Computing Klein modular forms
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1 Université de Bordeaux July 24, 2014 Paris 13 Workshop on Analytic Number Theory and Geometry
2 1 Motivation 2 3 4
3 Elliptic curves and automorphic forms Want a database of elliptic curves over number field F?
4 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?
5 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) = π ĜHom(π, L 2 (Γ\G)) H i (g, K;π E)
6 Automorphic forms for GL 2 Where do you find GL 2 automorphic forms?
7 Automorphic forms for GL 2 Where do you find GL 2 automorphic forms? H (GL 2 (Z F ), E)
8 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
9 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)
10 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) Call H i (O, E) a space of Klein modular forms.
11 Motivation for the Kleinian case Elliptic curves, abelian varieties of GL 2 type over number fields.
12 Motivation for the Kleinian case Elliptic curves, abelian varieties of GL 2 type over number fields. Torsion phenomenon.
13
14 Kleinian groups Motivation H := C+Cj where j 2 = 1 and jz = zj for all z C. The upper half-space H 3 := C+R >0 j. Metric ds 2 = dz 2 +dt 2, volume dv = t 2 dx dy dt t 3.
15 Kleinian groups Motivation H := C+Cj where j 2 = 1 and jz = zj for all z C. The upper half-space H 3 := C+R >0 j. Metric ds 2 = dz 2 +dt 2, volume dv = t 2 ( a b c d ) w := (aw + b)(cw + d) 1 for dx dy dt t 3. ( ) a b SL c d 2 (C).
16 Kleinian groups Motivation H := C+Cj where j 2 = 1 and jz = zj for all z C. The upper half-space H 3 := C+R >0 j. Metric ds 2 = dz 2 +dt 2, volume dv = t 2 ( a b c d ) w := (aw + b)(cw + d) 1 for dx dy dt t 3. Kleinian group: discrete subgroup of PSL 2 (C). Cofinite if it has finite covolume. ( ) a b SL c d 2 (C).
17 Quaternion algebras A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F. ) Explicitly, B = = F + Fi + Fj + Fij, i 2 = a, j 2 = b, ij = ji (char F 2). ( a,b F
18 Quaternion algebras A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F. ) Explicitly, B = = F + Fi + Fj + Fij, i 2 = a, j 2 = b, ij = ji (char F 2). ( a,b F Reduced norm: nrd(x + yi + zj + tij) = x 2 ay 2 bz 2 + abt 2.
19 Quaternion algebras A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F. ) Explicitly, B = = F + Fi + Fj + Fij, i 2 = a, j 2 = b, ij = ji (char F 2). ( a,b F Reduced norm: nrd(x + yi + zj + tij) = x 2 ay 2 bz 2 + abt 2. Z F integers of F number field. Order O B: subring, finitely generated Z F -module, OF = B.
20 Quaternion algebras A quaternion algebra B over a field F is a central simple algebra of dimension 4 over F. ) Explicitly, B = = F + Fi + Fj + Fij, i 2 = a, j 2 = b, ij = ji (char F 2). ( a,b F Reduced norm: nrd(x + yi + zj + tij) = x 2 ay 2 bz 2 + abt 2. Z F integers of F number field. Order O B: subring, finitely generated Z F -module, OF = 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.
21 Covolume formula Motivation F almost totally real number field: exactly one complex place. B/F Kleinian quaternion algebra: ramified at every real place.
22 Covolume formula Motivation 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).
23 Covolume formula Motivation 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) p ram. (N(p) 1) (4π 2 ) [F:Q] 1
24
25 Sketch of algorithm Motivation 1 Compute B and O
26 Sketch of algorithm Motivation 1 Compute B and O 2 Compute O 1
27 Sketch of algorithm Motivation 1 Compute B and O 2 Compute O 1 3 Compute H 1 (O 1 )
28 Sketch of algorithm Motivation 1 Compute B and O 2 Compute O 1 3 Compute H 1 (O 1 ) 4 Compute generator δ of ideal of norm p
29 Sketch of algorithm Motivation 1 Compute B and O 2 Compute O 1 3 Compute H 1 (O 1 ) 4 Compute generator δ of ideal of norm p 5 Compute Hecke operator T δ on H 1 (O 1 )
30 Fundamental domains Γ a Kleinian group. An open subset F H 3 is a fundamental domain if Γ F = H 3 F γf = for all 1 γ Γ.
31 Dirichlet domains Motivation
32 Dirichlet domains Motivation 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}}.
33 Dirichlet domains Motivation 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.
34 Structure of the Dirichlet domain
35 Structure of the Dirichlet domain Faces of the Dirichlet domain are grouped into pairs, corresponding to elements g, g 1 Γ;
36 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.
37 Poincaré s theorem Motivation Theorem (Poincaré) The elements corresponding to the faces are generators of Γ. The relations corresponding to the edge cycles generate all the relations among the generators.
38 Poincaré s theorem Motivation Theorem (Poincaré) 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.
39 Reduction algorithm x H 3, S Γ corresponding to the faces of D p (Γ).
40 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
41 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
42 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).
43 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 1 p, which will reduce to x = p, to write γ as a product of the generators.
44 Computation in hyperbolic space
45 Computation in hyperbolic space We have : an explicit formula for the hyperbolic distance;
46 Computation in hyperbolic space We have : an explicit formula for the hyperbolic distance; an explicit formula for bisectors;
47 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;
48 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.
49 Algorithm Motivation Suppose O maximal.
50 Algorithm Motivation Suppose O maximal. Enumerate elements of O 1 in a finite set S
51 Algorithm Motivation Suppose O maximal. Enumerate elements of O 1 in a finite set S Compute D = D p (S)
52 Algorithm Motivation 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)
53 Complexity Motivation Why would you care about the complexity?
54 Complexity Motivation Why would you care about the complexity? Bound on the size of the Dirichlet domain:
55 Complexity Motivation 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. gf f ǫ f.
56 Complexity Motivation 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. gf f ǫ f. diam(d p (Γ)) log Covol(Γ)
57 Complexity Motivation 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. gf f ǫ f. diam(d p (Γ)) log Covol(Γ) Proved complexity: Covol(Γ) O(1) Observed complexity: Covol(Γ) 2 Lower bound: Covol(Γ)
58 Group cohomology Motivation
59 Group cohomology Motivation Cocycles : Z 1 (Γ, E) := {φ : Γ E φ(gh) = φ(g)+g φ(h) g, h Γ}
60 Group cohomology Motivation 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 x g x : x E} Z 1 (Γ, E)
61 Group cohomology Motivation 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 x g x : x E} Z 1 (Γ, E) Cohomology : H 1 (Γ, E) := Z 1 (Γ, E)/B 1 (Γ, E)
62 Hecke operators Motivation Let δ Comm(Γ). H i (Γ, M)
63 Hecke operators Motivation Let δ Comm(Γ). H i (Γ, M) Res H i (Γ δγδ 1, M)
64 Hecke operators Motivation Let δ Comm(Γ). H i (Γ, M) Res H i (Γ δγδ 1, M) δ H i (δ 1 Γδ Γ, M)
65 Hecke operators Motivation Let δ Comm(Γ). H i (Γ, M) H i (Γ, M) Res H i (Γ δγδ 1, M) δ Cores H i (δ 1 Γδ Γ, M)
66 Hecke operators Motivation Let δ Comm(Γ). H i (Γ, M) Res T δ H i (Γ δγδ 1, M) δ H i (Γ, M) Cores H i (δ 1 Γδ Γ, M)
67 Principal ideal problem Recall: have to find a generator δ of ideal of norm p = (π).
68 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.
69 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.
70 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.
71
72 A quartic example Motivation Let F the unique quartic field of signature (2, 1) and discriminant 275. Let B be the unique quaternion algebra with discriminant 11Z F, ramified at every real place of F. Let O be a maximal order in B (it is unique up to conjugation). Then O 1 is a Kleinian group with covolume The fundamental domain I have computed has 310 faces and 924 edges.
73 A quartic example Motivation Let F the unique quartic field of signature (2, 1) and discriminant 275. Let B be the unique quaternion algebra with discriminant 11Z F, ramified at every real place of F. Let O be a maximal order in B (it is unique up to conjugation). Then O 1 is a Kleinian group with covolume The fundamental domain I have computed has 310 faces and 924 edges. Have a look at the fundamental domain!
74 A quartic example Motivation N(p) T p characteristic polynomial (x + 5)(x 2 5x 2) same (x + 5)(x 2 5x 2) (x + 8)(x 2 x 8) x(x + 4) same x(x + 4) 2 25 (x + 9)(x 2 x 74) 29 x(x 2 + 6x 24) 29 x(x 2 + 6x 24)
75 Non base-change classes Bianchi groups: Γ = SL 2 (Z F ) with F quadratic imaginary.
76 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.
77 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. Şengün, Rahm: extended their computations.
78 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. Şengün, Rahm: extended their computations. 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?
79 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. Şengün, Rahm: extended their computations. 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.
80 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. Şengün, Rahm: extended their computations. 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).
81 Everywhere good reduction abelian surfaces Bianchi group for F = Q( 223).
82 Everywhere good reduction abelian surfaces Bianchi group for F = Q( 223). Have a look at the fundamental domain!
83 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).
84 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). Şengün, Dembélé : the Jacobian J of the hyperelliptic curve y 2 = 33x x x x x x has good reduction everywhere, End(J) Q = Q( 2) and matches the Hecke eigenvalues.
85 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). Şengün, Dembélé : the Jacobian J of the hyperelliptic curve y 2 = 33x x x x x x has good reduction everywhere, End(J) Q = Q( 2) and matches the Hecke eigenvalues. Looking for more examples in Q( 455) and Q( 571).
86 Torsion Jacquet-Langlands Can also look at H 1 (Γ,Z) tor and H 1 (Γ,F p ).
87 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.
88 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.
89 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. Joint with H. Şengün: experimental verification with Hecke operators (still in progress).
90 Thank you for your attention!
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