Residual modular Galois representations
Residual modular Galois
representations: images and
applications
Samuele Anni
University of Warwick
London Number Theory Seminar
King’s College London, 20th May 2015
Residual modular Galois representations
Mod ` modular forms
1
Mod ` modular forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Local representation
5
Twist
6
Applications
Residual modular Galois representations
Mod ` modular forms
Let n be a positive integer. 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
Γ1 (n) acts on the complex upper half plane via fractional transformations.
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 .
The modular curve Y1 (n)C can be defined algebraically over Z [1/n].
Residual modular Galois representations
Mod ` modular forms
Let n ≥ 5 and k be positive integers and 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 )∗ .
The Z -subalgebra of EndC (S(Γ1 (n), k)C ) generated by Tp and diamond
operators hdi is the Hecke algebra T(n, k).
Analogous definition in characteristic zero and over any ring where n is
invertible.
Residual modular Galois representations
Mod ` 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 }.
T (n, k) will denote the associated Hecke algebra.
Residual modular Galois representations
Residual modular Galois representations
1
Mod ` modular forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Local representation
5
Twist
6
Applications
Residual modular Galois representations
Residual modular Galois representations
Theorem (Deligne, Serre, 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
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):
Example: for n = 1, k = 22 and ` = 23, the number field corresponding
to Pρf (Galois group isomorphic to PGL2 (F23 )) 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
8
+ 14586202192x + 29414918270x + 45332850431x − 6437110763x
3
2
4
− 111429920358x − 12449542097x + 93960798341x − 31890957224
Mascot, Zeng, Tian (#F ≤ 41).
12
Residual modular Galois representations
Residual modular Galois representations
Question
Can we compute the image of a residual modular Galois representation
without computing the representation?
Residual modular Galois representations
Image: an algorithm
1
Mod ` modular forms
2
Residual modular Galois representations
3
Image: an algorithm
Finite subgroup of PGL2 (F` )
Serre’s Conjecture
The algorithm
4
Local representation
5
Twist
6
Applications
Residual modular Galois representations
Image: an algorithm
Main ingredients:
1
Classification of possible images
2
Serre’s conjecture
Residual modular Galois representations
Image: an algorithm
Finite subgroup of PGL2 (F` )
1
Classification of possible images
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;
PSL2 (F`r ) 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
Image: an algorithm
Finite subgroup of PGL2 (F` )
The field of definition of the representation is the smallest field F ⊂ F`
over which ρf is equivalent to all its conjugate. The image of the
representation ρf is then a subgroup of GL2 (F).
Let Pρf : Gal(Q /Q ) → PGL2 (F) be the projective representation
associated to the representation ρf :
Gal(Q /Q )
ρf
/ GL2 (F)
π
Pρf
& PGL2 (F).
The representation Pρf can be defined on a different field than the field
of definition of the representation. This field is called the Dickson’s field
for the representation.
Residual modular Galois representations
Image: an algorithm
Serre’s Conjecture
2
Serre’s conjecture
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:
k(ρ)−1
det ρ = (ρ)χ`
,
where χ` is the cyclotomic character mod `.
Residual modular Galois representations
Image: an algorithm
The algorithm
The algorithm
Theorem
There is a polynomial time algorithm which takes as input:
n and k positive integers, n is given with its factorization;
` a prime not dividing n and such that 2 ≤ k ≤ ` + 1;
a character : (Z /nZ )∗ → C∗ ;
a morphism of ring f : T (n, k) → F` , i.e. the images of all diamond
operators and of the Tp operators up to a bound B(n, k),
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
Image: an algorithm
The algorithm
In almost all cases, the bound B(n, k) is the Sturm Bound for Γ0 (n) and
weight k:
Y k
1
k
B(n, 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
Image: an algorithm
The algorithm
For an input as in the previous slide we run into the following problem:
Problems
ρf can arise from lower level or weight:
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 level or weight: there exist a
character χ and g ∈ S(m, j)F` with m ≤ n
or j ≤ k such that
ρg ⊗ χ ∼
= ρf
Algorithm
Iteration “down to top”,
i.e. considering all
divisors of n: creation of
a database
Determine minimality
with respect to level and
with respect to weight.
Determine minimality
up to twisting.
Residual modular Galois representations
Image: an algorithm
The algorithm
Algorithm
Step 1 Iteration “down to
top”
Step 2 Determine minimality
with respect to level and
weight.
Step 3 Determine whether
reducible or irreducible.
Step 4 Determine minimality
up to twisting.
Step 5 Compute the
projective image
Step 6 Compute the image
Remarks
Reducibility is checked by comparing
with systems of eigenvalues coming
from specific Eisenstein series.
Compute the field of definition of the
representation: this can be done using
coefficients up to a finite bound.
Compute the Dickson’s field: this is
obtained twisting.
The projective image is determined by
excluding cases. Each case is related
to an explicit equality of mod `
modular forms or construction.
Residual modular Galois representations
Image: an algorithm
The algorithm
Setting (∗)
n and k positive integers;
` a prime not dividing n and such that 2 ≤ k ≤ ` + 1;
: (Z /nZ )∗ → C∗ a character;
f : T (n, k) → F` a morphism of rings;
ρf : GQ → GL2 (F` ) the unique, up to isomorphism, continuous
semi-simple representation attached to f ;
∗
∗
: (Z /nZ ) → F` be the character defined by det(ρf ) = χk−1
.
`
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;
Gi,p , with i ∈ Z >0 , the higher ramification subgroups;
Np (ρ) the valuation at p of the Artin conductor of ρ, where ρ is a
residual representation.
Residual modular Galois representations
Local representation
1
Mod ` modular forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Local representation
Local representation and conductor
5
Twist
6
Applications
Residual modular Galois representations
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).
(1) ⇒ ρf |Gp is reducible;
(2) ⇒ ρf |Gp is irreducible.
Residual modular Galois representations
Local representation
Local representation and conductor
From now on we will assume setting (∗).
Proposition
Let us suppose 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.
Can we deduce a computational criterion from this?
Residual modular Galois representations
Local representation
Local representation and conductor
Proposition
Assume 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
Local representation
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 the 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 ∗
χ
∗
ρf |G p ∼
, ρ f | Ip ∼
= 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` .
Residual modular Galois representations
Local representation
Local representation and conductor
Sketch of the proof
ρf |Ip ∼
=
1 | Ip
0
∗
.
1
If ρf |Ip is indecomposable (∗ =
6 0) then V Ip is either {0} if 1 is ramified,
1
or F` · ( 0 ) if 1 is unramified. The wild part of the conductor is equal to
the wild part of the conductor of 1 . Hence, we have that
(
1 = 1 + Np (1 )
if 1 is unramified,
Np (ρf ) =
2 + b(1 ) = 1 + Np (1 ) if 1 is ramified.
Since det(ρf ) = χk−1
, then det(ρf )|Ip = |Ip , so 1 |Ip = |Ip . Therefore,
`
we have that if ρf |Ip is indecomposable then Np (ρf ) = 1 + Np ().
The other case is analogous.
Residual modular Galois representations
Twist
1
Mod ` modular forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Local representation
5
Twist
Twisting by Dirichlet characters
The conductor of a twist
The system of eigenvalues of a twist
Example
Fields of definition
6
Applications
Residual modular Galois representations
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 level or
weight, i.e. there exist g ∈ S(m, j)F` with m ≤ n or j ≤ k and a
character χ such that ρg ⊗ χ ∼
= ρf .
Question
Can we compute twists?
For this talk, we limit ourselves to study twists of modular Galois
representations by Dirichlet characters with prime power conductor
coprime with the characteristic.
Residual modular Galois representations
Twist
The conductor of a twist
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 Dirichlet character used for
twisting.
Residual modular Galois representations
Twist
The conductor of a twist
Proposition
∗
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).
Residual modular Galois representations
Twist
The conductor of a twist
Proposition
Assume 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
Twist
The system of eigenvalues of a twist
It is also possible to know what is the system of eigenvalues associated
to the twist:
Proposition
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.
(a) ⇒ (ρf ⊗ χ)(Frobp ) 6= 0;
(b) ⇒ (ρf ⊗ χ)(Frobp ) = 0;
Residual modular Galois representations
Twist
The system of eigenvalues of a twist
Analogous result holds when f (Tp ) = 0, where p | N(ρf ):
Proposition
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
Twist
Example
Example: let n = 135 = 33 5. Let be the Dirichlet character modulo
9
.
135 of conductor 5 mapping 56 → 1, 82 → ζ36
S(135, 3, )new
two Galois orbits, the two Hecke eigenvalue fields are:
C
Q (x 16 + 217x 12 + 9264x 8 + 59497x 4 + 28561) and
Q (x 16 + 286x 12 + 16269x 8 + 85684x 4 + 62500).
We want to compute the image of the mod 7 representations attached to
the reduction of the forms in the first Galois orbit.
Applying the reduction map for all prime ideals above 7, we obtain the
following eigenvalue systems defined over F7 [x]/[x 2 + 6x + 4] ∼
= F7 [α]:
Residual modular Galois representations
Twist
Example
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
Twist
Example
Let us focus on f1 .
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
Residual modular Galois representations
Twist
Example
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 f1 attached to 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, the eigenvalues at primes p 6= 5 are
given by f (Tp )−1
f1 (p) while the eigenvalue at 5 is
−1
2
(−1
2 χ7 )(5) = 42 (5) = 5α + 5.
Residual modular Galois representations
Twist
Example
p
2
3
5
7
11
13
17
19
23
29
31
37
41
f1 (Tp )
α
0
5α + 4
0
α+3
6α
6α
6α + 4
3α + 4
2
4
α+6
5α + 1
f2 (Tp )
6α
0
2α + 3
0
6α + 4
6α
α
6α + 4
4α + 3
5
4
α+6
2α + 6
f3 (Tp )
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
5
4
6α
5α + 1
∼
ρf1 ⊗ −1
f1 = ρf3
ρf1 ⊗ −1
f1
α+6
0
5α + 5
0
α+3
α+6
6α + 1
α+3
3α
5
4
6α
5α + 1
Residual modular Galois representations
Twist
Example
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
Twist
Example
It is possible to show that
Pρf1 (GQ ) ∼
= PSL2 (F7 ) ⊂ PGL2 (F72 ),
and that the image of the Galois representation up to conjugation is
ρf1 (GQ ) ∼
=< α > SL2 (F7 ) ⊂ GL2 (F72 )
Residual modular Galois representations
Twist
Fields of definition
An application of the study of twists is the computation of the field of
definition of the representation and of the projective representation
associated.
Residual modular Galois representations
Twist
Fields of definition
Field of definition of the representation:
Proposition
Suppose that ρf is irreducible and does not arise from lower level or
weight. Then the field of definition of ρf is the smallest extension of F`
containing the elements of the set
S := {f (Tp )| p prime, p 6= ` and p ≤ B(n, k)} ∪ {f (hdi)| d ∈ (Z /nZ )∗ } ,
where B(n, k) is the Sturm bound for cusp forms for Γ0 (n).
Residual modular Galois representations
Twist
Fields of definition
Field of definition of the projective representation Pρf
Proposition
Let us suppose that ρf is irreducible, it does not arise from lower level or
weight and it is realized over F. Then the field of definition of Pρf is the
subfield of F fixed by
A : = σ ∈ Aut(F)| ∃τ : Gal(Q /Q ) → F∗ : ρσf ∼
= ρf ⊗ τ .
Residual modular Galois representations
Twist
Fields of definition
Proposition
Assume that ρf is irreducible, does not arise from lower level or weight
and is realized over F. Then the field of definition of Pρf is the smallest
extension of F` containing
(f (Tp ))2 /f (hpi)p k−1
for all p prime, p - n`.
This proposition is not effective: there is no bound for the number of
traces needed. Anyway, combining it with the previous, there is an
algorithm to compute the Dickson’s field.
Residual modular Galois representations
Applications
1
Mod ` modular forms
2
Residual modular Galois representations
3
Image: an algorithm
4
Local representation
5
Twist
6
Applications
Diophantine equations
Graphs and modularity lifting
Residual modular Galois representations
Applications
Diophantine equations
Proposition
Let p ≥ 5 be a prime and r ≥ 1 be an integer. Then there are no
solutions to
17r x p + 2y p = z 2
for non-zero coprime integers x, y , z.
Proof
If xy = ±1, then there is no solution (reduce modulo 8).
If xy 6= ±1, then, without loss of generality, we can assume that 17r x, 2y
and z are coprime, 1 ≤ r < p, and xy is odd. The solutions are then
related to the Frey curve
E : Y 2 + XY = X 3 + 2zX 2 + 2y p X
which has minimal discriminant 28 17r (xy 2 )p and conductor 27 Rad(17xy ).
Residual modular Galois representations
Applications
Diophantine equations
Proof.
It is possible to show that:
E does not have complex multiplication
E arises mod ` from some newform f ∈ S(27 17, 2) for primes `
dividing a constant given by an explicit recipe by Mazur. For
example ` = 3 works.
There are 16 Galois orbits of newforms of level 27 17 weight 2 which all
are irrational.
The mod 3 representation cannot arise from a lower level: this would
contradict Ribet’s Level-Lowering Theorem. It is irreducible because the
Frey curve does not admit 3 isogenies.
The field of definitions of the mod 3 Galois representations associated to
each of these newforms is larger than F3 : for all the newforms the
reduction of f (T5 ) ∈
/ F3 . Contradiction.
Residual modular Galois representations
Applications
Graphs and modularity lifting
Work on progress, joint with Vandita Patel (University of Warwick).
Idea: build graphs of congruences between modular forms and enrich
these graphs using datas coming form residual representations.
Vertices: newforms of levels and weights in a given set;
Edges: an eadge between two verices is drawn if a relation R holds.
Residual modular Galois representations
Applications
Graphs and modularity lifting
What kind of relations we are going to consider?
R1 : congruence modulo some prime `;
R2 : isomorphic mod ` Galois representations for some prime `;
R3 : isomorphic mod ` projective Galois representations for some
prime `.
until now we implemented an algorithm which draws graphs for R1 .
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois representations
Applications
Graphs and modularity lifting
Residual modular Galois
representations: images and
applications
Samuele Anni
University of Warwick
London Number Theory Seminar
King’s College London, 20th May 2015
Thanks!