over the
rationals ?
Gajda and Petersen (2011)
YES
September 2012 Warwick University
5/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Igusa (1959)
If trdeg K > 0 and g = 1, then (ρ`,A )`∈L is almost independent.
Serre (1972)
If trdeg K = 0 and dim A = 1,then (ρ`,A )`∈L is almost independent.
Serre (1986)
Same true (as for elliptic curves) for dim A > 1.
Question of Serre (1991)
Is the family (ρ`,A )`∈L almost independent, for K finitely gen. over the
rationals ?
Gajda and Petersen (2011)
YES
September 2012 Warwick University
5/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Igusa (1959)
If trdeg K > 0 and g = 1, then (ρ`,A )`∈L is almost independent.
Serre (1972)
If trdeg K = 0 and dim A = 1,then (ρ`,A )`∈L is almost independent.
Serre (1986)
Same true (as for elliptic curves) for dim A > 1.
Question of Serre (1991)
Is the family (ρ`,A )`∈L almost independent, for K finitely gen. over the
rationals ?
Gajda and Petersen (2011)
YES
September 2012 Warwick University
5/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Igusa (1959)
If trdeg K > 0 and g = 1, then (ρ`,A )`∈L is almost independent.
Serre (1972)
If trdeg K = 0 and dim A = 1,then (ρ`,A )`∈L is almost independent.
Serre (1986)
Same true (as for elliptic curves) for dim A > 1.
Question of Serre (1991)
Is the family (ρ`,A )`∈L almost independent, for K finitely gen. over the
rationals ?
Gajda and Petersen (2011)
YES
September 2012 Warwick University
5/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
Igusa (1959)
If trdeg K > 0 and g = 1, then (ρ`,A )`∈L is almost independent.
Serre (1972)
If trdeg K = 0 and dim A = 1,then (ρ`,A )`∈L is almost independent.
Serre (1986)
Same true (as for elliptic curves) for dim A > 1.
Question of Serre (1991)
Is the family (ρ`,A )`∈L almost independent, for K finitely gen. over the
rationals ?
Gajda and Petersen (2011)
YES
September 2012 Warwick University
5/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
(3) More general
K − a fin. gen. field over Q
X /K − a separated scheme of finite type over K
(q)
q
η`,X : GK −→ Glb (Q` ) = Aut(Het
(XK̄ , Q` )) the associated Galois
representation, where b is the qth Betti number.
Serre and Illusie (2010)
(q)
If trdeg K = 0, then the family (η`,X )`∈L is almost independent.
Question of Serre and Illusie (2010)
Is this still true if trdeg K > 0 ?
Relation with the Tate module - as Galois modules:
1
T` (A) ⊗ Q` = Het
(Ǎ, Q` (1)).
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
(3) More general
K − a fin. gen. field over Q
X /K − a separated scheme of finite type over K
(q)
q
η`,X : GK −→ Glb (Q` ) = Aut(Het
(XK̄ , Q` )) the associated Galois
representation, where b is the qth Betti number.
Serre and Illusie (2010)
(q)
If trdeg K = 0, then the family (η`,X )`∈L is almost independent.
Question of Serre and Illusie (2010)
Is this still true if trdeg K > 0 ?
Relation with the Tate module - as Galois modules:
1
T` (A) ⊗ Q` = Het
(Ǎ, Q` (1)).
September 2012 Warwick University
6/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
(3) More general
K − a fin. gen. field over Q
X /K − a separated scheme of finite type over K
(q)
q
η`,X : GK −→ Glb (Q` ) = Aut(Het
(XK̄ , Q` )) the associated Galois
representation, where b is the qth Betti number.
Serre and Illusie (2010)
(q)
If trdeg K = 0, then the family (η`,X )`∈L is almost independent.
Question of Serre and Illusie (2010)
Is this still true if trdeg K > 0 ?
Relation with the Tate module - as Galois modules:
1
T` (A) ⊗ Q` = Het
(Ǎ, Q` (1)).
September 2012 Warwick University
6/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Definitions and Notation
(3) More general
K − a fin. gen. field over Q
X /K − a separated scheme of finite type over K
(q)
q
η`,X : GK −→ Glb (Q` ) = Aut(Het
(XK̄ , Q` )) the associated Galois
representation, where b is the qth Betti number.
Serre and Illusie (2010)
(q)
If trdeg K = 0, then the family (η`,X )`∈L is almost independent.
Question of Serre and Illusie (2010)
Is this still true if trdeg K > 0 ?
Relation with the Tate module - as Galois modules:
1
T` (A) ⊗ Q` = Het
(Ǎ, Q` (1)).
September 2012 Warwick University
6/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
Plan
1
Definitions and Notation
2
Results
3
Proof sketch of the corollary
4
Monodromies for abelian varieties
5
An application to arithmetic
6
Proofs
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
Theorem A (G and Sebastian Petersen, to appear in Compositio)
If K is a finitely gen. field of char. zero and X is a separated scheme
(q)
of finite type over K , then the family (η`,X )`∈L is almost independent.
Important ingredients of the proof (more details below):
• Theorem B, below - an extension of Serre’s criterion for linear
independence of the family (η` )`∈L . We use the classical paper by Katz
and Lang on the π1et , for X smooth and proper
• for non smooth X we use alterations of de Jong - as in Katz and Laumon
paper (1996).
• a specialization argument - reduction to the number field case (= Serre’s
theorem of 2010).
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
Theorem A (G and Sebastian Petersen, to appear in Compositio)
If K is a finitely gen. field of char. zero and X is a separated scheme
(q)
of finite type over K , then the family (η`,X )`∈L is almost independent.
Important ingredients of the proof (more details below):
• Theorem B, below - an extension of Serre’s criterion for linear
independence of the family (η` )`∈L . We use the classical paper by Katz
and Lang on the π1et , for X smooth and proper
• for non smooth X we use alterations of de Jong - as in Katz and Laumon
paper (1996).
• a specialization argument - reduction to the number field case (= Serre’s
theorem of 2010).
September 2012 Warwick University
8/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
Theorem A (G and Sebastian Petersen, to appear in Compositio)
If K is a finitely gen. field of char. zero and X is a separated scheme
(q)
of finite type over K , then the family (η`,X )`∈L is almost independent.
Important ingredients of the proof (more details below):
• Theorem B, below - an extension of Serre’s criterion for linear
independence of the family (η` )`∈L . We use the classical paper by Katz
and Lang on the π1et , for X smooth and proper
• for non smooth X we use alterations of de Jong - as in Katz and Laumon
paper (1996).
• a specialization argument - reduction to the number field case (= Serre’s
theorem of 2010).
September 2012 Warwick University
8/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
Theorem A (G and Sebastian Petersen, to appear in Compositio)
If K is a finitely gen. field of char. zero and X is a separated scheme
(q)
of finite type over K , then the family (η`,X )`∈L is almost independent.
Important ingredients of the proof (more details below):
• Theorem B, below - an extension of Serre’s criterion for linear
independence of the family (η` )`∈L . We use the classical paper by Katz
and Lang on the π1et , for X smooth and proper
• for non smooth X we use alterations of de Jong - as in Katz and Laumon
paper (1996).
• a specialization argument - reduction to the number field case (= Serre’s
theorem of 2010).
September 2012 Warwick University
8/
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
It follows (from Theorem A):
Corollary (G and S.Petersen)
If A/K is an abelian variety over a fin. gen. field of zero char, then
there exists a finite extension E/K such that the family of division fields
(E(A[l ∞ ]))`∈L is E−linearly disjoint.
Remark (work in progress in positive char) a similar theorem holds
over char. p > 0:
Theorem (G.Böckle, G.W., S.Petersen, 2012)
(q)
For K a fin. gen. field of char. p > 0, then the family (η`,X )`∈L is almost
independent over the field F̄p K .
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Results
It follows (from Theorem A):
Corollary (G and S.Petersen)
If A/K is an abelian variety over a fin. gen. field of zero char, then
there exists a finite extension E/K such that the family of division fields
(E(A[l ∞ ]))`∈L is E−linearly disjoint.
Remark (work in progress in positive char) a similar theorem holds
over char. p > 0:
Theorem (G.Böckle, G.W., S.Petersen, 2012)
(q)
For K a fin. gen. field of char. p > 0, then the family (η`,X )`∈L is almost
independent over the field F̄p K .
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Plan
1
Definitions and Notation
2
Results
3
Proof sketch of the corollary
4
Monodromies for abelian varieties
5
An application to arithmetic
6
Proofs
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
For E, any field extension of Q we define the constant field of E:
κE := {x ∈ E :
x is algebraic over Q}
For an algebraic extension E/K we have:
E
Q̄E
κE
κE K
Q̄K
κK
K
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Wojciech Gajda Adam Mickiewicz UniversityPozna
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() function fields and independence of `-adic representations37
Proof sketch of the corollary
For E, any field extension of Q we define the constant field of E:
κE := {x ∈ E :
x is algebraic over Q}
For an algebraic extension E/K we have:
E
Q̄E
κE
κE K
Q̄K
κK
K
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
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over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Denote
K - the function field of an affine, normal Q-variety S
S (E) - the normalization of S in E
Call
• E/K constant if κE K = E
• E/K geometric if κE = κK
0
• E/K unramified along S if the map S (E ) −→ S is etale for every
finite subextension E 0 /K
• KS,nr - the maximal unramified along S ext. of K
• Snr - the normalization of S in KS,nr .
Note that
π1et (S) = Gal(KS,nr /K )
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Denote
K - the function field of an affine, normal Q-variety S
S (E) - the normalization of S in E
Call
• E/K constant if κE K = E
• E/K geometric if κE = κK
0
• E/K unramified along S if the map S (E ) −→ S is etale for every
finite subextension E 0 /K
• KS,nr - the maximal unramified along S ext. of K
• Snr - the normalization of S in KS,nr .
Note that
π1et (S) = Gal(KS,nr /K )
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Denote
K - the function field of an affine, normal Q-variety S
S (E) - the normalization of S in E
Call
• E/K constant if κE K = E
• E/K geometric if κE = κK
0
• E/K unramified along S if the map S (E ) −→ S is etale for every
finite subextension E 0 /K
• KS,nr - the maximal unramified along S ext. of K
• Snr - the normalization of S in KS,nr .
Note that
π1et (S) = Gal(KS,nr /K )
September 2012 Warwick University
12 /
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Denote
K - the function field of an affine, normal Q-variety S
S (E) - the normalization of S in E
Call
• E/K constant if κE K = E
• E/K geometric if κE = κK
0
• E/K unramified along S if the map S (E ) −→ S is etale for every
finite subextension E 0 /K
• KS,nr - the maximal unramified along S ext. of K
• Snr - the normalization of S in KS,nr .
Note that
π1et (S) = Gal(KS,nr /K )
September 2012 Warwick University
12 /
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Denote
K - the function field of an affine, normal Q-variety S
S (E) - the normalization of S in E
Call
• E/K constant if κE K = E
• E/K geometric if κE = κK
0
• E/K unramified along S if the map S (E ) −→ S is etale for every
finite subextension E 0 /K
• KS,nr - the maximal unramified along S ext. of K
• Snr - the normalization of S in KS,nr .
Note that
π1et (S) = Gal(KS,nr /K )
September 2012 Warwick University
12 /
Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Facts (EGA V)
For an abelian variety A/K
• (replacing S by an affine open, if necessary) A extends to an
abelian scheme A −→ S (i.e., Aη = A)
• A[n] −→ S is a finite etale group scheme (since residue chars of
S are 0)
Hence each
ρl,A : GK −→ Gln (Ql )
factors through
π1et (S) = G(KS,nr /K )
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Facts (EGA V)
For an abelian variety A/K
• (replacing S by an affine open, if necessary) A extends to an
abelian scheme A −→ S (i.e., Aη = A)
• A[n] −→ S is a finite etale group scheme (since residue chars of
S are 0)
Hence each
ρl,A : GK −→ Gln (Ql )
factors through
π1et (S) = G(KS,nr /K )
September 2012 Warwick University
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Wojciech Gajda Adam Mickiewicz UniversityPozna
Abelianń,
varieties
POLAND
over
() function fields and independence of `-adic representations37
Proof sketch of the corollary
Facts (EGA V)
For an abelian variety A/K
• (replacing S by an affine open, if necessary) A extends to an
abelian scheme A −→ S (i.e., Aη = A)
• A[n] −→ S is a finite etale group scheme (since residue chars of
S are 0)
Hence each
ρl,A : GK −→ Gln (Ql )
factors through
π1et (S) = G(KS,nr /K )
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Proof sketch of the corollary
Facts (EGA V)
For an abelian variety A/K
• (replacing S by an affine open, if necessary) A extends to an
abelian scheme A −→ S (i.e., Aη = A)
• A[n] −→ S is a finite etale group scheme (since residue chars of
S are 0)
Hence each
ρl,A : GK −→ Gln (Ql )
factors through
π1et (S) = G(KS,nr /K )
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Proof sketch of the corollary
• for P ∈ S a closed point
AP := A ×S k (P) - the corr. fibre at P over the number field k (P)
Pnr - a point of Snr over P. Easy to check: k (Pnr ) = Q̄. We have a
specialization diagram:
G(KS,nr /K )
ρ`,A
O
/ Aut(V` (A))
(,→)
D(Pnr ) := DKS,nr /K (Pnr )
≡
G(k (Pnr )/k (P))
ρ`,AP
≡
/ Aut(V` (AP ))
Serre’s result (1986) implies that (ρ`,AP )`∈L is almost independent,
hence (ρ`,A |D(Pnr ))`∈L is almost independent.
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Proof sketch of the corollary
• for P ∈ S a closed point
AP := A ×S k (P) - the corr. fibre at P over the number field k (P)
Pnr - a point of Snr over P. Easy to check: k (Pnr ) = Q̄. We have a
specialization diagram:
G(KS,nr /K )
ρ`,A
O
/ Aut(V` (A))
(,→)
D(Pnr ) := DKS,nr /K (Pnr )
≡
G(k (Pnr )/k (P))
ρ`,AP
≡
/ Aut(V` (AP ))
Serre’s result (1986) implies that (ρ`,AP )`∈L is almost independent,
hence (ρ`,A |D(Pnr ))`∈L is almost independent.
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Proof sketch of the corollary
• for P ∈ S a closed point
AP := A ×S k (P) - the corr. fibre at P over the number field k (P)
Pnr - a point of Snr over P. Easy to check: k (Pnr ) = Q̄. We have a
specialization diagram:
G(KS,nr /K )
ρ`,A
O
/ Aut(V` (A))
(,→)
D(Pnr ) := DKS,nr /K (Pnr )
≡
G(k (Pnr )/k (P))
ρ`,AP
≡
/ Aut(V` (AP ))
Serre’s result (1986) implies that (ρ`,AP )`∈L is almost independent,
hence (ρ`,A |D(Pnr ))`∈L is almost independent.
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Proof sketch of the corollary
• for P ∈ S a closed point
AP := A ×S k (P) - the corr. fibre at P over the number field k (P)
Pnr - a point of Snr over P. Easy to check: k (Pnr ) = Q̄. We have a
specialization diagram:
G(KS,nr /K )
ρ`,A
O
/ Aut(V` (A))
(,→)
D(Pnr ) := DKS,nr /K (Pnr )
≡
G(k (Pnr )/k (P))
ρ`,AP
≡
/ Aut(V` (AP ))
Serre’s result (1986) implies that (ρ`,AP )`∈L is almost independent,
hence (ρ`,A |D(Pnr ))`∈L is almost independent.
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Proof sketch of the corollary
• for P ∈ S a closed point
AP := A ×S k (P) - the corr. fibre at P over the number field k (P)
Pnr - a point of Snr over P. Easy to check: k (Pnr ) = Q̄. We have a
specialization diagram:
G(KS,nr /K )
ρ`,A
O
/ Aut(V` (A))
(,→)
D(Pnr ) := DKS,nr /K (Pnr )
≡
G(k (Pnr )/k (P))
ρ`,AP
≡
/ Aut(V` (AP ))
Serre’s result (1986) implies that (ρ`,AP )`∈L is almost independent,
hence (ρ`,A |D(Pnr ))`∈L is almost independent.
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Proof sketch of the corollary
Remarks
• in terms of fields this means:
K`1 K`2 . . . K`s ∩ K`s+1
is geometric over K
(after a finite extension), for any `1 < `2 < · · · < `s+1 in L
• We need (and prove) a much stronger fact:
K`1 K`2 . . . K`s ∩ K`s+1 = K
(after a finite extension), for any `1 < `2 < · · · < `s+1 in L
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Proof sketch of the corollary
Remarks
• in terms of fields this means:
K`1 K`2 . . . K`s ∩ K`s+1
is geometric over K
(after a finite extension), for any `1 < `2 < · · · < `s+1 in L
• We need (and prove) a much stronger fact:
K`1 K`2 . . . K`s ∩ K`s+1 = K
(after a finite extension), for any `1 < `2 < · · · < `s+1 in L
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Proof sketch of the corollary
Remarks
• in terms of fields this means:
K`1 K`2 . . . K`s ∩ K`s+1
is geometric over K
(after a finite extension), for any `1 < `2 < · · · < `s+1 in L
• We need (and prove) a much stronger fact:
K`1 K`2 . . . K`s ∩ K`s+1 = K
(after a finite extension), for any `1 < `2 < · · · < `s+1 in L
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Proof sketch of the corollary
Theorem B (G and S.Petersen)
Let S be a normal Q-variety and let K = Q(S).
Consider a family of continuous representations
η` : π1et (S) −→ Gln (Z` ),
where ` ∈ L
(∗) Assume
there is a point P̂ ∈ Snr such that (η` |D)`∈L is almost independent for
an open subgroup D ⊂ D(P̂).
Then the family (η` )`∈L is almost independent.
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Proof sketch of the corollary
Theorem B (G and S.Petersen)
Let S be a normal Q-variety and let K = Q(S).
Consider a family of continuous representations
η` : π1et (S) −→ Gln (Z` ),
where ` ∈ L
(∗) Assume
there is a point P̂ ∈ Snr such that (η` |D)`∈L is almost independent for
an open subgroup D ⊂ D(P̂).
Then the family (η` )`∈L is almost independent.
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Proof sketch of the corollary
Theorem B (G and S.Petersen)
Let S be a normal Q-variety and let K = Q(S).
Consider a family of continuous representations
η` : π1et (S) −→ Gln (Z` ),
where ` ∈ L
(∗) Assume
there is a point P̂ ∈ Snr such that (η` |D)`∈L is almost independent for
an open subgroup D ⊂ D(P̂).
Then the family (η` )`∈L is almost independent.
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Proof sketch of the corollary
Theorem B (G and S.Petersen)
Let S be a normal Q-variety and let K = Q(S).
Consider a family of continuous representations
η` : π1et (S) −→ Gln (Z` ),
where ` ∈ L
(∗) Assume
there is a point P̂ ∈ Snr such that (η` |D)`∈L is almost independent for
an open subgroup D ⊂ D(P̂).
Then the family (η` )`∈L is almost independent.
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Proof sketch of the corollary
Theorem B (G and S.Petersen)
Let S be a normal Q-variety and let K = Q(S).
Consider a family of continuous representations
η` : π1et (S) −→ Gln (Z` ),
where ` ∈ L
(∗) Assume
there is a point P̂ ∈ Snr such that (η` |D)`∈L is almost independent for
an open subgroup D ⊂ D(P̂).
Then the family (η` )`∈L is almost independent.
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Proof sketch of the corollary
Remarks
• Theorem B may seem surprising, since D(P̂) is far from being
open in the group π1et (S)
• Theorem B implies Theorem A by the base change theorems in
cohomology due to Katz and Laumon (1996) and by Serre and
Illusie result (2010).
• Theorem A implies Corollary for abelian varieties
• Proof of Theorem B is a blend of geometric class field theory
(Katz and Lang (1986)) and the group theory results of M.Nori for
subgroups in Gln ’s of finite fields.
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Proof sketch of the corollary
Remarks
• Theorem B may seem surprising, since D(P̂) is far from being
open in the group π1et (S)
• Theorem B implies Theorem A by the base change theorems in
cohomology due to Katz and Laumon (1996) and by Serre and
Illusie result (2010).
• Theorem A implies Corollary for abelian varieties
• Proof of Theorem B is a blend of geometric class field theory
(Katz and Lang (1986)) and the group theory results of M.Nori for
subgroups in Gln ’s of finite fields.
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Proof sketch of the corollary
Remarks
• Theorem B may seem surprising, since D(P̂) is far from being
open in the group π1et (S)
• Theorem B implies Theorem A by the base change theorems in
cohomology due to Katz and Laumon (1996) and by Serre and
Illusie result (2010).
• Theorem A implies Corollary for abelian varieties
• Proof of Theorem B is a blend of geometric class field theory
(Katz and Lang (1986)) and the group theory results of M.Nori for
subgroups in Gln ’s of finite fields.
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Proof sketch of the corollary
Remarks
• Theorem B may seem surprising, since D(P̂) is far from being
open in the group π1et (S)
• Theorem B implies Theorem A by the base change theorems in
cohomology due to Katz and Laumon (1996) and by Serre and
Illusie result (2010).
• Theorem A implies Corollary for abelian varieties
• Proof of Theorem B is a blend of geometric class field theory
(Katz and Lang (1986)) and the group theory results of M.Nori for
subgroups in Gln ’s of finite fields.
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Monodromies for abelian varieties
Plan
1
Definitions and Notation
2
Results
3
Proof sketch of the corollary
4
Monodromies for abelian varieties
5
An application to arithmetic
6
Proofs
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Monodromies for abelian varieties
Problem
Compute images of ρ`,A
¯ and ρ`,A in terms of linear algebraic groups
Serre (1972)
If A/F is a non-CM elliptic curve (where F is a # field), then
Im ρ`,A
¯ = GL2 (F` ) for almost all `.
Serre (1986)
If A/F is a principally polarized Abelian variety, g = 2, 6, or is an odd
integer, and End A = Z, then (Im ρ`,A
¯ )0 = Sp2g (F` ) for almost all `.
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Monodromies for abelian varieties
Problem
Compute images of ρ`,A
¯ and ρ`,A in terms of linear algebraic groups
Serre (1972)
If A/F is a non-CM elliptic curve (where F is a # field), then
Im ρ`,A
¯ = GL2 (F` ) for almost all `.
Serre (1986)
If A/F is a principally polarized Abelian variety, g = 2, 6, or is an odd
integer, and End A = Z, then (Im ρ`,A
¯ )0 = Sp2g (F` ) for almost all `.
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Monodromies for abelian varieties
Problem
Compute images of ρ`,A
¯ and ρ`,A in terms of linear algebraic groups
Serre (1972)
If A/F is a non-CM elliptic curve (where F is a # field), then
Im ρ`,A
¯ = GL2 (F` ) for almost all `.
Serre (1986)
If A/F is a principally polarized Abelian variety, g = 2, 6, or is an odd
integer, and End A = Z, then (Im ρ`,A
¯ )0 = Sp2g (F` ) for almost all `.
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Monodromies for abelian varieties
Ribet (1976)
If A/F is a principally polarized GL2 −Abelian variety (i.e.,
End A ⊗ Q = E is a tot. real number field [E : Q] = g), then
Im ρ`,A
¯ = {M ∈ GL2 (OE /`) :
det M ∈ F×
` }
for almost all `.
Less classical result:
C.Hall (2009)
Let A/F be a principally polarized Abelian variety and End A = Z.
Assume that there is a prime of OF at which A has semistable
reduction of toric dimension one. Then (Im ρ`,A
¯ )0 = Sp2g (F` ) for
almost all `.
Theorems of Serre and Hall extend to finitely generated fields.
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Monodromies for abelian varieties
Ribet (1976)
If A/F is a principally polarized GL2 −Abelian variety (i.e.,
End A ⊗ Q = E is a tot. real number field [E : Q] = g), then
Im ρ`,A
¯ = {M ∈ GL2 (OE /`) :
det M ∈ F×
` }
for almost all `.
Less classical result:
C.Hall (2009)
Let A/F be a principally polarized Abelian variety and End A = Z.
Assume that there is a prime of OF at which A has semistable
reduction of toric dimension one. Then (Im ρ`,A
¯ )0 = Sp2g (F` ) for
almost all `.
Theorems of Serre and Hall extend to finitely generated fields.
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Monodromies for abelian varieties
Ribet (1976)
If A/F is a principally polarized GL2 −Abelian variety (i.e.,
End A ⊗ Q = E is a tot. real number field [E : Q] = g), then
Im ρ`,A
¯ = {M ∈ GL2 (OE /`) :
det M ∈ F×
` }
for almost all `.
Less classical result:
C.Hall (2009)
Let A/F be a principally polarized Abelian variety and End A = Z.
Assume that there is a prime of OF at which A has semistable
reduction of toric dimension one. Then (Im ρ`,A
¯ )0 = Sp2g (F` ) for
almost all `.
Theorems of Serre and Hall extend to finitely generated fields.
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Monodromies for abelian varieties
Assume:
K - a finitely generated field (over its prime field), e.g.,
K = F (S) the function field of a smooth, geom. connected variety S
over a global field k
A/K - an abelian variety over K , e.g.,
A → S is an abelian scheme over S, and Aη = A.
Problem
Compute images of Galois representations attached to A at `
in terms of linear algebraic groups (A over a finitely generated field K ).
A related question
Compare images of Galois at the generic Aη and special fibres
As at closed points s ∈ S.
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Monodromies for abelian varieties
Assume:
K - a finitely generated field (over its prime field), e.g.,
K = F (S) the function field of a smooth, geom. connected variety S
over a global field k
A/K - an abelian variety over K , e.g.,
A → S is an abelian scheme over S, and Aη = A.
Problem
Compute images of Galois representations attached to A at `
in terms of linear algebraic groups (A over a finitely generated field K ).
A related question
Compare images of Galois at the generic Aη and special fibres
As at closed points s ∈ S.
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Monodromies for abelian varieties
Assume:
K - a finitely generated field (over its prime field), e.g.,
K = F (S) the function field of a smooth, geom. connected variety S
over a global field k
A/K - an abelian variety over K , e.g.,
A → S is an abelian scheme over S, and Aη = A.
Problem
Compute images of Galois representations attached to A at `
in terms of linear algebraic groups (A over a finitely generated field K ).
A related question
Compare images of Galois at the generic Aη and special fibres
As at closed points s ∈ S.
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Monodromies for abelian varieties
Abelian varieties of Hall type
Definition
A/K is of Hall type if:
(1) End A = Z and
(2) there is a discrete valuation v at K such that A has
semistable reduction of toric dimension one at v
Recall:
Condition (2) means that there is an exact sequence of group
schemes:
1 −→ T −→ Nvo −→ B −→ 0
T = Gm - a torus over κ(v ) = Ov /mv
B - an abelian variety over κ(v )
N −→ Spec Ov - the Neron model of A over Ov = integers of Kv ,
Nv = N ⊗Ov κ(v ) - the special fibre.
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Monodromies for abelian varieties
Abelian varieties of Hall type
Definition
A/K is of Hall type if:
(1) End A = Z and
(2) there is a discrete valuation v at K such that A has
semistable reduction of toric dimension one at v
Recall:
Condition (2) means that there is an exact sequence of group
schemes:
1 −→ T −→ Nvo −→ B −→ 0
T = Gm - a torus over κ(v ) = Ov /mv
B - an abelian variety over κ(v )
N −→ Spec Ov - the Neron model of A over Ov = integers of Kv ,
Nv = N ⊗Ov κ(v ) - the special fibre.
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Monodromies for abelian varieties
Abelian varieties of Hall type
Definition
A/K is of Hall type if:
(1) End A = Z and
(2) there is a discrete valuation v at K such that A has
semistable reduction of toric dimension one at v
Recall:
Condition (2) means that there is an exact sequence of group
schemes:
1 −→ T −→ Nvo −→ B −→ 0
T = Gm - a torus over κ(v ) = Ov /mv
B - an abelian variety over κ(v )
N −→ Spec Ov - the Neron model of A over Ov = integers of Kv ,
Nv = N ⊗Ov κ(v ) - the special fibre.
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Monodromies for abelian varieties
Example - C.Hall
Let
f ∈ Z[x] - a monic, square-free polynomial, deg f = n ≥ 5
Cf - smooth, projective curve; affine part y 2 = f (x)
A = Jac(Cf )
Properties
(a) Zarhin proved in 2007: If Gal(Spl(f )/Q) = Sn , then End A = Z.
(b) A has semistable reduction of toric dim. one at a prime p :
if f̄ = f1 f2 mod p, (f1 , f2 ) = 1 and f1 = (x − α)2 with α ∈ Fp , and f2
is square-free of degree n−2.
Kowalski proved using the large sieve (2009): most polynomials f
have properties (a) and (b).
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Monodromies for abelian varieties
Example - C.Hall
Let
f ∈ Z[x] - a monic, square-free polynomial, deg f = n ≥ 5
Cf - smooth, projective curve; affine part y 2 = f (x)
A = Jac(Cf )
Properties
(a) Zarhin proved in 2007: If Gal(Spl(f )/Q) = Sn , then End A = Z.
(b) A has semistable reduction of toric dim. one at a prime p :
if f̄ = f1 f2 mod p, (f1 , f2 ) = 1 and f1 = (x − α)2 with α ∈ Fp , and f2
is square-free of degree n−2.
Kowalski proved using the large sieve (2009): most polynomials f
have properties (a) and (b).
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Monodromies for abelian varieties
Results
Theorem C (Arias-de-Reyna, G., to appear in JPAA)
Every abelian variety A/K of Hall type defined over a finitely
generated field (of arbitrary characteristic) has big monodromy,
i.e., the image Im ρ`,A
¯ contains Sp2g (F` ), for ` >> 0.
• Hall proved Theorem C for number fields in 2009.
Proof of Theorem C uses:
- A[l] is a simple Fl [Im ρ̄l ]-module
- Im ρ̄l contains a transvection, i.e., an unipotent endomorphism of
2g
Fl with dim Eig(u, 1) = 2g − 1
- a group theory result of Hall (replacing Lie algebras)
- induction over the transcedence deg. of K
- a technically tricky specialization argument (if char= 0).
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Monodromies for abelian varieties
Results
Theorem C (Arias-de-Reyna, G., to appear in JPAA)
Every abelian variety A/K of Hall type defined over a finitely
generated field (of arbitrary characteristic) has big monodromy,
i.e., the image Im ρ`,A
¯ contains Sp2g (F` ), for ` >> 0.
• Hall proved Theorem C for number fields in 2009.
Proof of Theorem C uses:
- A[l] is a simple Fl [Im ρ̄l ]-module
- Im ρ̄l contains a transvection, i.e., an unipotent endomorphism of
2g
Fl with dim Eig(u, 1) = 2g − 1
- a group theory result of Hall (replacing Lie algebras)
- induction over the transcedence deg. of K
- a technically tricky specialization argument (if char= 0).
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Monodromies for abelian varieties
Results
Theorem C (Arias-de-Reyna, G., to appear in JPAA)
Every abelian variety A/K of Hall type defined over a finitely
generated field (of arbitrary characteristic) has big monodromy,
i.e., the image Im ρ`,A
¯ contains Sp2g (F` ), for ` >> 0.
• Hall proved Theorem C for number fields in 2009.
Proof of Theorem C uses:
- A[l] is a simple Fl [Im ρ̄l ]-module
- Im ρ̄l contains a transvection, i.e., an unipotent endomorphism of
2g
Fl with dim Eig(u, 1) = 2g − 1
- a group theory result of Hall (replacing Lie algebras)
- induction over the transcedence deg. of K
- a technically tricky specialization argument (if char= 0).
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Plan
1
Definitions and Notation
2
Results
3
Proof sketch of the corollary
4
Monodromies for abelian varieties
5
An application to arithmetic
6
Proofs
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Notation
A/K an abelian variety over a field K
GKe := GK × GK × · · · × GK ,
K̄ <σ> ,
K (σ) :=
for σ ∈
by coordinates of σ.
GKe
for
e≥1
the subgroup < σ >⊂ GK is generated
Geyer-Jarden conjecture (1978)
(a) For almost all σ ∈ GK , the group A(K (σ))Tors is infinite
- moreover A(K (σ))[l] 6= 0 for infinitely many l
(almost - in the sense of Haar measure on GK )
(b) For e ≥ 2 and almost all σ ∈ GKe , the group A(K (σ))Tors is finite.
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Notation
A/K an abelian variety over a field K
GKe := GK × GK × · · · × GK ,
K̄ <σ> ,
K (σ) :=
for σ ∈
by coordinates of σ.
GKe
for
e≥1
the subgroup < σ >⊂ GK is generated
Geyer-Jarden conjecture (1978)
(a) For almost all σ ∈ GK , the group A(K (σ))Tors is infinite
- moreover A(K (σ))[l] 6= 0 for infinitely many l
(almost - in the sense of Haar measure on GK )
(b) For e ≥ 2 and almost all σ ∈ GKe , the group A(K (σ))Tors is finite.
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Notation
A/K an abelian variety over a field K
GKe := GK × GK × · · · × GK ,
K̄ <σ> ,
K (σ) :=
for σ ∈
by coordinates of σ.
GKe
for
e≥1
the subgroup < σ >⊂ GK is generated
Geyer-Jarden conjecture (1978)
(a) For almost all σ ∈ GK , the group A(K (σ))Tors is infinite
- moreover A(K (σ))[l] 6= 0 for infinitely many l
(almost - in the sense of Haar measure on GK )
(b) For e ≥ 2 and almost all σ ∈ GKe , the group A(K (σ))Tors is finite.
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Geyer-Jarden conjecture - state of art
Geyer-Jarden, 1978
(a) and (b) true for elliptic curves
Jacobson-Jarden, 2001
(b) is true over K of char = 0
Geyer-Jarden, 2005
(a) holds true for K a number field, over a finite extension L/K ,
i.e., there is L/K s.t. A(L(σ))Tors is infinite for almost all σ ∈ GL .
D.Zywina proved in 2010 that one can take L = K .
GJC has been open for dim A > 1 and char > 0.
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Geyer-Jarden conjecture - state of art
Geyer-Jarden, 1978
(a) and (b) true for elliptic curves
Jacobson-Jarden, 2001
(b) is true over K of char = 0
Geyer-Jarden, 2005
(a) holds true for K a number field, over a finite extension L/K ,
i.e., there is L/K s.t. A(L(σ))Tors is infinite for almost all σ ∈ GL .
D.Zywina proved in 2010 that one can take L = K .
GJC has been open for dim A > 1 and char > 0.
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Theorem D(Arias-de-Reyna, G., Petersen, to appear in Math. Nachr.)
The GJC holds true for all abelian varieties with big monodromy.
In particular Theorem D ( + Theorem C + extension of Serre’s
theorem to fin. gen. fields) imply:
Corollary
The GJC is true:
(1) for A/K of Hall type, for K fin. gen. (of arbitrary characteristic)
(2) for A/K of dim 2, 6 or odd, with End A = Z, for K fin. gen. and
char K = 0.
Important proof ingredients of Theorem D:
- the classical lemma of Borel and Cantelli of measure theory
- counting of certain symplectic matricies (e.g., with eigenvalue 1) in
the Galois image - a delicate matter, if char > 0.
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Theorem D(Arias-de-Reyna, G., Petersen, to appear in Math. Nachr.)
The GJC holds true for all abelian varieties with big monodromy.
In particular Theorem D ( + Theorem C + extension of Serre’s
theorem to fin. gen. fields) imply:
Corollary
The GJC is true:
(1) for A/K of Hall type, for K fin. gen. (of arbitrary characteristic)
(2) for A/K of dim 2, 6 or odd, with End A = Z, for K fin. gen. and
char K = 0.
Important proof ingredients of Theorem D:
- the classical lemma of Borel and Cantelli of measure theory
- counting of certain symplectic matricies (e.g., with eigenvalue 1) in
the Galois image - a delicate matter, if char > 0.
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Theorem D(Arias-de-Reyna, G., Petersen, to appear in Math. Nachr.)
The GJC holds true for all abelian varieties with big monodromy.
In particular Theorem D ( + Theorem C + extension of Serre’s
theorem to fin. gen. fields) imply:
Corollary
The GJC is true:
(1) for A/K of Hall type, for K fin. gen. (of arbitrary characteristic)
(2) for A/K of dim 2, 6 or odd, with End A = Z, for K fin. gen. and
char K = 0.
Important proof ingredients of Theorem D:
- the classical lemma of Borel and Cantelli of measure theory
- counting of certain symplectic matricies (e.g., with eigenvalue 1) in
the Galois image - a delicate matter, if char > 0.
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Theorem D(Arias-de-Reyna, G., Petersen, to appear in Math. Nachr.)
The GJC holds true for all abelian varieties with big monodromy.
In particular Theorem D ( + Theorem C + extension of Serre’s
theorem to fin. gen. fields) imply:
Corollary
The GJC is true:
(1) for A/K of Hall type, for K fin. gen. (of arbitrary characteristic)
(2) for A/K of dim 2, 6 or odd, with End A = Z, for K fin. gen. and
char K = 0.
Important proof ingredients of Theorem D:
- the classical lemma of Borel and Cantelli of measure theory
- counting of certain symplectic matricies (e.g., with eigenvalue 1) in
the Galois image - a delicate matter, if char > 0.
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Proofs
Plan
1
Definitions and Notation
2
Results
3
Proof sketch of the corollary
4
Monodromies for abelian varieties
5
An application to arithmetic
6
Proofs
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Proofs
Proof sketch of part (b) of GJC, for A as above:
Claim
Let e ≥ 2 and A/K is an abelian variety over fin. gen. field with big
monodromy. Then for almost all σ ∈ GKe there exists only finitely many
primes l s.t.
A(Ksep (σ)[l]) 6= 0.
Lemma of Borel-Cantelli
Let {Xl }l∈L be a sequence of measurable sets in a measure space
(X , µ) s.t. µ(X ) = 1.
P
(b) If l∈L µ(Xl ) < ∞, then almost every x ∈ X (outside of a set of
measure zero) belongs to at most finitely many of Xl0 s.
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Proofs
Proof sketch of part (b) of GJC, for A as above:
Claim
Let e ≥ 2 and A/K is an abelian variety over fin. gen. field with big
monodromy. Then for almost all σ ∈ GKe there exists only finitely many
primes l s.t.
A(Ksep (σ)[l]) 6= 0.
Lemma of Borel-Cantelli
Let {Xl }l∈L be a sequence of measurable sets in a measure space
(X , µ) s.t. µ(X ) = 1.
P
(b) If l∈L µ(Xl ) < ∞, then almost every x ∈ X (outside of a set of
measure zero) belongs to at most finitely many of Xl0 s.
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Proofs
Q
(a) If µ(∩
X
)
=
l∈I
l
l∈I µ(Xl ) for any finite I ⊂ L
P
and l∈L µ(Xl ) = ∞,
then almost every x ∈ X belongs to infinitely many of the Xl0 s.
To prove Claim we take in (b) of Borel-Cantelli Lemma:
L = {primes}
Xl := {σ ∈ GKe :
A(K (σ))[l] 6= 0},
where σ := (σ1 , . . . , σe ).
It is enough to show:
X
µ(Xl ) < ∞.
l prime
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Q
(a) If µ(∩
X
)
=
l∈I
l
l∈I µ(Xl ) for any finite I ⊂ L
P
and l∈L µ(Xl ) = ∞,
then almost every x ∈ X belongs to infinitely many of the Xl0 s.
To prove Claim we take in (b) of Borel-Cantelli Lemma:
L = {primes}
Xl := {σ ∈ GKe :
A(K (σ))[l] 6= 0},
where σ := (σ1 , . . . , σe ).
It is enough to show:
X
µ(Xl ) < ∞.
l prime
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Proofs
Q
(a) If µ(∩
X
)
=
l∈I
l
l∈I µ(Xl ) for any finite I ⊂ L
P
and l∈L µ(Xl ) = ∞,
then almost every x ∈ X belongs to infinitely many of the Xl0 s.
To prove Claim we take in (b) of Borel-Cantelli Lemma:
L = {primes}
Xl := {σ ∈ GKe :
A(K (σ))[l] 6= 0},
where σ := (σ1 , . . . , σe ).
It is enough to show:
X
µ(Xl ) < ∞.
l prime
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Proofs
We have:
S
Xl = P∈A[l]−{0} {σ ∈ GKe : σi (P) = P,
=
S
e
P∈A[l]−{0} GK (P)
for all 1 ≤ i ≤ e}
= ∪P̄∈P(A[l]) GKe (P) .
We know:
µ(GKe (P) ) = [K (P) : K ]−e
Since A[l] is a simple Fl (GK )-module, we get easily:
[K (P) : K ]−1 ≤ [K (A[l]) : K ]−1/2g .
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Proofs
We have:
S
Xl = P∈A[l]−{0} {σ ∈ GKe : σi (P) = P,
=
S
e
P∈A[l]−{0} GK (P)
for all 1 ≤ i ≤ e}
= ∪P̄∈P(A[l]) GKe (P) .
We know:
µ(GKe (P) ) = [K (P) : K ]−e
Since A[l] is a simple Fl (GK )-module, we get easily:
[K (P) : K ]−1 ≤ [K (A[l]) : K ]−1/2g .
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Proofs
We have:
S
Xl = P∈A[l]−{0} {σ ∈ GKe : σi (P) = P,
=
S
e
P∈A[l]−{0} GK (P)
for all 1 ≤ i ≤ e}
= ∪P̄∈P(A[l]) GKe (P) .
We know:
µ(GKe (P) ) = [K (P) : K ]−e
Since A[l] is a simple Fl (GK )-module, we get easily:
[K (P) : K ]−1 ≤ [K (A[l]) : K ]−1/2g .
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Proofs
We have:
S
Xl = P∈A[l]−{0} {σ ∈ GKe : σi (P) = P,
=
S
e
P∈A[l]−{0} GK (P)
for all 1 ≤ i ≤ e}
= ∪P̄∈P(A[l]) GKe (P) .
We know:
µ(GKe (P) ) = [K (P) : K ]−e
Since A[l] is a simple Fl (GK )-module, we get easily:
[K (P) : K ]−1 ≤ [K (A[l]) : K ]−1/2g .
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Proofs
Hence:
µ(Xl ) <
P
P̄∈P(A[l]) [K (A[l])
: K ]−e/2g =
l 2g −1
−e/2g .
l−1 |Gl |
But:
Gl ⊃ Sp2g (Fl )
for all l > l0
(big monodromy)
and:
sl := |Sp2g (Fl )| = l g
2
Qg
i=1 (l
2i
− 1)
and by an easy check:
P
P
l 2g −1 −e/2g
≤ l>l0 l −2 < ∞.
l>l0 l−1 sl
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Proofs
Hence:
µ(Xl ) <
P
P̄∈P(A[l]) [K (A[l])
: K ]−e/2g =
l 2g −1
−e/2g .
l−1 |Gl |
But:
Gl ⊃ Sp2g (Fl )
for all l > l0
(big monodromy)
and:
sl := |Sp2g (Fl )| = l g
2
Qg
i=1 (l
2i
− 1)
and by an easy check:
P
P
l 2g −1 −e/2g
≤ l>l0 l −2 < ∞.
l>l0 l−1 sl
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Proofs
Hence:
µ(Xl ) <
P
P̄∈P(A[l]) [K (A[l])
: K ]−e/2g =
l 2g −1
−e/2g .
l−1 |Gl |
But:
Gl ⊃ Sp2g (Fl )
for all l > l0
(big monodromy)
and:
sl := |Sp2g (Fl )| = l g
2
Qg
i=1 (l
2i
− 1)
and by an easy check:
P
P
l 2g −1 −e/2g
≤ l>l0 l −2 < ∞.
l>l0 l−1 sl
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Proofs
Hence:
µ(Xl ) <
P
P̄∈P(A[l]) [K (A[l])
: K ]−e/2g =
l 2g −1
−e/2g .
l−1 |Gl |
But:
Gl ⊃ Sp2g (Fl )
for all l > l0
(big monodromy)
and:
sl := |Sp2g (Fl )| = l g
2
Qg
i=1 (l
2i
− 1)
and by an easy check:
P
P
l 2g −1 −e/2g
≤ l>l0 l −2 < ∞.
l>l0 l−1 sl
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Proofs
THANK YOU
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Proofs
an easy proof
Proposition (Serre)
If A/K is a principally polarized Abelian variety over K of char. zero,
g = 2, 6, or is an odd integer, and End A = Z, then A
(Im ρ̄l )0 = Sp2g (Fl )
for almost all l.
• Proposition extends Serre’s theorem of 1986 to finitely generated
fields of zero characteristic
• A similar result for elliptic curves is true over Fp (t) (proven by
Igusa in the 50th).
• Question Does Igusa’s theorem hold true in higher dimensions
(say, for A over Fp (t)) ?
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Proofs
an easy proof
Proposition (Serre)
If A/K is a principally polarized Abelian variety over K of char. zero,
g = 2, 6, or is an odd integer, and End A = Z, then A
(Im ρ̄l )0 = Sp2g (Fl )
for almost all l.
• Proposition extends Serre’s theorem of 1986 to finitely generated
fields of zero characteristic
• A similar result for elliptic curves is true over Fp (t) (proven by
Igusa in the 50th).
• Question Does Igusa’s theorem hold true in higher dimensions
(say, for A over Fp (t)) ?
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Proofs
an easy proof
Proposition (Serre)
If A/K is a principally polarized Abelian variety over K of char. zero,
g = 2, 6, or is an odd integer, and End A = Z, then A
(Im ρ̄l )0 = Sp2g (Fl )
for almost all l.
• Proposition extends Serre’s theorem of 1986 to finitely generated
fields of zero characteristic
• A similar result for elliptic curves is true over Fp (t) (proven by
Igusa in the 50th).
• Question Does Igusa’s theorem hold true in higher dimensions
(say, for A over Fp (t)) ?
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Proofs
an easy proof
Proposition (Serre)
If A/K is a principally polarized Abelian variety over K of char. zero,
g = 2, 6, or is an odd integer, and End A = Z, then A
(Im ρ̄l )0 = Sp2g (Fl )
for almost all l.
• Proposition extends Serre’s theorem of 1986 to finitely generated
fields of zero characteristic
• A similar result for elliptic curves is true over Fp (t) (proven by
Igusa in the 50th).
• Question Does Igusa’s theorem hold true in higher dimensions
(say, for A over Fp (t)) ?
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Proofs
Proof of Proposition (by induction on trdeg(K /Q)):
trdeg(K /Q) = 0 -
the open image theorem of Serre.
Assume that trdeg(K /Q) = d > 0
F /Q - fin. gen. field, trdeg(F /Q) = d − 1
C/F - smooth curve over F with F (C) = K
A - extends to an abelian group scheme A −→ C.
R.Noot: there exists a closed c ∈ C s.t.:
Ac := A ×C F (c) is an abelian variety over F (c)
End Ac = End A = Z
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Proofs
Proof of Proposition (by induction on trdeg(K /Q)):
trdeg(K /Q) = 0 -
the open image theorem of Serre.
Assume that trdeg(K /Q) = d > 0
F /Q - fin. gen. field, trdeg(F /Q) = d − 1
C/F - smooth curve over F with F (C) = K
A - extends to an abelian group scheme A −→ C.
R.Noot: there exists a closed c ∈ C s.t.:
Ac := A ×C F (c) is an abelian variety over F (c)
End Ac = End A = Z
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Proofs
Proof of Proposition (by induction on trdeg(K /Q)):
trdeg(K /Q) = 0 -
the open image theorem of Serre.
Assume that trdeg(K /Q) = d > 0
F /Q - fin. gen. field, trdeg(F /Q) = d − 1
C/F - smooth curve over F with F (C) = K
A - extends to an abelian group scheme A −→ C.
R.Noot: there exists a closed c ∈ C s.t.:
Ac := A ×C F (c) is an abelian variety over F (c)
End Ac = End A = Z
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Proofs
By induction
Gal(F (c)(Ac [l])/F (c)) = GSp(A[l]),
for l >> 0.
Let
v - discrete valuation on K attached to c
wl - extension of v to K (A[l]).
Then
Gal(F (c)(Ac [l])/F (c)) ∼
= Gal(F (wl )/F (v )) ∼
=
∼
= D(wl ) ⊂ Gal(K (A[l])/K ) ⊂ GSp(A[l])
for l >> 0 which implies the claim.
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Proofs
By induction
Gal(F (c)(Ac [l])/F (c)) = GSp(A[l]),
for l >> 0.
Let
v - discrete valuation on K attached to c
wl - extension of v to K (A[l]).
Then
Gal(F (c)(Ac [l])/F (c)) ∼
= Gal(F (wl )/F (v )) ∼
=
∼
= D(wl ) ⊂ Gal(K (A[l])/K ) ⊂ GSp(A[l])
for l >> 0 which implies the claim.
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Proofs
By induction
Gal(F (c)(Ac [l])/F (c)) = GSp(A[l]),
for l >> 0.
Let
v - discrete valuation on K attached to c
wl - extension of v to K (A[l]).
Then
Gal(F (c)(Ac [l])/F (c)) ∼
= Gal(F (wl )/F (v )) ∼
=
∼
= D(wl ) ⊂ Gal(K (A[l])/K ) ⊂ GSp(A[l])
for l >> 0 which implies the claim.
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