Residual modular Galois representations
A local-global principle for isogenies over number fields
Images of modular Galois
representations mod `
Samuele Anni
Universiteit Leiden - Université Bordeaux 1
Leiden, ALGANT meeting;
23rd February 2013
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
1
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
2
A local-global principle for isogenies of
prime degree over number fields
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Let us fix a positive integer n ∈ Z >0 .
Definition
The congruence subgroup Γ1 (n) of SL2 (Z ) is the subgroup given by
a b
Γ1 (n) =
∈ SL2 (Z ) : a ≡ d ≡ 1 , c ≡ 0 mod n .
c d
The integer n is called level of the congruence subgroup.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Over the upper half plane:
H = {z ∈ C| Im(z) > 0}
we can define an action of Γ1 (n) via
fractional transformations:
Γ1 (n) × H
→ H
(γ, z) 7→ γ(z) =
az + b
cz + d
a b
where γ =
.
c d
Moreover, if n ≥ 4 then Γ1 (n) acts freely
on H .
Samuele Anni
Fundamental domain for Γ1 (n), Verrill webpage.
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Definition
We define the modular curve Y1 (n)C to be the non-compact Riemann
surface obtained giving on Γ1 (n)\H the complex structure induced by the
quotient map.
Let X1 (n)C be the compactification of Y1 (n)C .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Definition
We define the modular curve Y1 (n)C to be the non-compact Riemann
surface obtained giving on Γ1 (n)\H the complex structure induced by the
quotient map.
Let X1 (n)C be the compactification of Y1 (n)C .
Fact: Y1 (n)C can be defined algebraically over Q (in fact over Z [1/n]).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
The group GL+
2 (Q ) acts on H via fractional transformation, and its
action has a particular behaviour with respect to Γ1 (n).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
The group GL+
2 (Q ) acts on H via fractional transformation, and its
action has a particular behaviour with respect to Γ1 (n).
Γ1 (n)
Proposition
∀g ∈ GL+
2 (Q ) the discrete groups
g Γ1 (n)g −1 and Γ1 (n) are commensurable,
i.e. g Γ1 (n)g −1 ∩ Γ1 (n) is a subgroup of
finite index in g Γ1 (n)g −1 and Γ1 (n).
Samuele Anni
g −1 Γ1 (n)g
'
Γ1 (n)
H
/H
Y1 (n)C
% Y1 (n)C
g
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
We can define operators on Y1 (n) through the correspondences given
before:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
We can define operators on Y1 (n) through the correspondences given
before:
the Hecke
operators Tp for every prime p, using
1 0
g=
∈ GL+
2 (Q ) ;
0 p
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
We can define operators on Y1 (n) through the correspondences given
before:
the Hecke
operators Tp for every prime p, using
1 0
g=
∈ GL+
2 (Q ) ;
0 p
the diamond
operators
hdi for every d ∈ (Z /nZ )∗ , using
−1
d
0
g=
∈ GL+
2 (Q ).
0
d
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
For n ≥ 5, ` prime not dividing n, and k integer we define, à la Katz:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
For n ≥ 5, ` prime not dividing n, and k integer we define, à la Katz:
mod ` cusp forms
S(n, k)F` = H0 (X1 (n)F` , ω ⊗k (− Cusps)).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
For n ≥ 5, ` prime not dividing n, and k integer we define, à la Katz:
mod ` cusp forms
S(n, k)F` = H0 (X1 (n)F` , ω ⊗k (− Cusps)).
S(n, k)F` is a finite dimensional F` -vector space, equipped with Hecke
operators Tn (n ≥ 1) and diamond operators hdi for every d ∈ (Z /nZ )∗ .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
For n ≥ 5, ` prime not dividing n, and k integer we define, à la Katz:
mod ` cusp forms
S(n, k)F` = H0 (X1 (n)F` , ω ⊗k (− Cusps)).
S(n, k)F` is a finite dimensional F` -vector space, equipped with Hecke
operators Tn (n ≥ 1) and diamond operators hdi for every d ∈ (Z /nZ )∗ .
Analogous definition in characteristic zero and over any ring where n is
invertible.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
One may think that mod ` modular forms come from reduction of
characteristic zero modular forms mod `:
S(n, k)Z [1/n] → S(n, k)F` .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
One may think that mod ` modular forms come from reduction of
characteristic zero modular forms mod `:
S(n, k)Z [1/n] → S(n, k)F` .
Unfortunately, this map is not surjective for k = 1.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
One may think that mod ` modular forms come from reduction of
characteristic zero modular forms mod `:
S(n, k)Z [1/n] → S(n, k)F` .
Unfortunately, this map is not surjective for k = 1.
Even worse: given a character : (Z /nZ )∗ → C∗ the map
S(n, k, )OK → S(n, k, )F
is not always surjective even if k > 1, where OK is the ring of integers
of the number field where is defined, F` ⊆ F and
S(n, k, )OK = {f ∈ S(n, k)OK | ∀d ∈ (Z /nZ )∗ , hdi f = (d)f }.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Definition
We define the Hecke algebra T(n, k) of S(n, k)C as the Z -subalgebra of
EndC (S(Γ1 (n), k)C ) generated by the Hecke operators Tp for every prime
p and the diamond operators hdi for every d ∈ (Z /nZ )∗ .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Definition
We define the Hecke algebra T(n, k) of S(n, k)C as the Z -subalgebra of
EndC (S(Γ1 (n), k)C ) generated by the Hecke operators Tp for every prime
p and the diamond operators hdi for every d ∈ (Z /nZ )∗ .
Fact: T(n, k) is finitely generated as Z -module.
We can associate a Hecke algebra T(n, k, ) to each S(n, k, )C .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Shimura, Deligne)
Let n and k be positive integers. Let F be a finite field of characteristic
`, ` - n, and f : T(n, k) F a surjective morphism of rings.
Then there is a continuous semi-simple representation:
ρf : Gal(Q /Q ) → GL2 (F) that is unramified outside n` such that for all
p not dividing n` we have, in F:
Tr(ρf (Frobp )) = f (Tp ) and det(ρf (Frobp )) = f (hpi)p k−1 .
Such a ρf is unique up to isomorphism.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Shimura, Deligne)
Let n and k be positive integers. Let F be a finite field of characteristic
`, ` - n, and f : T(n, k) F a surjective morphism of rings.
Then there is a continuous semi-simple representation:
ρf : Gal(Q /Q ) → GL2 (F) that is unramified outside n` such that for all
p not dividing n` we have, in F:
Tr(ρf (Frobp )) = f (Tp ) and det(ρf (Frobp )) = f (hpi)p k−1 .
Such a ρf is unique up to isomorphism.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Shimura, Deligne)
Let n and k be positive integers. Let F be a finite field of characteristic
`, ` - n, and f : T(n, k) F a surjective morphism of rings.
Then there is a continuous semi-simple representation:
ρf : Gal(Q /Q ) → GL2 (F) that is unramified outside n` such that for all
p not dividing n` we have, in F:
Tr(ρf (Frobp )) = f (Tp ) and det(ρf (Frobp )) = f (hpi)p k−1 .
Such a ρf is unique up to isomorphism.
Computing ρf is “difficult”,
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Shimura, Deligne)
Let n and k be positive integers. Let F be a finite field of characteristic
`, ` - n, and f : T(n, k) F a surjective morphism of rings.
Then there is a continuous semi-simple representation:
ρf : Gal(Q /Q ) → GL2 (F) that is unramified outside n` such that for all
p not dividing n` we have, in F:
Tr(ρf (Frobp )) = f (Tp ) and det(ρf (Frobp )) = f (hpi)p k−1 .
Such a ρf is unique up to isomorphism.
Computing ρf is “difficult”, but theoretically it can be done in
polynomial time in n, k, #F.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Question
Can we compute the image of a residual modular Galois representation
without computing the representation?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Main ingredients:
Theorem (Dickson)
Let ` be an odd prime and H a finite subgroup of PGL2 (F` ). Then a
conjugate of H is one of the following groups:
a finite subgroups of the upper triangular matrices;
SL2 (F`r )/{±1} or PGL2 (F`r ) for r ∈ Z >0 ;
a dihedral group D2n with n ∈ Z >1 , (`, n) = 1;
or it is isomorphic to A4 , S4 or A5 .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Khare, Wintenberger, Dieulefait) Serre’s Conjecture
Let ` be a prime number and let ρ : Gal(Q /Q ) → GL2 (F` ) be an odd,
absolutely irreducible, continuous representation.
Then ρ is modular of level N(ρ), weight k(ρ) and character (ρ).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Khare, Wintenberger, Dieulefait) Serre’s Conjecture
Let ` be a prime number and let ρ : Gal(Q /Q ) → GL2 (F` ) be an odd,
absolutely irreducible, continuous representation.
Then ρ is modular of level N(ρ), weight k(ρ) and character (ρ).
N(ρ) (the level) is the Artin conductor away from `.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Khare, Wintenberger, Dieulefait) Serre’s Conjecture
Let ` be a prime number and let ρ : Gal(Q /Q ) → GL2 (F` ) be an odd,
absolutely irreducible, continuous representation.
Then ρ is modular of level N(ρ), weight k(ρ) and character (ρ).
N(ρ) (the level) is the Artin conductor away from `.
k(ρ) (the weight) is given by a recipe in terms of ρ|I` .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Theorem (Khare, Wintenberger, Dieulefait) Serre’s Conjecture
Let ` be a prime number and let ρ : Gal(Q /Q ) → GL2 (F` ) be an odd,
absolutely irreducible, continuous representation.
Then ρ is modular of level N(ρ), weight k(ρ) and character (ρ).
N(ρ) (the level) is the Artin conductor away from `.
k(ρ) (the weight) is given by a recipe in terms of ρ|I` .
∗
(ρ) : (Z /N(ρ)Z )∗ → F` is given by:
det ◦ρ = (ρ)χk(ρ)−1 .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Samuele Anni
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Algorithm
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Algorithm
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Algorithm
Input:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Algorithm
Input:
n positive integer;
` prime such that (n, `) = 1;
k positive integer such that 2 ≤ k ≤ ` + 1;
a character : (Z /nZ )∗ → C∗ ;
a morphism of ring f : T(n, k, ) → F` ;
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Algorithm
Input:
n positive integer;
` prime such that (n, `) = 1;
k positive integer such that 2 ≤ k ≤ ` + 1;
a character : (Z /nZ )∗ → C∗ ;
a morphism of ring f : T(n, k, ) → F` ;
Output:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Algorithm
Input:
n positive integer;
` prime such that (n, `) = 1;
k positive integer such that 2 ≤ k ≤ ` + 1;
a character : (Z /nZ )∗ → C∗ ;
a morphism of ring f : T(n, k, ) → F` ;
Output:
Image of the associated Galois representation ρf , up to conjugacy as
subgroup of GL2 (F` ).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
How do we want to construct this algorithm?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
How do we want to construct this algorithm?
Iteration “down to top”, i.e. considering all divisors of n (and all twists).
Determination of the projective image. Determination of the image.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
How do we want to construct this algorithm?
Iteration “down to top”, i.e. considering all divisors of n (and all twists).
Determination of the projective image. Determination of the image.
How many Tp ?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
How do we want to construct this algorithm?
Iteration “down to top”, i.e. considering all divisors of n (and all twists).
Determination of the projective image. Determination of the image.
How many Tp ?
Bound linear in n (and k): Sturm Bound at level n, which we will denote
as SB(n, k) (at the moment the known bound is ∼ `5 n3 ).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Results
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Results
We have studied the field of definition of the representation, and of
the projective representation:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Results
We have studied the field of definition of the representation, and of
the projective representation: in both cases we can determine such
fields computing operators up SB(n, k).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Results
We have studied the field of definition of the representation, and of
the projective representation: in both cases we can determine such
fields computing operators up SB(n, k).
We have a result about twists of representation: this will speed up
the computation of the projective image.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Results
We have studied the field of definition of the representation, and of
the projective representation: in both cases we can determine such
fields computing operators up SB(n, k).
We have a result about twists of representation: this will speed up
the computation of the projective image.
We are studying methods for switching characteristic in the case of
“exceptional” projective image, i.e. projective image isomorphic to
A4 , S4 or A5 . Idea: construct a database in characteristic 2, 3 and 5
to have information in larger characteristic.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Example: projective image S4 in characteristic 3.
Ideas:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Example: projective image S4 in characteristic 3.
Ideas:
a modular representation which has S4 as projective image in
characteristic 3 has “big” projective image i.e. PGL2 (F3 ) ∼
= S4 ;
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Example: projective image S4 in characteristic 3.
Ideas:
a modular representation which has S4 as projective image in
characteristic 3 has “big” projective image i.e. PGL2 (F3 ) ∼
= S4 ;
from mod 3 modular forms with projective image S4 , we want to
construct characteristic 0 forms;
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Example: projective image S4 in characteristic 3.
Ideas:
a modular representation which has S4 as projective image in
characteristic 3 has “big” projective image i.e. PGL2 (F3 ) ∼
= S4 ;
from mod 3 modular forms with projective image S4 , we want to
construct characteristic 0 forms;
use these forms to decide about projective image S4 in characteristic
larger than 3.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Input:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Input:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Input:
n positive integer, (n, 3) = 1;
k ∈ {2, 3, 4};
a character : (Z /nZ )∗ → C∗ ;
a morphism of rings f : T(n, k, ) → F3 .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Suppose the algorithm has certified that ρf is absolutely irreducible and
that Pρf ∼
= S4 . Suppose also that f is “minimal” with respect to weight,
level and twisting.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Suppose the algorithm has certified that ρf is absolutely irreducible and
that Pρf ∼
= S4 . Suppose also that f is “minimal” with respect to weight,
level and twisting. What else do we know?
Field of definition of the representation: F;
Field of definition of the projective representation: F3 ;
Data on the local components;
Image of the representation: ρf (Gal(Q /Q )) ⊆ F∗ · GL2 (F3 ).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Suppose the algorithm has certified that ρf is absolutely irreducible and
that Pρf ∼
= S4 . Suppose also that f is “minimal” with respect to weight,
level and twisting. What else do we know?
Field of definition of the representation: F;
Field of definition of the projective representation: F3 ;
Data on the local components;
Image of the representation: ρf (Gal(Q /Q )) ⊆ F∗ · GL2 (F3 ).
Let β : F∗ · GL2 (F3 ) → GL2 (OK ) be a faithful irreducible 2-dimensional
representation, where OK is the ring of integers of a number field.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Samuele Anni
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
Samuele Anni
β
)
/ GL2 (OK )
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Yes, we can bound it.
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) for all p?
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) for all p?
Yes for the primes dividing the level and 3
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) for all p?
Yes for the primes dividing the level and 3
No for the unramified primes! Problem: distinguish elements in GL2 (F3 )
using only traces and determinants is not possible.
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
By Serre’s conjecture there exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) for all p?
Yes for the primes dividing the level and 3
No for the unramified primes! Problem: distinguish elements in GL2 (F3 )
using only traces and determinants is not possible.
Solution:
check in characteristic 2 and 5.
Samuele Anni
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
ρfπβ
Samuele Anni
β
'
/ GL2 (OK )
π
,
GL2 (F2 )
Images of modular Galois representations mod `
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
Residual modular Galois representations
A local-global principle for isogenies over number fields
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
'
/ GL2 (OK )
π
ρfπβ
,
GL2 (F2 )
ρfπβ (Gal(Q /Q )) ⊆ F0∗ × GL2 (F2 )
Pρfπβ (Gal(Q /Q )) ∼
= S3
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Can we determine the level of fπβ ?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Can we determine the level of fπβ ?
Yes, we can bound it.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Can we determine the level of fπβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) using fπβ (Tp ), fπβ (hpi) for all p?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Can we determine the level of fπβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) using fπβ (Tp ), fπβ (hpi) for all p?
Yes for the primes dividing the level and 3.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Can we determine the level of fπβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) using fπβ (Tp ), fπβ (hpi) for all p?
Yes for the primes dividing the level and 3.
For the unramified primes there is still a problem but we have candidates
i.e. a finite list of mod 2 modular forms with prescribed properties.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Modular curves and Modular Forms
Residual modular Galois representations
Image of Residual modular Galois representations
Algorithm
Example: projective image S4 in characteristic 3
There exists a mod 2 modular form fπβ such that ρfπβ ∼
= π ◦ β ◦ ρf .
Can we determine the level of fπβ ?
Yes, we can bound it.
Can we determine fβ (Tp ), fβ (hpi) using fπβ (Tp ), fπβ (hpi) for all p?
Yes for the primes dividing the level and 3.
For the unramified primes there is still a problem but we have candidates
i.e. a finite list of mod 2 modular forms with prescribed properties.
How can we solve this problem?
For each candidate we have a power series in characteristic 0. All power
series are defined over the same ring of integers so we can reduce them
modulo 5 and check if the list we obtain does occur as eigenvalue system
or not. Claim: only one power series is a modular form.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
1
Residual modular Galois representations
2
A local-global principle for isogenies of
prime degree over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Definition
Let E be an elliptic curve defined on a number field K , and let ` be a
prime number. If p is a prime of K where E has good reduction, p not
dividing `, we say that E admits an `-isogeny locally at p if the Néron
model of E over the ring of integer of Kp admits an `-isogeny.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Definition
Let E be an elliptic curve defined on a number field K , and let ` be a
prime number. If p is a prime of K where E has good reduction, p not
dividing `, we say that E admits an `-isogeny locally at p if the Néron
model of E over the ring of integer of Kp admits an `-isogeny.
Question
Let E be an elliptic curve defined over a number field K , and let ` be a
prime number, if E admits an `-isogeny locally at a set of primes with
density one then does E admit an `-isogeny over K ?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Theorem (Sutherland)
Let E be an elliptic curve
over a number field K and let ` be a
qdefined
−1
prime number. Assume
`∈
/ K , and suppose E /K admits an
`
`-isogeny locally at a set of primes with density one.
Then E admits an `-isogeny over a quadratic extension of K .
Moreover, if ` ≡ 1 mod 4 or ` < 7, E admits an `-isogeny defined over K .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Definition
Let K be a number field, let E be an elliptic curve over K and ` a prime
number, a pair (`, j(E )) is said to be exceptional for K if E /K admits
an `-isogeny locally everywhere but not over K .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Definition
Let K be a number field, let E be an elliptic curve over K and ` a prime
number, a pair (`, j(E )) is said to be exceptional for K if E /K admits
an `-isogeny locally everywhere but not over K .
Sutherland proved the following result:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Definition
Let K be a number field, let E be an elliptic curve over K and ` a prime
number, a pair (`, j(E )) is said to be exceptional for K if E /K admits
an `-isogeny locally everywhere but not over K .
Sutherland proved the following result:
Theorem (Sutherland)
The pair (7, 2268945/128) is the only exceptional pair for Q .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Theorem (Sutherland)
Let E be an elliptic curve
over a number field K and let ` be a
qdefined
−1
prime number. Assume
`∈
/ K , and suppose E /K admits an
`
`-isogeny locally at a set of primes with density one.
Then E admits an `-isogeny over a quadratic extension of K .
Moreover, if ` ≡ 1 mod 4 or ` < 7, E admits an `-isogeny defined over K .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Remark
Let (`, j(E )) be an exceptional pair for the number field K and let
G = ρE ,` (Gal(Q /K )). Then G is a subgroup of GL2 (F` ) such that
|P1 (F` )g | > 0 for all g ∈ G but |P1 (F` )G | = 0.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Remark
Let (`, j(E )) be an exceptional pair for the number field K and let
G = ρE ,` (Gal(Q /K )). Then G is a subgroup of GL2 (F` ) such that
|P1 (F` )g | > 0 for all g ∈ G but |P1 (F` )G | = 0.
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Remark
Let (`, j(E )) be an exceptional pair for the number field K and let
G = ρE ,` (Gal(Q /K )). Then G is a subgroup of GL2 (F` ) such that
|P1 (F` )g | > 0 for all g ∈ G but |P1 (F` )G | = 0.
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Remark
Let (`, j(E )) be an exceptional pair for the number field K and let
G = ρE ,` (Gal(Q /K )). Then G is a subgroup of GL2 (F` ) such that
|P1 (F` )g | > 0 for all g ∈ G but |P1 (F` )G | = 0.
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
ζ` is in K if and only if G is contained in SL2 (F` );
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Remark
Let (`, j(E )) be an exceptional pair for the number field K and let
G = ρE ,` (Gal(Q /K )). Then G is a subgroup of GL2 (F` ) such that
|P1 (F` )g | > 0 for all g ∈ G but |P1 (F` )G | = 0.
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
ζ` is in K if and only if G is contained in SL2 (F` );
H, the q
projective image of G , is contained in SL2 (F` )/ {±1} if and
−1
only if
` is in K .
`
Hence, the study of the local-global principle
about `-isogenies over an
q
−1
`
belonging to K or not.
`
arbitrary number field K depends on
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Lemma (Sutherland)
Let G be a subgroup of GL2 (F` ) whose image H in PGL2 (F` ) is not
contained in SL2 (F` )/ {±1}. Suppose |P1 (F` )g | > 0 for all g ∈ G but
|P1 (F` )G | = 0.
Then ` ≡ 3 mod 4 and the following holds:
(1) H is dihedral of order 2n, where n > 1 is an odd divisor of (`−1)/2;
(2) G is properly contained in the normalizer of a split Cartan subgroup;
(3) P1 (F` )/G contains an orbit of size 2.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Let (`,
qj(E )) be an exceptional pair for the number field K , and assume
−1
that
`∈
/ K . Let G be ρE ,` (Gal(Q /K )) and let H be its image in
`
PGL2 (F` ). Let C ⊂ G be the preimage of the maximal cyclic subgroup
of H. Then
det(C ) ∈ (F∗` )2 .
where (F∗` )2 denotes the group of squares in F∗` .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Let (`,
qj(E )) be an exceptional pair for the number field K , and assume
−1
that
`∈
/ K . Let G be ρE ,` (Gal(Q /K )) and let H be its image in
`
PGL2 (F` ). Let C ⊂ G be the preimage of the maximal cyclic subgroup
of H. Then
det(C ) ∈ (F∗` )2 .
where (F∗` )2 denotes the group of squares in F∗` .
Proposition (A.)
q
−1
Let (`, j(E )) be an exceptional pair for the number field K with
`
`
√
not belonging to K . Then E admits an `-isogeny over K ( −`) (and
actually, two such isogenies).
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Main Theorem (A.)
Let (`, j(E )) be an q
exceptional pair for the number field K of degree d
−1
over Q , such that
`∈
/ K . Then ` ≡ 3 mod 4 and
`
7 ≤ ` ≤ 6d+1.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Main Theorem (A.)
Let (`, j(E )) be an q
exceptional pair for the number field K of degree d
−1
over Q , such that
`∈
/ K . Then ` ≡ 3 mod 4 and
`
7 ≤ ` ≤ 6d+1.
Remark
This theorem implies the result obtained by Sutherland in the case
K = Q.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Let us assume that
q
−1
`
Preliminaries and introduction to the problem
New results
Finiteness result
` belongs to K .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Let us assume that
q
−1
`
Preliminaries and introduction to the problem
New results
Finiteness result
` belongs to K .
Lemma (A.)
Let G be a subgroup of GL2 (F` ) whose image H in PGL2 (F` ) is
contained in SL2 (F` )/ {±1}. Suppose |P1 (F` )g | > 0 for all g ∈ G but
|P1 (F` )G | = 0. Then ` ≡ 1 mod 4 and one of the followings holds:
(1) H is dihedral of order 2n, where n ∈ Z >1 is a divisor of `−1;
(2) H is isomorphic to one of the following exceptional groups: A4 , S4 or
A5 .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Corollary (A.)
Let G be a subgroup of GL2 (F` ) whose image H in PGL2 (F` ) is
contained in SL2 (F` )/ {±1}. Suppose |P1 (F` )g | > 0 for all g ∈ G but
|P1 (F` )G | = 0. If H is dihedral of order 2n, where n ∈ Z >1 is a divisor of
`−1, then G is properly contained in the normalizer of a split Cartan
subgroup and P1 (F` )/G contains an orbit of size 2.
Corollary (A.)
Let G be a subgroup of GL2 (F` ) whose image H in PGL2 (F` ) is
contained in SL2 (F` )/ {±1}. Suppose |P1 (F` )g | > 0 for all g ∈ G but
|P1 (F` )G | = 0. Then:
if H is isomorphic to A4 then ` ≡ 1 mod 12;
if H is isomorphic to S4 then ` ≡ 1 mod 24;
if H is isomorphic to A5 then ` ≡ 1 mod 60.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Let E be an elliptic curve defined over a number q
field K of degree d over
−1
Q and let ` be a prime number. Let us suppose
` ∈ K . Suppose
`
E /K admits an `-isogeny locally at a set of primes with density one.
Then:
(1) if `≡3 mod 4 the elliptic curve E admits a global `-isogeny over K ;
(2) if `≡1 mod 4 the elliptic curve E admits an `-isogeny over L, finite
extension of K , which can ramify only at primes dividing the
conductor of E and `. Moreover, if `≡−1 mod 3 or if ` ≥ 60d+1,
then E admits an `-isogeny over a quadratic extension L of K .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Question
Let K be a number field and let ` be a prime number, how many
exceptional pairs (`, j(E )) do exist over K ?
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Question
Let K be a number field and let ` be a prime number, how many
exceptional pairs (`, j(E )) do exist over K ?
Proposition (A.)
Let (`, j(E )) be an exceptional pair for the number field K of degree d
over Q and discriminant ∆. Then
` ≤ max {∆, 6d+1} .
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Given a number field K :
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Given a number field K :
if ` = 2, 3 then there exists no exceptional pair;
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Given a number field K :
if ` = 2, 3 then there exists no exceptional pair;
there exist infinitely many
√ exceptional pairs (5, j(E )) for the number
field K if and only if 5 belongs to K ;
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Proposition (A.)
Given a number field K :
if ` = 2, 3 then there exists no exceptional pair;
there exist infinitely many
√ exceptional pairs (5, j(E )) for the number
field K if and only if 5 belongs to K ;
if ` > 7, then the number of exceptional pairs (`, j(E )) is finite.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
The local-global principle for 7-isogenies leads us to a dichotomy between
a finite and an infinite number of counterexamples according to the rank
of a specific elliptic curve that we call the Elkies-Sutherland curve:
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
The local-global principle for 7-isogenies leads us to a dichotomy between
a finite and an infinite number of counterexamples according to the rank
of a specific elliptic curve that we call the Elkies-Sutherland curve:
Proposition (A.)
If ` = 7 then the number of exceptional pairs (7, j(E )) for a number field
K , is finite or infinite, depending on the rank of the elliptic curve
E 0 : y 2 = x 3 − 1715x + 33614
being respectively 0 or positive.
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Further directions
Generalization for simple abelian varieties of dimension d over Q
which are principally polarized i.e. study of the subgroups of
P GSp2d (F` )...
Generalization for abelian varieties of GL2 -type;
Generalization for isogenies of prime power degree;
Generalization for isogenies of degree given by products of primes;
...
Samuele Anni
Images of modular Galois representations mod `
Residual modular Galois representations
A local-global principle for isogenies over number fields
Preliminaries and introduction to the problem
New results
Finiteness result
Images of modular Galois
representations mod `
Samuele Anni
Universiteit Leiden - Université Bordeaux 1
Leiden, ALGANT meeting;
23rd February 2013
Thanks!
Samuele Anni
Images of modular Galois representations mod `