Plan to finish a first complete HMF over Q(sqrt(5)) of weight (2,2)
implementation that works and is very fast. Goal is to finish this in
less than 24 from now!
[x] P^1(O_F/N) using fast residue rings
[x] is_square
[x] sqrt
[x] local splitting for O_F/P^n
[x] Mat2x2(O_F/N) -- matrix object with optimal action on P^1(O_F/N) elts.
[x] 120 icosians
[x] Dividing P^1 into orbits for action of R^*.
[x] write fast version of code for hecke operator T_P for odd P
[x] See this for ModN_Reduction. # TODO: This algorithm only makes sense when trying to compute splitting at one prime, but we're applying it to a product of primes!
Example: P = IcosiansModP1ModN(F.primes_above(5) * F.ideal(3))
[ ] code for hecke operator T_2 (hardcode norm 2 elts); easy once have splittings at all primes
[ ] make computation of the alpha's for computing Hecke operators fast, so we can compute T_p for p < 10000 (say).
[ ] refactor sqrt5_fast.pyx -- it's too big.
[ ] BUG:
Good example:
sage: P = sqrt5_fast.IcosiansModP1ModN(F.ideal(8)); P
The 2 orbits for the action of the Icosian group on Projective line modulo the ideal (8) of norm 64
sage: o = P.compute_std_to_rep_table_debug(); o
Found representative number 0: (0, 1)
It has an orbit of size 60
Found representative number 1: (1, 1)
It has an orbit of size 31
[[0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 24, 25, 27, 28, 30, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 73, 74, 75, 76, 78, 79], [1, 3, 12, 13, 14, 22, 23, 26, 29, 31, 32, 38, 43, 49, 52, 60, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79]]
sage: P.cardinality()
2
sage: P.
P.cardinality P.compute_std_to_rep_table_debug
P.compute_std_to_rep_table P.hecke_matrix
sage: Proj
ProjectiveGeometryDesign ProjectiveHypersurface ProjectiveSpace
sage: P1
P1List P1ModList P1NFList P1_orbits P1_orbits2
sage: sqrt5_fast.ProjectiveLineModN(F.ideal(8)).cardinality()
80
-----------------------------------
sage: P = IcosiansModP1ModN(F.ideal(9))
sage: o = P.compute_std_to_rep_table_debug(); o
Found representative number 0: (0, 1)
It has an orbit of size 61
Found representative number 1: (1 + g, 1)
It has an orbit of size 35
which violates orbit stabilizer theorem; plus orbits aren't disjoint!
sage: P = IcosiansModP1ModN(F.ideal(27))
sage: P.cardinality() # it should be 14
17
sage: time T = P.hecke_matrix(F.primes_above(5)[0])
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s
sage: T.fcp()
(x - 6) * x^6 * (x^10 - 34*x^8 + 2*x^7 + 366*x^6 - 56*x^5 - 1302*x^4 + 323*x^3 + 239*x^2 + 6*x - 12)
sage: t5 = P.hecke_matrix(F.primes_above(5)[0])
sage: t57 = P.hecke_matrix(F.primes_above(7)[0])
sage: t7 = P.hecke_matrix(F.primes_above(7)[0])
sage: t5*t7 == t7*t5
False
3 0
7 1
9 1
11 3
13 3
17 5
19 7
21 9
23 9
27 16
29 15
31 19
33 23
37 23
39 29
41 31
43 31
47 37
49 44
51 49
53 47
57 67
59 59
61 214
63 83
67 75
69 89
71 87
73 89
77 119
79 107
81 159
83 115
87 436
89 135
91 143
93 629
97 157
99 287
compared to:
> for N in [2..100] do if GCD(N,10) eq 1 then print N, Dimension(HilbertCuspForms(F, Integers(F)*N)); end if; end for;
3 0
7 1
9 1
11 3
13 3
17 5
19 7
21 9
23 9
27 13
29 15
31 19
33 23
37 23
39 29
41 31
43 31
47 37
49 41
51 49
53 47
57 67
59 59
61 67
63 75
67 75
69 89
71 87
73 89
77 119
79 107
81 121
83 115
87 151
89 135
91 143
93 171
97 157
99 215
