Twists and residual modular Galois representations
Twists and residual modular Galois
representations
Samuele Anni
University of Warwick
Building Bridges, Bristol
10th July 2014
Twists and residual modular Galois representations
Modular curves and Modular Forms
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Local representation
4
Twist
5
Examples
Twists and residual modular Galois representations
Modular curves and Modular Forms
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 ) : n | a−1 , n | c .
c d
The integer n is called level of the congruence subgroup.
Twists and residual modular Galois representations
Modular curves and Modular Forms
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 .
Escher, Reducing Lizards Tessellation
Twists and residual modular Galois representations
Modular curves and Modular Forms
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]).
Twists and residual modular Galois representations
Modular curves and Modular Forms
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
For every g ∈ GL+
2 (Q ), the discrete groups
g Γ1 (n)g −1 and Γ1 (n) are commensurable
g −1 Γ1 (n)g
'
Γ1 (n)
H
/H
Y1 (n)C
% Y1 (n)C
g
Twists and residual modular Galois representations
Modular curves and Modular Forms
We 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
a b
g=
∈ Γ0 (n), where Γ0 (n) is the set of matrices in SL2 (Z )
c d
which are upper triangular modulo n.
Twists and residual modular Galois representations
Modular curves and Modular Forms
For n ≥ 5 and k positive integers, let ` be a prime not dividing n.
Following Katz, we define the space of mod ` cusp forms as
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.
Twists and residual modular Galois representations
Modular curves and Modular Forms
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.
OK is the ring of integers of the number field where is defined, F is the
residue field at ` and is the reduction of S(n, k, )OK = {f ∈ S(n, k)OK | ∀d ∈ (Z /nZ )∗ , hdi f = (d)f }.
Twists and residual modular Galois representations
Modular curves and Modular Forms
Definition
The Hecke algebra T(n, k) of S(n, k)C is the Z -subalgebra of
EndC (S(Γ1 (n), k)C ) generated by Hecke operators Tp for every prime p
and by diamond operators hdi for every d ∈ (Z /nZ )∗ .
Fact:
T(n, k) is finitely generated as Z -module.
Given a character : (Z /nZ )∗ → C∗ , we associate a Hecke algebra
T (n, k) to each S(n, k, )C :
S(n, k, )C = {f ∈ S(n, k)C | ∀d ∈ (Z /nZ )∗ , hdi f = (d)f }.
Twists and residual modular Galois representations
Residual modular Galois representations
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Local representation
4
Twist
5
Examples
Twists and residual modular Galois representations
Residual modular Galois representations
Theorem (Deligne, Shimura)
Let n and k be positive integers. Let F be a finite field of characteristic
`, with ` not dividing 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),
unramified outside n`, such that for all p not dividing n` we have:
Trace(ρf (Frobp )) = f (Tp ) and det(ρf (Frobp )) = f (hpi)p k−1 in F.
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:
Edixhoven, Couveignes, de Jong, Merkl, Bruin, Bosman (#F ≤ 32);
Mascot, Zeng, Tian (#F ≤ 41).
Twists and residual modular Galois representations
Residual modular Galois representations
Question
Can we compute the image of a residual modular Galois representation
without computing the representation?
Theorem (A.)
There is a polynomial time algorithm which takes as input:
n and k be positive integers;
` be a prime number not dividing n, such that 2 ≤ k ≤ ` + 1;
a character : (Z /nZ )∗ → C∗ ;
a morphism of ring f : T (n, k) → F` , and in particular the images
of all diamond operators and of the Tp operators up to a bound B,
and gives as output the image of the associated Galois representation ρf ,
up to conjugacy as subgroup of GL2 (F` ) without computing ρf .
Twists and residual modular Galois representations
Residual modular Galois representations
In almost all cases, the bound B is the Sturm Bound for Γ0 (n) and
weight k:
Y 1
k
k
·n·
1+
· n log log n
12
p
12
p|n prime
In the cases when this bound is not enough, then the Sturm Bound for
Γ0 (nq 2 ) and weight k, where q is the smallest odd prime not dividing n,
is the required bound.
Twists and residual modular Galois representations
Residual modular Galois representations
Some of the problems studied:
ρf can arise from lower level or weight, i.e. there exists
g ∈ S(m, j)F` with m ≤ n or j ≤ k such that ρg ∼
= ρf
ρf can arise as twist of a representation of lower conductor, i.e.
there exist g ∈ S(m, j)F` with m ≤ n or j ≤ k and a Dirichlet
character χ such that ρg ⊗ χ ∼
= ρf
Twists and residual modular Galois representations
Residual modular Galois representations
One of the principal ingredients:
Theorem (Khare, Wintenberger, Dieulefait, Kisin),
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 .
Twists and residual modular Galois representations
Residual modular Galois representations
Setting (∗)
n and k be positive integers;
` be a prime number not dividing n, such that 2 ≤ k ≤ ` + 1;
: (Z /nZ )∗ → C∗ be a character;
f : T (n, k) → F` be a morphism of rings;
ρf : GQ → GL2 (F` ) be the unique, up to isomorphism, continuous
semi-simple representation attached to f ;
∗
∗
: (Z /nZ ) → F` be the character defined by (a) = f (hai) for all
a ∈ (Z /nZ )∗ .
Let p be a prime dividing n`. Let us denote by
Gp = Gal(Q p /Q p ) ⊂ GQ the decomposition subgroup at p;
Ip the inertia subgroup, It the tame inertia subgroup;
Gi,p , with i ∈ Z >0 , the higher ramification subgroups (Ip = G0,p ).
Notation: given a residual representation ρ, we will denote as Np (ρ) the
valuation at p of the Artin conductor of ρ.
Twists and residual modular Galois representations
Local representation
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Local representation
Local representation at `
Local representation at primes dividing the level
4
Twist
5
Examples
Twists and residual modular Galois representations
Local representation
Local representation at `
Local representation at `
Theorem (Deligne)
Assume setting (∗). Suppose that f (T` ) 6= 0. Then ρf |G` is reducible,
and up to conjugation in GL2 (F` ), we have
k−1
∗
χ` λ((`)/f (T` ))
ρf |G` ∼
=
0
λ(f (T` ))
where λ(a) is the unramified character of G` taking Frob` ∈ G` /I` to a,
∗
for any a ∈ F` .
Twists and residual modular Galois representations
Local representation
Local representation at `
Theorem (Fontaine)
Assume setting (∗). Suppose that f (T` ) = 0. Then ρf |G` is irreducible,
and up to conjugation in GL2 (F` ), we have
0k−1
ϕ
0
ρf |I` ∼
=
0
ϕk−1
∗
where ϕ0 , ϕ : It → F` are the two fundamental characters of level 2.
Twists and residual modular Galois representations
Local representation
Local representation at primes dividing the level
Local representation at primes dividing the level
Theorem (Gross-Vignéras, Serre: Conjecture 3.2.6? )
Let ρ : GQ → GL(V ) be a continuous, odd, irreducible representation of
the absolute Galois group over Q to a 2-dimensional F` -vector space V .
Let n = N(ρ) and k = k(ρ), let f ∈ S(n, k)F` be an eigenform such that
ρf ∼
= ρ. Let p be a prime divisor of `n.
(1) If f (Tp ) 6= 0, then there exists a stable line D ⊂ V for the action of
Gp , the decomposition subgroup at p, such that the inertia group at
p acts trivially on V /D. Moreover, f (Tp ) is equal to the eigenvalue
of Frobp which acts on V /D.
(2) If f (Tp ) = 0, then there exists no stable line D ⊂ V as in (1).
Twists and residual modular Galois representations
Twist
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Local representation
4
Twist
Local representation and conductor
Twisting by Dirichlet characters
5
Examples
Twists and residual modular Galois representations
Twist
Local representation and conductor
Proposition (A.)
Assume setting (∗) and that ρf is irreducible and it does not arise from
lower level. Let p be a prime dividing n such that f (Tp ) 6= 0.
Then ρf |Gp is decomposable if and only if ρf |Ip is decomposable.
This proposition is proved using representation theory.
Twists and residual modular Galois representations
Twist
Local representation and conductor
Proposition (A.)
Assume setting (∗) and that ρf is irreducible and it does not arise from
lower level. Let p be a prime dividing n, such that f (Tp ) 6= 0. Then:
(a) ρf |Ip is decomposable if and only if Np (ρf ) = Np ();
(b) ρf |Ip is indecomposable if and only if Np (ρf ) = 1 + Np ().
Twists and residual modular Galois representations
Twist
Local representation and conductor
Sketch of the proof
The valuation of N(ρf ) at p is given by:
Np (ρf ) =
X
i≥0
1
dim(V /V Gi,p ) = dim(V /V Ip ) + b(V ),
[G0,p : Gi,p ]
where V is the two-dimensional F` -vector space underlying the
representation, V Gi,p is its subspace of invariants under Gi,p , and b(V ) is
the wild part of the conductor.
Since f (Tp ) 6= 0, the representation restricted to the decomposition
group at p is reducible. Hence, after conjugation,
k−1
1 |Ip ∗ ∼ |Ip ∗
χ
∗
ρf |Gp ∼
, ρf |Ip ∼
= 1 `
=
=
0 1
0
1
0
2
where 1 and 2 are characters of Gp with 2 unramified, χ` is the mod `
cyclotomic character and ∗ belongs to F` .
Twists and residual modular Galois representations
Twist
Local representation and conductor
Remark
If ρf |Ip is indecomposable then the image of inertia at p is of order
divisible by ` and so the image cannot be exceptional, hence it is big.
Twists and residual modular Galois representations
Twist
Twisting by Dirichlet characters
Let n be a positive integer. Any Dirichlet character of conductor n can
be decomposed into local characters, one for each prime divisor of n.
With no loss of generality, we reduce ourselves to study twists of modular
Galois representations with Dirichlet characters with prime power
conductor.
Twists and residual modular Galois representations
Twist
Twisting by Dirichlet characters
Question
What is the conductor of the twist?
Shimura gave an upper bound:
lcm(cond(χ)2 , n)
where n is the level of the form and χ is the character used for twisting.
Twists and residual modular Galois representations
Twist
Twisting by Dirichlet characters
Proposition (A.)
Assume setting (∗). Let p be a prime not dividing n`. Let
∗
χ : (Z /p i Z )∗ → F` , for i > 0, be a non-trivial character. Then
Np (ρf ⊗ χ) = 2Np (χ).
Twists and residual modular Galois representations
Twist
Twisting by Dirichlet characters
Proposition (A.)
Assume setting (∗) and that ρf is irreducible and it does not arise from
lower level. Let p be a prime dividing n and suppose that f (Tp ) 6= 0.
∗
Let χ : (Z /p i Z )∗ → F` , for i > 0, be a non-trivial character. Then
Np (ρf ⊗ χ) = Np (χ) + Np (χ).
Twists and residual modular Galois representations
Twist
Twisting by Dirichlet characters
It is also possible to know what is the system of eigenvalues associated to
the twist:
Proposition (A.)
Assume setting (∗). Suppose that ρf is irreducible and that N(ρf ) = n.
Let p be a prime dividing n and suppose that f (Tp ) 6= 0. Let χ from
∗
(Z /p i Z )∗ to F` , with i > 0, be a non-trivial character. Then
(a) if ρf |Ip is decomposable then the representation ρf ⊗ χ restricted to
Gp , the decomposition group at p, admits a stable line with
unramified quotient if and only if Np (ρf ⊗ χ) = Np (ρf );
(b) if ρf |Ip is indecomposable then the representation ρf ⊗ χ restricted to
Gp does not admit any stable line with unramified quotient.
Twists and residual modular Galois representations
Twist
Twisting by Dirichlet characters
Proposition (A.)
Assume setting (∗). Suppose that ρf is irreducible and that N(ρf ) = n.
Let p be a prime dividing n and suppose that f (Tp ) = 0. Then:
(a) if ρf |Gp is reducible then there exists a mod ` modular form g of
weight k and level at most np and a non-trivial character
∗
χ : (Z /p i Z )∗ → F` with i > 0 such that g (Tp ) 6= 0 and
∼
ρg = ρf ⊗ χ;
(b) if ρf |Gp is irreducible then for any non-trivial character
∗
χ : (Z /p i Z )∗ → F` with i > 0 the representation ρf ⊗ χ restricted
to Gp does not admit any stable line with unramified quotient.
Twists and residual modular Galois representations
Examples
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Local representation
4
Twist
5
Examples
Example 1
Example 2
Example 3
Twists and residual modular Galois representations
Examples
Example 1
Example 1
Let n = 135 = 33 5. Let be the Dirichlet character modulo 135 of
9
.
conductor 5 mapping 56 → 1, 82 → ζ36
two Galois orbits: the two Hecke eigenvalue fields are:
S(135, 3, )new
C
Q (x 16 + 217x 12 + 9264x 8 + 59497x 4 + 28561) and
Q (x 16 + 286x 12 + 16269x 8 + 85684x 4 + 62500).
In the first one, applying the reduction map for all prime ideals above 7,
we obtain the following eigenvalue systems defined over
F72 ∼
= F7 [x]/[x 2 + 6x + 4] ∼
= F7 [α]:
Twists and residual modular Galois representations
Examples
Example 1
p
f1 (Tp )
f2 (Tp )
f3 (Tp )
2
3
5
7
11
13
17
19
23
α
0
5α + 4
0
α+3
6α
6α
6α + 4
3α + 4
6α
0
2α + 3
0
6α + 4
6α
α
6α + 4
4α + 3
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
f1 ,f2 and f3 are mod 7 modular forms of level 135 and weight 3.
It is easy to verify that the corresponding representations are of minimal
level and weight.
Twists and residual modular Galois representations
Examples
Example 1
Let us focus on f1 .
Since 5 divides the level and f1 (T5 ) 6= 0:
2
1 χ7 ∗ ∼ −1
χ2
∼
ρf1 |G5 =
= 2 f1 7
0
2
0
0
2
The character of f1 has conductor 5 hence the representation is reducible
and decomposable, so ∗ = 0. Moreover 2 (5) = f1 (T5 ).
If we twist by −1
f1 then we have:
∼
(ρf1 ⊗ −1
f1 )|G5 =
2
−1
2 χ7
0
0
2 −1
f1
hence we know the eigenvalue at 5. The computation on the conductor
−1
2
gives that the conductor is 135 and (−1
2 χ7 )(5) = 42 (5) = 5α + 5.
Twists and residual modular Galois representations
Examples
Example 1
p
f1 (Tp )
f2 (Tp )
f3 (Tp )
ρf1 ⊗ −1
f1
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
α
0
5α + 4
0
α+3
6α
6α
6α + 4
3α + 4
2
4
α+6
5α + 1
α
2α
6α
0
2α + 3
0
6α + 4
6α
α
6α + 4
4α + 3
5
4
α+6
2α + 6
α
5α
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
5
4
6α
5α + 1
6α + 1
2α + 5
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
5
4
6α
5α + 1
6α + 1
2α + 5
Twists and residual modular Galois representations
Examples
Example 1
The level is also divisible by 3. But f1 (T3 ) = 0.
The local representation at 3 is irreducible: in order to prove this we have
to check all lover levels and possible twist.
In this case it is easy since the newforms space is empty in most cases.
The argument is similar for all levels, so we will just show what happens
for level 15.
p
f1 (Tp )
g1 (Tp )
g2 (Tp )
g3 (Tp )
g4 (Tp )
2
3
5
7
11
13
α
0
5α + 4
0
α+3
6α
6α + 6
6α
3α + 5
4α + 5
6α + 1
4α + 5
α+5
α+6
4α + 1
3α + 2
α
3α + 2
4α + 1
α
2α + 1
2
α
5
3α + 5
6α + 1
5α + 3
2
6α + 1
5
Twists and residual modular Galois representations
Examples
Example 1
It is possible to show that the image of the Galois representation up to
conjugation is
ρf1 (GQ ) ∼
=< α > SL2 (F7 ) ⊂ GL2 (F72 )
(f (Tp ))2 (f (p)p 2 )−1 7→ 6, 0, 6, 4, 5, 4, 5, 3, 1, 1, 6, 4, 3
Twists and residual modular Galois representations
Examples
Example 2
Example 2
Let us consider S(40, 2, τ )new
where τ is the quadratic character
C
conductor 40 mapping 31 → 1,21 → −1, 17 → −1.
As before, by reduction get mod 7 eigenvalue systems, for example fi for
i = 1, 2, 3, 4:
p
f1 (Tp )
f2 (Tp )
f3 (Tp )
f4 (Tp )
2
3
5
7
11
13
17
19
23
6
3
2
6
4
0
2
3
1
1
4
1
1
4
0
5
3
6
5
3
6
1
3
0
5
4
6
2
4
5
6
3
0
2
4
1
Twists and residual modular Galois representations
Examples
Example 2
Let χ be the character over F7 of conductor 8 such that χf1 has
conductor 4. Since f1 (T2 ) 6= 0, then we have that:
N2 (ρf ⊗ χ) = N2 (χf1 ) + N2 (χ) = 2 + 3 = 5 6= 6.
Hence, let us check the twist and the eigenvalue systems at level 160:
N2 (ρf ⊗ χ) = 25 5 = 160:
p
g1 (Tp )
g2 (Tp )
g3 (Tp )
g4 (Tp )
ρf1 ⊗ χ
ρf2 ⊗ χ
ρf3 ⊗ χ
ρf4 ⊗ χ
2
3
5
7
11
13
17
19
23
0
4
6
6
4
0
5
3
1
0
3
5
1
4
0
2
3
6
0
4
2
1
3
0
2
4
6
0
3
1
6
3
0
5
4
1
0
3
5
1
4
0
2
3
6
0
4
6
6
4
0
5
3
1
0
3
1
6
3
0
5
4
1
0
4
2
1
3
0
2
4
6
Twists and residual modular Galois representations
Examples
Example 2
All the systems given by gi are reduction of the same form.
Also in this case it is possible to prove that the image of the Galois
representation is big: in this case, it is isomorphic to GL2 (F7 ), up to
conjugation.
Twists and residual modular Galois representations
Examples
Example 3
Example 3
S(7, 3)new
, S(49, 3)new
C
C
The Hecke eigenvalue fields in newformspace at level 49 are given by:
Q (x 2 − 3x + 9)
Q (x 4 − 4x 3 + 22x 2 − 44x + 167)
Q (x 8 + 4x 7 − 6x 6 − 96x 5 + 225x 4 + 1336x 3 − 514x 2 − 5948x + 7399)
Number Field with defining polynomial
x 54 + 5x 53 + 43x 52 + 169x 51 + · · · + 248413945171829320x 16 +
· · · + 348605788594202619x 8 + · · · + 10909749546081
Number Field with defining polynomial
x 96 + 13x 95 + 69x 94 + 174x 93 + 16x 92 − 1672x 91 + . . .
while in level 7 . . . the Hecke eigenvalue field is Q .
Twists and residual modular Galois representations
Examples
Example 3
Let F52 ∼
= F5 [β].
= F5 [x]/[x 2 + 2x + 4] ∼
Let χ be Dirichlet character over F52 modulo 49 of conductor 7 mapping
3 → 3β.
p
f (Tp )
g (Tp )
ρf ⊗ χ
2
3
5
7
11
13
17
19
23
29
2
0
0
3
4
0
0
0
3
1
β
0
0
0
3β+1
0
0
0
4β
1
β
0
0
0
3β+1
0
0
0
4β
1
In this case we have that Pρf ∼
= A4 and ρf (GQ ) ∼
= F∗5 π −1 (A4 ), where A4
is the alternating group of 4 elements.
Twists and residual modular Galois representations
Examples
Example 3
Twists and residual modular Galois
representations
Samuele Anni
University of Warwick
Building Bridges, Bristol
10th July 2014
Thanks!