Residual modular Galois representations: an algorithmic approach
Residual modular Galois
representations:
an algorithmic approach
Samuele Anni
University of Warwick
Algebra and Number Theory Seminar
University College Dublin
22nd September 2014
Residual modular Galois representations: an algorithmic approach
Modular curves and Modular Forms
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Twist
5
Projective image S4 : a construction
Residual modular Galois representations: an algorithmic approach
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.
Residual modular Galois representations: an algorithmic approach
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
Residual modular Galois representations: an algorithmic approach
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]).
Residual modular Galois representations: an algorithmic approach
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
Residual modular Galois representations: an algorithmic approach
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.
Residual modular Galois representations: an algorithmic approach
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.
Residual modular Galois representations: an algorithmic approach
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: let : (Z /nZ )∗ → C∗ be a character, whose image is
contained in OK , ring of integers of the number field K , then the map
S(n, k, )OK → S(n, k, )F
is not always surjective even if k > 1, where 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 }
Residual modular Galois representations: an algorithmic approach
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 }.
Residual modular Galois representations: an algorithmic approach
Residual modular Galois representations
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Twist
5
Projective image S4 : a construction
Residual modular Galois representations: an algorithmic approach
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.
Residual modular Galois representations: an algorithmic approach
Residual modular Galois representations
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).
For example: n = 1, k = 22 and ` = 23 then the number field
corresponding to Pρf is given by:
x
24
− 2x
23
+ 115x
+ 1837654x
16
+ 279370098x
22
+ 23x
21
− 800032x
11
+ 1909x
15
20
+ 22218x
+ 9856374x
+ 1464085056x
7
10
14
19
+ 9223x
+ 52362168x
9
18
13
+ 121141x
− 32040725x
+ 1129229689x + 3299556862x
6
5
17
+ 14586202192x + 29414918270x + 45332850431x − 6437110763x
3
2
4
− 111429920358x − 12449542097x + 93960798341x − 31890957224
Mascot, Zeng, Tian (#F ≤ 41).
12
8
Residual modular Galois representations: an algorithmic approach
Residual modular Galois representations
Question
Can we compute the image of a residual modular Galois representation
without computing the representation?
Why one should be interested in this?
To obtain explicit answer to question proposed by Serre, Ribet... for
example, uniform bounds for surjectivity of residual representations
for elliptic curves and abelian varieties.
The inverse Galois problem.
Diophantine equations.
In lifting theorems for representations and chain of congruences
techniques one of the hypothesis is big residual image.
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Twist
5
Projective image S4 : a construction
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
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 subgroup 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 .
Definition
If G := ρf (Gal(Q /Q )) has order prime to ` we call the image
exceptional.
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
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 .
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
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 .
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
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.
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
Problems
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 .
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
Algorithm
Step 1 Iteration “down to
top”
Remarks
Step 2 Determine minimality
with respect to level and
weight.
Reducibility is checked by comparing
with systems of eigenvalues coming
from specific Eisenstein series.
Step 3 Determine whether
reducible or irreducible.
The projective image is determined by
excluding cases. Each exceptional
case is related to a particular equality
of mod ` modular forms or a
particular construction.
Step 4 Determine minimality
up to twisting.
Step 5 Compute the
projective image
Step 6 Compute the image
Residual modular Galois representations: an algorithmic approach
Image: an algorithm
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;
Given a residual representation ρ, we will denote as Np (ρ) the valuation
at p of the Artin conductor of ρ.
Residual modular Galois representations: an algorithmic approach
Twist
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Twist
Local representation
Twisting by Dirichlet characters
Example 1
Example 2
Example 3
5
Projective image S4 : a construction
Residual modular Galois representations: an algorithmic approach
Twist
Local representation
Theorem (Gross-Vignéras-Fontaine,
Serre: Conjecture 3.2.6? )
Let
the
Let
Let
ρ : GQ → GL(V ) be a continuous, odd, irreducible representation of
absolute Galois group over Q to a 2-dimensional F` -vector space V .
f ∈ S(N(ρ), k(ρ))F` be an eigenform such that ρf ∼
= ρ.
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 , such that Ip 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).
Residual modular Galois representations: an algorithmic approach
Twist
Local representation
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.
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 ().
Residual modular Galois representations: an algorithmic approach
Twist
Twisting by Dirichlet characters
Assume setting (∗) and that ρf is irreducible and it does not arise from
lower level or weight.
Then ρf can still arise as twist of a representation of lower conductor or
weight, 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 .
Question
Can we fill in the database for arbitrary twists?
Equivalently, what is the relation between the conductors, the weights
and the eigenvalue systems corresponding to a representation and its
twist by a character?
Residual modular Galois representations: an algorithmic approach
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.
For this talk, we limit ourselves to study twists of modular Galois
representations with Dirichlet characters with prime power conductor and
unramified at `.
Residual modular Galois representations: an algorithmic approach
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.
Residual modular Galois representations: an algorithmic approach
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).
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).
Residual modular Galois representations: an algorithmic approach
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 , 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.
Residual modular Galois representations: an algorithmic approach
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.
Residual modular Galois representations: an algorithmic approach
Twist
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]:
Residual modular Galois representations: an algorithmic approach
Twist
Example 1
p
2
3
5
7
11
13
17
19
23
f1 (Tp )
α
0
5α + 4
0
α+3
6α
6α
6α + 4
3α + 4
f2 (Tp )
6α
0
2α + 3
0
6α + 4
6α
α
6α + 4
4α + 3
f3 (Tp )
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.
Residual modular Galois representations: an algorithmic approach
Twist
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 ) = 5α + 4.
If we twist by −1
f1 then we have:
∼
(ρf1 ⊗ −1
f1 )|G5 =
2
−1
2 χ7
0
0
2 −1
f1
The conductor of the twist is 135 and the eigenvalue at 5 is
−1
2
(−1
2 χ7 )(5) = 42 (5) = 5α + 5.
Residual modular Galois representations: an algorithmic approach
Twist
Example 1
p
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
f1 (Tp )
α
0
5α + 4
0
α+3
6α
6α
6α + 4
3α + 4
2
4
α+6
5α + 1
α
2α
f2 (Tp )
6α
0
2α + 3
0
6α + 4
6α
α
6α + 4
4α + 3
5
4
α+6
2α + 6
α
5α
f3 (Tp )
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
5
4
6α
5α + 1
6α + 1
2α + 5
ρf1 ⊗ −1
f1
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
5
4
6α
5α + 1
6α + 1
2α + 5
Residual modular Galois representations: an algorithmic approach
Twist
Example 1
The level is also divisible by 3. But f1 (T3 ) = 0.
The local representation at 3 is irreducible: to prove this claim 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, let us what happens for level 15.
p
2
3
5
7
11
13
f1 (Tp )
α
0
5α + 4
0
α+3
6α
g1 (Tp )
6α + 6
6α
3α + 5
4α + 5
6α + 1
4α + 5
g2 (Tp )
α+5
α+6
4α + 1
3α + 2
α
3α + 2
g3 (Tp )
4α + 1
α
2α + 1
2
α
5
g4 (Tp )
3α + 5
6α + 1
5α + 3
2
6α + 1
5
Residual modular Galois representations: an algorithmic approach
Twist
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 )
Residual modular Galois representations: an algorithmic approach
Twist
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
2
3
5
7
11
13
17
19
23
f1 (Tp )
6
3
2
6
4
0
2
3
1
f2 (Tp )
1
4
1
1
4
0
5
3
6
f3 (Tp )
5
3
6
1
3
0
5
4
6
f4 (Tp )
2
4
5
6
3
0
2
4
1
Residual modular Galois representations: an algorithmic approach
Twist
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
2
3
5
7
11
13
17
19
23
g1 (Tp )
0
4
6
6
4
0
5
3
1
g2 (Tp )
0
3
5
1
4
0
2
3
6
g3 (Tp )
0
4
2
1
3
0
2
4
6
g4 (Tp )
0
3
1
6
3
0
5
4
1
ρf1 ⊗ χ
0
3
5
1
4
0
2
3
6
ρf2 ⊗ χ
0
4
6
6
4
0
5
3
1
ρf3 ⊗ χ
0
3
1
6
3
0
5
4
1
ρf4 ⊗ χ
0
4
2
1
3
0
2
4
6
Residual modular Galois representations: an algorithmic approach
Twist
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.
Residual modular Galois representations: an algorithmic approach
Twist
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 .
Residual modular Galois representations: an algorithmic approach
Twist
Example 3
Let F52 ∼
= F5 [x]/[x 2 + 2x + 4] ∼
= F5 [β].
Let χ be Dirichlet character over F52 modulo 49 of conductor 7 mapping
3 → 3β.
p
2
3
5
7
11
13
17
19
23
29
f (Tp )
2
0
0
3
4
0
0
0
3
1
g (Tp )
β
0
0
0
3β+1
0
0
0
4β
1
ρf ⊗ χ
β
0
0
0
3β+1
0
0
0
4β
1
In this case we have that Pρf ∼
= A4 and ρf (GQ ) ∼
= F∗5 A˜4 .
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
1
Modular curves and Modular Forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Twist
5
Projective image S4 : a construction
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
In the cases where the projective image is isomorphic to S4 , A4 or A5 , in
the algorithm we proceed using characteristic switching technique.
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.
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
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 .
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
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 2-dimensional representation,
where OK is the ring of integers of a number field.
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
)
/ GL2 (OK )
There exists fβ of weight 1 such that ρfβ ∼
= β ◦ ρf .
Can we determine the level of fβ ?
Yes, studying the local representation at primes dividing n and at 3.
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.
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
ρfβ
Gal(Q /Q )
ρf
/ F∗ GL2 (F3 )
β
ρfπβ
'
/ GL2 (OK )
ρfπβ (Gal(Q /Q )) ⊆ F0∗ × GL2 (F2 )
Pρfπβ (Gal(Q /Q )) ∼
= S3
π
,
GL2 (F2 )
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
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. If this method
does not work use Schaeffer’s Algorithm.
Residual modular Galois representations: an algorithmic approach
Projective image S4 : a construction
Residual modular Galois
representations:
an algorithmic approach
Samuele Anni
University of Warwick
Algebra and Number Theory Seminar
University College Dublin
22nd September 2014
Thanks!