Computing Klein modular forms
Computing Klein modular forms
Aurel Page
Universit ´e de Bordeaux
July 24, 2014
Paris 13
Workshop on Analytic Number Theory and Geometry
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1
2
3
4
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Elliptic curves and automorphic forms
Want a database of elliptic curves over number field F ?
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Elliptic curves and automorphic forms
Want a database of elliptic curves over number field F ?
Start with GL
2 automorphic forms!
Where do you find automorphic forms?
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Elliptic curves and automorphic forms
Want a database of elliptic curves over number field F ?
Start with GL
2 automorphic forms!
Where do you find automorphic forms?
In the cohomology of arithmetic groups!
Matsushima’s formula: Γ discrete cocompact subgroup of connected Lie group G , E representation of G .
H i
(Γ , E )
M
Hom ( π, L
2
(Γ \ G )) ⊗ H i
( g , K ; π ⊗ E )
π ∈ b
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Automorphic forms for GL
2
Where do you find GL
2 automorphic forms?
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Automorphic forms for GL
2
Where do you find GL
2 automorphic forms?
H ∗ ( GL
2
( Z
F
) , E )
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Automorphic forms for GL
2
Where do you find GL
2 automorphic forms?
H ∗ ( GL
2
( Z
F
) , E )
Jacquet–Langlands: H ∗ ( O × , E ) , O order in a quaternion algebra
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Automorphic forms for GL
2
Where do you find GL
2 automorphic forms?
H ∗ ( GL
2
( Z
F
) , E )
Jacquet–Langlands: H ∗ ( O × , E ) , O order in a quaternion algebra
Kleinian case: O × ⊂ GL
2
( C ) , H
1
( O ×
, E )
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Automorphic forms for GL
2
Where do you find GL
2 automorphic forms?
H ∗ ( GL
2
( Z
F
) , E )
Jacquet–Langlands: H ∗ ( O × , E ) , O order in a quaternion algebra
Kleinian case: O
Call H i
( O
× ⊂ GL
2
( C ) , H
1
( O ×
, E )
×
, E ) a space of Klein modular forms .
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Motivation for the Kleinian case
Elliptic curves, abelian varieties of GL
2 fields.
type over number
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Motivation for the Kleinian case
Elliptic curves, abelian varieties of GL
2 fields.
type over number
Torsion phenomenon .
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Arithmetic Kleinian groups
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Kleinian groups
H := C + C j where j
2
= − 1 and jz = ¯ for all z ∈ C .
The upper half-space H 3
Metric d s
2 =
| d z | 2 + d t t 2
2
:= C
, volume d
+
V
R
> 0
= j .
d x d y d t t 3
.
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Kleinian groups
H := C + C j where j
2
= − 1 and jz = ¯ for all z ∈ C .
The upper half-space H 3
Metric d s
2 =
| d z | 2 + d t t 2
2
:= C
, volume d
+
V
R
> 0
= j .
d x d y d t t 3
.
a b c d
· w := ( aw + b )( cw + d )
− 1 for a b c d
∈ SL
2
( C ) .
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Kleinian groups
H := C + C j where j
2
= − 1 and jz = ¯ for all z ∈ C .
The upper half-space H 3
Metric d s
2 =
| d z | 2 + d t t 2
2
:= C
, volume d
+
V
R
> 0
= j .
d x d y d t t 3
.
a b c d
· w := ( aw + b )( cw + d )
− 1 for a b c d
∈ SL
2
( C ) .
Kleinian group: discrete subgroup of PSL
2
( C ) .
Cofinite if it has finite covolume.
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Quaternion algebras
A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F .
Explicitly, B
(char F = 2).
= a , b
F
= F + Fi + Fj + Fij , i
2 = a , j
2 = b , ij = − ji
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Quaternion algebras
A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F .
Explicitly, B =
(char F = 2).
a , b
F
= F + Fi + Fj + Fij , i
2 = a , j
2 = b , ij = −
Reduced norm: nrd ( x + yi + zj + tij ) = x
2 − ay
2 − bz
2
+ abt
2
.
ji
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Quaternion algebras
A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F .
Explicitly, B =
(char F = 2).
a , b
F
= F + Fi + Fj + Fij , i
2 = a , j
2 = b , ij = −
Reduced norm: nrd ( x + yi + zj + tij ) = x
2 − ay
2 − bz
2
+ abt
2
.
ji
Z
F integers of F number field.
Order O ⊂ B : subring, finitely generated Z
F
-module, O F = B .
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Quaternion algebras
A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F .
Explicitly, B =
(char F = 2).
a , b
F
= F + Fi + Fj + Fij , i
2 = a , j
2 = b , ij = −
Reduced norm: nrd ( x + yi + zj + tij ) = x
2 − ay
2 − bz
2
+ abt
2
.
ji
Z
F integers of F number field.
Order O ⊂ B : subring, finitely generated Z
F
-module, O F = B .
A place v of F is split or ramified according as whether B ⊗
F
F v is M
2
( F v
) or a division algebra.
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Covolume formula
F almost totally real number field: exactly one complex place.
B / F Kleinian quaternion algebra: ramified at every real place.
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Covolume formula
F almost totally real number field: exactly one complex place.
B / F Kleinian quaternion algebra: ramified at every real place.
O order in B , Γ image of reduced norm one group O 1 in PSL
2
( C ) .
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Covolume formula
F almost totally real number field: exactly one complex place.
B / F Kleinian quaternion algebra: ramified at every real place.
O order in B , Γ image of reduced norm one group O 1 in PSL
2
( C ) .
Theorem
Γ is a cofinite Kleinian group.
It is cocompact iff B is a division algebra.
If O is maximal then
Covol (Γ) =
| ∆
F
| 3 / 2
ζ
F
( 2 )
( 4 π
Q
2 ) p ram.
( N ( p ) − 1 )
[ F : Q ] − 1
·
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Algorithms
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Sketch of algorithm
1 Compute B and O
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Sketch of algorithm
1
2
Compute B and O
Compute O 1
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Sketch of algorithm
1
2
3
Compute B and O
Compute O 1
Compute H 1 ( O 1 )
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Sketch of algorithm
1
2
3
4
Compute B and O
Compute O 1
Compute H 1 ( O 1 )
Compute generator δ of ideal of norm p
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Sketch of algorithm
1
2
3
4
5
Compute B and O
Compute O 1
Compute H 1 ( O 1 )
Compute generator δ of ideal of norm p
Compute Hecke operator T
δ on H
1
( O 1
)
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Fundamental domains
Γ a Kleinian group. An open subset F ⊂ H 3 domain if is a fundamental
Γ · F = H 3
F ∩ γ F = ∅ for all 1 = γ ∈ Γ .
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Dirichlet domains
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Dirichlet domains
Let p ∈ H 3 with trivial stabilizer in Γ . The Dirichlet domain
D p
(Γ) := { x ∈ X | d ( x , p ) < d ( γ · x , p ) ∀ γ ∈ Γ \ { 1 }}
= { x ∈ X | d ( x , p ) < d ( x , γ
− 1 · p ) ∀ γ ∈ Γ \ { 1 }} .
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Dirichlet domains
Let p ∈ H 3 with trivial stabilizer in Γ . The Dirichlet domain
D p
(Γ) := { x ∈ X | d ( x , p ) < d ( γ · x , p ) ∀ γ ∈ Γ \ { 1 }}
= { x ∈ X | d ( x , p ) < d ( x , γ
− 1 · p ) ∀ γ ∈ Γ \ { 1 }} .
is a fundamental domain for Γ that is a hyperbolic polyhedron.
If Γ is cofinite, D p
(Γ) has finitely many faces.
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Structure of the Dirichlet domain
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Structure of the Dirichlet domain
Faces of the Dirichlet domain are grouped into pairs , corresponding to elements g , g
− 1 ∈ Γ ;
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Structure of the Dirichlet domain
Faces of the Dirichlet domain are grouped into pairs , corresponding to elements g , g
− 1 ∈ Γ ;
Edges of the domain are grouped into cycles , product of corresponding elements in Γ has finite order .
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Poincar ´e’s theorem
Theorem (Poincar ´e)
The elements corresponding to the faces are generators of Γ . The relations corresponding to the edge cycles generate all the relations among the generators.
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Poincar ´e’s theorem
Theorem (Poincar ´e)
The elements corresponding to the faces are generators of Γ . The relations corresponding to the edge cycles generate all the relations among the generators.
If a partial Dirichlet domain D p
( S ) has a face-pairing and cycles of edges, then it is a fundamental domain for the group generated by S.
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Reduction algorithm x ∈ H 3 , S ⊂ Γ corresponding to the faces of D p
(Γ) .
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Reduction algorithm x ∈ H 3 , S ⊂ Γ corresponding to the faces of D p
(Γ) .
if possible x ← gx for some g ∈ S s.t. d ( x , p ) decreases
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Reduction algorithm x ∈ H 3 , S ⊂ Γ corresponding to the faces of D p
(Γ) .
if possible x ← gx for some g ∈ S s.t. d ( x , p ) decreases repeat
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Reduction algorithm x ∈ H 3 , S ⊂ Γ corresponding to the faces of D p
(Γ) .
if possible x ← gx for some g ∈ S s.t. d ( x , p ) decreases repeat
→ point x
′ = g k
· · · g
1 x s.t.
x
′ ∈ D p
(Γ) .
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Reduction algorithm x ∈ H 3 , S ⊂ Γ corresponding to the faces of D p
(Γ) .
if possible x ← gx for some g ∈ S s.t. d ( x , p ) decreases repeat
→ point x
′ = g k
· · · g
1 x s.t.
x
′ ∈ D p
(Γ) .
In particular for γ ∈ Γ , take x = γ to x ′ = p , to write γ
− 1 p , which will reduce as a product of the generators.
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Computation in hyperbolic space
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Computation in hyperbolic space
We have : an explicit formula for the hyperbolic distance;
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Computation in hyperbolic space
We have : an explicit formula for the hyperbolic distance; an explicit formula for bisectors;
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Computation in hyperbolic space
We have : an explicit formula for the hyperbolic distance; an explicit formula for bisectors; an algorithm for computing intersections of half-spaces;
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Computation in hyperbolic space
We have : an explicit formula for the hyperbolic distance; an explicit formula for bisectors; an algorithm for computing intersections of half-spaces; an algorithm for computing the volume of a polyhedron.
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Algorithm
Suppose O maximal.
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Algorithm
Suppose O maximal.
Enumerate elements of O 1 in a finite set S
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Algorithm
Suppose O maximal.
Enumerate elements of O 1 in a finite set S
Compute D = D p
( S )
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Algorithm
Suppose O maximal.
Enumerate elements of O 1 in a finite set S
Compute D = D p
( S )
Repeat until D has a face-pairing and Vol ( D ) < 2 Covol (Γ)
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Complexity
Why would you care about the complexity?
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Complexity
Why would you care about the complexity?
Bound on the size of the Dirichlet domain:
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Complexity
Why would you care about the complexity?
Bound on the size of the Dirichlet domain:
SL
2
( C ) action on π = L 2
0
(Γ \ SL
2
( C )) has property ( τ ) :
There are explicit Q ⊂ SL
2
( C ) compact and ǫ > 0, s.t.
for all f ∈ π , there exists g ∈ Q s.t.
k gf − f k ≥ ǫ k f k .
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Complexity
Why would you care about the complexity?
Bound on the size of the Dirichlet domain:
SL
2
( C ) action on π = L 2
0
(Γ \ SL
2
( C )) has property ( τ ) :
There are explicit Q ⊂ SL
2
( C ) compact and ǫ > 0, s.t.
for all f ∈ π , there exists g ∈ Q s.t.
k gf − f k ≥ ǫ k f k .
⇒ diam ( D p
(Γ)) ≪ log Covol (Γ)
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Complexity
Why would you care about the complexity?
Bound on the size of the Dirichlet domain:
SL
2
( C ) action on π = L 2
0
(Γ \ SL
2
( C )) has property ( τ ) :
There are explicit Q ⊂ SL
2
( C ) compact and ǫ > 0, s.t.
for all f ∈ π , there exists g ∈ Q s.t.
k gf − f k ≥ ǫ k f k .
⇒ diam ( D p
(Γ)) ≪ log Covol (Γ)
Proved complexity: Covol (Γ)
O ( 1 )
Observed complexity: Covol (Γ) 2
Lower bound: Covol (Γ)
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Group cohomology
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Group cohomology
Cocycles :
Z
1
(Γ , E ) := { φ : Γ → E | φ ( gh ) = φ ( g ) + g · φ ( h ) ∀ g , h ∈ Γ }
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Group cohomology
Cocycles :
Z
1
(Γ , E ) := { φ : Γ → E | φ ( gh ) = φ ( g ) + g · φ ( h ) ∀ g , h ∈ Γ }
Coboundaries :
B
1
(Γ , E ) := { φ x
: g 7→ x − g · x : x ∈ E } ⊂ Z
1
(Γ , E )
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Group cohomology
Cocycles :
Z
1
(Γ , E ) := { φ : Γ → E | φ ( gh ) = φ ( g ) + g · φ ( h ) ∀ g , h ∈ Γ }
Coboundaries :
B
1
(Γ , E ) := { φ x
: g 7→ x − g · x : x ∈ E } ⊂ Z
1
(Γ , E )
Cohomology :
H
1
(Γ , E ) := Z
1
(Γ , E ) / B
1
(Γ , E )
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Hecke operators
Let δ ∈ Comm (Γ) .
H i (Γ , M )
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Hecke operators
Let δ ∈ Comm (Γ) .
H i (Γ , M )
Res y
H i
(Γ ∩ δ Γ δ
− 1
, M )
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Hecke operators
Let δ ∈ Comm (Γ) .
H i (Γ , M )
Res y
H i
(Γ ∩ δ Γ δ
− 1
, M ) −−−−→ H i
( δ
− 1
Γ δ ∩ Γ , M )
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Hecke operators
Let δ ∈ Comm (Γ) .
H i (Γ
Res y
, M ) H i (Γ , M ) x
Cores
H i
(Γ ∩ δ Γ δ
− 1
, M ) −−−−→ H i
( δ
− 1
Γ δ ∩ Γ , M )
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Hecke operators
Let δ ∈ Comm (Γ) .
H i (Γ
Res y
, M )
T
−−−−→ H i (Γ , M ) x
Cores
H i
(Γ ∩ δ Γ δ
− 1
, M ) −−−−→ H i
( δ
− 1
Γ δ ∩ Γ , M )
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Principal ideal problem
Recall: have to find a generator δ of ideal of norm p = ( π ) .
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Principal ideal problem
Recall: have to find a generator δ of ideal of norm p = ( π ) .
Search in { nrd = π } / O 1
: complexity ∆
O ( 1 )
B
.
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Principal ideal problem
Recall: have to find a generator δ of ideal of norm p = ( π ) .
Search in { nrd = π } / O 1
: complexity ∆
O ( 1 )
B
.
Buchmann’s algorithm over number fields: subexponential under GRH.
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Principal ideal problem
Recall: have to find a generator δ of ideal of norm p = ( π ) .
Search in { nrd = π } / O 1
: complexity ∆
O ( 1 )
B
.
Buchmann’s algorithm over number fields: subexponential under GRH.
Adapt Buchmann’s algorithm over a quaternion algebra: heuristically subexponential, efficient in practice.
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Examples
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A quartic example
Let F the unique quartic field of signature ( 2 , 1 ) and discriminant − 275. Let B be the unique quaternion algebra with discriminant 11 Z maximal order in
F
, ramified at every real place of F . Let O be a
B (it is unique up to conjugation). Then O 1 is a Kleinian group with covolume 93 .
72 . . .
The fundamental domain I have computed has 310 faces and 924 edges.
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A quartic example
Let F the unique quartic field of signature ( 2 , 1 ) and discriminant − 275. Let B be the unique quaternion algebra with discriminant 11 Z maximal order in
F
, ramified at every real place of F . Let O be a
B (it is unique up to conjugation). Then O 1 is a Kleinian group with covolume 93 .
72 . . .
The fundamental domain I have computed has 310 faces and 924 edges.
Have a look at the fundamental domain!
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A quartic example
N ( p )
9
9
16
19
19
25
29
29
T p characteristic polynomial
2 − 4 − 5
− 2 3 − 2
0 0 − 5 same
( x + 5 )( x
2 − 5 x − 2 )
1 4 − 11
2 0 − 6
0 0 − 8
− 4 0 4
0 − 4 2
0 0 0 same
—
—
—
(
( x x
+
+
5
8 x
)(
)(
( x x x
2
2
+
− 5 x − 2 )
−
4 ) x
2
− 8 )
Aurel Page x ( x + 4 ) 2
( x + 9 )( x
2 − x − 74 ) x ( x 2 + 6 x − 24 ) x ( x
2
+ 6 x
Non base-change classes
Bianchi groups: Γ = SL
2
( Z
F
) with F quadratic imaginary.
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Non base-change classes
Bianchi groups: Γ = SL
2
( Z
F
) with F quadratic imaginary.
Finis, Grunewald, Tirao: formula for the base-change subspace in H i
(Γ , E k
) . Remark: very few non base-change classes.
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Non base-change classes
Bianchi groups: Γ = SL
2
( Z
F
) with F quadratic imaginary.
Finis, Grunewald, Tirao: formula for the base-change subspace in H i
(Γ , E k
) . Remark: very few non base-change classes.
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Non base-change classes
Bianchi groups: Γ = SL
2
( Z
F
) with F quadratic imaginary.
Finis, Grunewald, Tirao: formula for the base-change subspace in H i
(Γ , E k
) . Remark: very few non base-change classes.
Question (Modified Maeda’s conjecture)
Does the set of non-lifted cuspidal newforms in H i (Γ , E k
) , modulo twins, forms one Galois orbit? Is it true for k ≫ 0?
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Non base-change classes
Bianchi groups: Γ = SL
2
( Z
F
) with F quadratic imaginary.
Finis, Grunewald, Tirao: formula for the base-change subspace in H i
(Γ , E k
) . Remark: very few non base-change classes.
Question (Modified Maeda’s conjecture)
Does the set of non-lifted cuspidal newforms in H i (Γ , E k
) , modulo twins, forms one Galois orbit? Is it true for k ≫ 0?
Experiment : always dimension 2 space, except ( d , k ) = ( − 199 , 2 ) : dimension 4.
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Non base-change classes
Bianchi groups: Γ = SL
2
( Z
F
) with F quadratic imaginary.
Finis, Grunewald, Tirao: formula for the base-change subspace in H i
(Γ , E k
) . Remark: very few non base-change classes.
Question (Modified Maeda’s conjecture)
Does the set of non-lifted cuspidal newforms in H i (Γ , E k
) , modulo twins, forms one Galois orbit? Is it true for k ≫ 0?
Experiment : always dimension 2 space, except ( d , k ) = ( − 199 , 2 ) : dimension 4. Two twin Galois orbits with coefficients in Q ( 13 ) , swapped by Gal ( F / Q ) .
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Everywhere good reduction abelian surfaces
Bianchi group for F = Q (
√
− 223 ) .
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Everywhere good reduction abelian surfaces
Bianchi group for F = Q (
√
− 223 ) .
Have a look at the fundamental domain!
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Everywhere good reduction abelian surfaces
Bianchi group for F = Q (
√
− 223 ) .
Have a look at the fundamental domain!
Cuspidal subspace in H
1
(Γ , C ) : dimension 2, one Galois orbit in Q ( 2 ) .
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Everywhere good reduction abelian surfaces
Bianchi group for F = Q (
√
− 223 ) .
Have a look at the fundamental domain!
Cuspidal subspace in H
1
(Γ , C ) : dimension 2, one Galois orbit in Q ( 2 ) .
S J of the hyperelliptic curve y
2
= 33 x
−
6 + 110
√
2788739 x
−
2
223 x 5 + 36187 x
+ 652936
√
−
4
223 x
− 28402
+
√
− 223
14157596 x 3 has good reduction everywhere, End matches the Hecke eigenvalues.
( J ) ⊗ Q
∼
Q (
√
2 ) and
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Everywhere good reduction abelian surfaces
Bianchi group for F = Q (
√
− 223 ) .
Have a look at the fundamental domain!
Cuspidal subspace in H
1
(Γ , C ) : dimension 2, one Galois orbit in Q ( 2 ) .
S J of the hyperelliptic curve y
2
= 33 x
−
6 + 110
√
2788739 x
−
2
223 x 5 + 36187 x
+ 652936
√
−
4
223 x
− 28402
+
√
− 223
14157596 x 3 has good reduction everywhere, End ( J ) ⊗ Q matches the Hecke eigenvalues.
Looking for more examples in Q (
∼
Q (
√
2 ) and
√
− 455 ) and Q (
√
− 571 ) .
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Torsion Jacquet-Langlands
Can also look at H
1
(Γ , Z ) tor and H
1
(Γ , F p
) .
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Torsion Jacquet-Langlands
Can also look at H
1
(Γ , Z ) tor and H
1
(Γ , F p
) .
Jacquet–Langlands: relation between (co)homology of Γ
0
( N ) in a Bianchi group and that of Γ coming from a division algebra of discriminant N .
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Torsion Jacquet-Langlands
Can also look at H
1
(Γ , Z ) tor and H
1
(Γ , F p
) .
Jacquet–Langlands: relation between (co)homology of Γ
0
( N ) in a Bianchi group and that of Γ coming from a division algebra of discriminant N .
Calegari, Venkatesh: numerical torsion Jacquet-Langlands correspondence.
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Torsion Jacquet-Langlands
Can also look at H
1
(Γ , Z ) tor and H
1
(Γ , F p
) .
Jacquet–Langlands: relation between (co)homology of Γ
0
( N ) in a Bianchi group and that of Γ coming from a division algebra of discriminant N .
Calegari, Venkatesh: numerical torsion Jacquet-Langlands correspondence.
operators (still in progress).
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Thank you for your attention !
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