2020-05-26 00:10:24 +02:00
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using RamanujanGraphs
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using LinearAlgebra
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using Nemo
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2020-09-03 12:36:58 +02:00
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using ArgParse
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2020-05-26 00:10:24 +02:00
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2020-06-09 16:18:00 +02:00
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using Logging
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using Dates
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2020-05-26 00:10:24 +02:00
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2020-06-09 16:18:00 +02:00
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include("src/nemo_utils.jl")
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2020-05-26 00:10:24 +02:00
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2020-06-09 16:18:00 +02:00
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function SL2p_gens(p)
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2020-09-01 22:12:59 +02:00
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if p == 31
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a, b = let
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a = SL₂{p}([8 14; 4 11])
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b = SL₂{p}([23 0; 14 27])
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@assert isone(a^10)
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@assert isone(b^10)
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2020-09-03 14:11:29 +02:00
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a, b
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end
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elseif p == 41
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a, b = let
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a = SL₂{p}([0 28; 19 35])
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b = SL₂{p}([38 27; 2 9])
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@assert isone(a^10)
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@assert isone(b^10)
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2020-09-01 22:12:59 +02:00
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a, b
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end
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elseif p == 109
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2020-06-09 10:52:55 +02:00
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a, b = let
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a = SL₂{p}([0 1; 108 11])
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b = SL₂{p}([57 2; 52 42])
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2020-05-26 00:10:24 +02:00
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@assert isone(a^10)
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@assert isone(b^10)
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a, b
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end
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elseif p == 131
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a, b = let
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2020-05-26 00:10:24 +02:00
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a = SL₂{p}([-58 -24; -58 46])
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b = SL₂{p}([0 -3; 44 -12])
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@assert isone(a^10)
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@assert isone(b^10)
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a, b
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end
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else
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@warn "no special set of generators for prime $p"
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2020-06-09 10:52:55 +02:00
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a, b = let
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2020-05-26 00:10:24 +02:00
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a = SL₂{p}(1, 0, 1, 1)
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b = SL₂{p}(1, 1, 0, 1)
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a, b
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end
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end
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2020-06-09 16:18:00 +02:00
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return a,b
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2020-05-26 00:10:24 +02:00
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end
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2020-06-09 16:18:00 +02:00
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function adjacency(ϱ, CC, a, b)
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A = matrix(CC, ϱ(a))
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B = matrix(CC, ϱ(b))
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2020-05-26 00:10:24 +02:00
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2020-06-09 16:18:00 +02:00
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return sum(A^i for i = 1:4) + sum(B^i for i = 1:4)
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end
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2020-05-26 00:10:24 +02:00
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2020-09-03 12:36:58 +02:00
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function parse_our_args()
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s = ArgParseSettings()
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@add_arg_table! s begin
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"-p"
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help = "the prime p for which to use PSL(2,p)"
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arg_type = Int
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required = true
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"-a"
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help = "generator a (optional)"
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"-b"
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help = "generator b (optional)"
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"--ab"
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help = "array of generators a and b (optional)"
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end
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result = parse_args(s)
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for key in ["a", "b", "ab"]
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val = get(result, key, "")
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if val != nothing
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result[key] = eval(Meta.parse(val))
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else
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delete!(result, key)
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end
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end
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val = get(result, "ab", "")
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if val != ""
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result["a"] = val[1]
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result["b"] = val[2]
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end
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result
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end
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parsed_args = parse_our_args()
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2020-06-09 16:18:00 +02:00
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const p = try
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p = parsed_args["p"]
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2020-06-09 16:18:00 +02:00
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RamanujanGraphs.Primes.isprime(p)
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p
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catch ex
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2020-09-01 22:12:59 +02:00
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@error "You need to provide a prime, ex: `julia adj_psl2_eigvals.jl -p 31`"
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2020-06-09 16:18:00 +02:00
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rethrow(ex)
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end
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2020-05-26 00:10:24 +02:00
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2020-06-09 16:18:00 +02:00
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const LOGFILE = "SL(2,$p)_eigvals_$(now()).log"
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open(joinpath("log", LOGFILE), "w") do io
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with_logger(SimpleLogger(io)) do
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CC = AcbField(128)
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a,b = SL2p_gens(p)
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a = SL₂{p}(get(parsed_args, "a", a))
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b = SL₂{p}(get(parsed_args, "b", b))
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@info "a = " a
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@info "b = " b
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2020-06-09 16:18:00 +02:00
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Borel_cosets = let p = p, (a,b) = (a,b)
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SL2p, sizes =
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RamanujanGraphs.generate_balls([a, b, inv(a), inv(b)], radius = 21)
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@assert sizes[end] == RamanujanGraphs.order(SL₂{p})
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RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p}))
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2020-05-26 00:10:24 +02:00
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end
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2020-09-03 12:36:58 +02:00
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all_large_evs = []
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let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
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2020-06-08 16:18:02 +02:00
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2020-06-09 16:18:00 +02:00
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for j = 0:(p-1)÷4
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h = PrincipalRepr(
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α => root_of_unity(CC, (p - 1) ÷ 2, j),
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Borel_cosets,
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)
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@time adj = adjacency(h, CC, a, b)
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2020-06-08 16:18:02 +02:00
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2020-06-09 16:18:00 +02:00
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try
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@time ev = let evs = safe_eigvals(adj)
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_count_multiplicites(evs)
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end
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2020-09-03 12:36:58 +02:00
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all_large_evs = vcat(all_large_evs, [Float64(real(x[1])) for x in ev[1:2]])
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2020-06-09 16:18:00 +02:00
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@info "Principal Series Representation $j" ev[1:2] ev[end]
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catch ex
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@error "Principal Series Representation $j failed" ex
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ex isa InterruptException && rethrow(ex)
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end
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2020-05-26 00:10:24 +02:00
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end
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2020-06-09 16:18:00 +02:00
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end
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let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)),
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β = RamanujanGraphs.generator_min(QuadraticExt(α))
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2020-05-26 00:10:24 +02:00
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2020-06-09 16:18:00 +02:00
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if p % 4 == 1
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ub = (p - 1) ÷ 4
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ζ = root_of_unity(CC, (p + 1) ÷ 2, 1)
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else # p % 4 == 3
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ub = (p + 1) ÷ 4
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ζ = root_of_unity(CC, (p + 1), 1)
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2020-05-26 00:10:24 +02:00
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end
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2020-06-09 16:18:00 +02:00
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for k = 1:ub
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h = DiscreteRepr(
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RamanujanGraphs.GF{p}(1) => root_of_unity(CC, p),
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β => ζ^k,
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)
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@time adj = adjacency(h, CC, a, b)
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try
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@time ev = let evs = safe_eigvals(adj)
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_count_multiplicites(evs)
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end
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2020-09-03 12:36:58 +02:00
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all_large_evs = vcat(all_large_evs, [Float64(real(x[1])) for x in ev[1:2]])
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2020-06-09 16:18:00 +02:00
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@info "Discrete Series Representation $k" ev[1:2] ev[end]
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catch ex
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@error "Discrete Series Representation $k : failed" ex
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ex isa InterruptException && rethrow(ex)
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end
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end
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2020-05-26 00:10:24 +02:00
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end
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2020-09-03 12:36:58 +02:00
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all_large_evs = sort(all_large_evs, rev=true)
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λ = all_large_evs[2]
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ε = (λ - 3)/5
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α = acos(ε)
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α_deg = α/pi*180
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@info "Numerically" λ ε α α_deg
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2020-06-09 16:18:00 +02:00
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end # with_logger
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end # open(logfile)
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2020-05-26 00:10:24 +02:00
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2020-06-09 10:52:55 +02:00
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#
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2020-05-26 00:10:24 +02:00
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# using RamanujanGraphs.LightGraphs
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# using Arpack
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#
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2020-06-09 10:52:55 +02:00
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# Γ, eigenvalues = let p = 109,
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# a = PSL₂{p}([ 0 1; 108 11]),
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# b = PSL₂{p}([ 57 2; 52 42])
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2020-05-26 00:10:24 +02:00
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#
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# S = unique([[a^i for i in 1:4]; [b^i for i in 1:4]])
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#
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# @info "Generating set S of $(eltype(S))" S
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# @time Γ, verts, vlabels, elabels =
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# RamanujanGraphs.cayley_graph(RamanujanGraphs.order(PSL₂{p}), S)
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#
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# @assert all(LightGraphs.degree(Γ,i) == length(S) for i in vertices(Γ))
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# @assert LightGraphs.nv(Γ) == RamanujanGraphs.order(PSL₂{p})
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# A = adjacency_matrix(Γ)
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# @time eigenvalues, _ = eigs(A, nev=5)
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# @show Γ eigenvalues
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# Γ, eigenvalues
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# end
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#
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# let p = 131,
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# a = PSL₂{p}([-58 -24; -58 46]),
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# b = PSL₂{p}([0 -3; 44 -12])
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#
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# S = unique([[a^i for i in 1:4]; [b^i for i in 1:4]])
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#
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# @info "Generating set S of $(eltype(S))" S
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# @time Γ, verts, vlabels, elabels =
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# RamanujanGraphs.cayley_graph(RamanujanGraphs.order(PSL₂{p}), S)
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#
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2020-05-26 00:10:24 +02:00
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# @assert all(LightGraphs.degree(Γ,i) == length(S) for i in vertices(Γ))
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# @assert LightGraphs.nv(Γ) == RamanujanGraphs.order(PSL₂{p})
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2020-05-26 00:10:24 +02:00
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# A = adjacency_matrix(Γ)
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# @time eigenvalues, _ = eigs(A, nev=5)
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# @show Γ eigenvalues
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# Γ, eigenvalues
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# end
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