data-analysis-odd-level.sagews

# Load mod 2 multiplicity data. Each line has the form "N m1 m2" where
# N is an odd positive integer, and m1 and m2 equal the multiplicities of 0 and 1
# as roots of the charpoly of T_2 on S_2(Gamma_0(p), F_2).

f = open("mod2-01mults.txt", "r")
l = []
d = {}
for s in f:
    if s.isspace(): continue
    try:
        N, m0, m1 = s.split(" ")
    except ValueError: print s
    N = Integer(N); m0 = Integer(m0); m1 = Integer(m1)
    l.append((N, m0, m1))
    d[N] = (m0, m1)
f.close()
cutoff = l[-1][0] + 1
print("Multiplicity data up to " + str(cutoff) + " loaded successfully.")

Multiplicity data up to 200000 loaded successfully.

# QA: the levels are the odd integers in the range [1, cutoff].
print([i[0] for i in l] == srange(1, cutoff, 2))

True

# Compute new multiplicities by accounting for degeneration maps.
# QA: these multiplicities must be nonnegative.
l2 = []
d2 = {}
for N in srange(1, cutoff, 2):
    (m0, m1) = d[N]
    for j in N.divisors():
        if j<N:
            s = number_of_divisors(N//j)
            (t0, t1) = d2[j]
            m0 -= s*t0; m1 -= s*t1
    if m0<0 or m1<0: print N
    l2.append((N, m0, m1))
    d2[N] = (m0, m1)
print("New multiplicities computed successfully.")

New multiplicities computed successfully.

# Plots depicting nonzero multiplicities of 0, 1.
l0 = []; l1 = []
i0 = 0; i1 = 0
for N in range(1, cutoff, 2):
    (m0, m1) = d2[N]
    if m0 != 0: i0 += 1 
    if m1 != 0: i1 += 1
    if N>1000:
        l0.append((log(N), 2*i0/(N+1)))
        l1.append((log(N), 2*i1/(N+1)))
show(points(l0, color='darkgreen', pointsize=2, aspect_ratio=40),
     points(l1, color='darkgreen', pointsize=2, aspect_ratio=50))

#cutoff = 5000

def is_eligible_conductor(N):
    if valuation(N, 2) > 8: return False
    N2 = N.odd_part()
    if valuation(N2, 3) > 5: return False
    N3 = N//3^valuation(N2,3)
    t = factor(N3)
    for i in t:
        if i[1] >= 3: return False
    return True

def h(N):
    if is_square(N): return 1
    K = QuadraticField(N)
    return K.class_number()

def hodd(N):
    if is_square(N): return 1
    K = QuadraticField(N)
    return K.class_number().odd_part()

def hmodord2(N):
    if is_square(N): return 1
    K = QuadraticField(N)
    G = K.class_group()
    I = K.prime_above(2)
    return (G.order() // G(I).order()).odd_part()

def h2(N):
    if is_square(N): return 1
    K = QuadraticField(N)
    h = K.class_number()
    u = K.units()[0]
    v = (u-1)/2
    return(3*h if v.is_integral() else h)

# Test lower bound for 1-multiplicity for N == 1 (mod 8).
i=j=0
for N in srange(1, cutoff, 2):
    (m0, m1) = d2[N]
    if N%8 == 1 and N.is_squarefree():
        t1 = m1 - (hmodord2(N)-1 + 2*(hodd(-N) - 1) + 1)
        if t1 < 0: print("Negative:", N)
        if N.is_prime():
            j += 1
            if t1 == 0: i += 1
            if valuation(h(-N),2) > 2 and t1 == 0: print N
print("No more examples.")
print("For prime levels, proportion where equality holds: " + str(1.0*i/j))

('Negative:', 1) No more examples. For prime levels, proportion where equality holds: 0.330370370370370

# Test lower bound for 0-multiplicity for N == 3 (mod 8).
i=j=0
for N in srange(1, cutoff, 2):
    (m0, m1) = d2[N]
    if N%8 == 3:
        t0 = m0 - h(-N)
        if t0 < 0: print("Negative:", N)
        if N.is_prime():
            j += 1
            if t0 == 0: i += 1
print("No more examples.")
print("For prime levels, proportion where equality holds: " + str(1.0*i/j))

('Negative:', 3) No more examples. For prime levels, proportion where equality holds: 0.470300333704116

# Test lower bound for 1-multiplicity for N == 3 (mod 8).
i=j=0
for N in srange(1, cutoff, 2):
    (m0, m1) = d2[N]
    if N%8 == 3:
        t1 = m1 - (hodd(N) + hodd(-N) - 2)
#        t1 -= 2^(len(factor(N))-1) - 1
        if not N.is_prime(): t1 -= 1
        if t1 < 0: print N
        if N.is_squarefree() and len(factor(N)) >= 2: #N.is_prime():
            j += 1
            if t1 == 0: i += 1
print("No more examples.")
print("For prime levels, proportion where equality holds: " + str(1.0*i/j))

27 No more examples. For prime levels, proportion where equality holds: 0.000507292327203551 27 No more examples. For prime levels, proportion where equality holds: 0.000507292327203551

# Test lower bound for 0-multiplicity for N == 5 (mod 8).
i=j=0
for N in srange(1, cutoff, 2):
    (m0, m1) = d2[N]
    if N%8 == 5 and N.is_squarefree():
        t0 = m0 - (h2(N) - h(N))
        if t0 <0: print N
        if N.is_prime():
            j += 1
            if t0 == 0: i += 1
print("No more examples.")
print("For prime levels, proportion where equality holds: " + str(1.0*i/j))

No more examples. For prime levels, proportion where equality holds: 0.774772777654622

# Test lower bound for 1-multiplicity for N == 5 (mod 8).
i=j=0
for N in srange(1, cutoff, 2):
    (m0, m1) = d2[N]
    if N%8 == 5:
        t1 = m1 - (hodd(N)-1 + hodd(-N)-1)
        t1 -= 2^(len(factor(N))-1) - 1
        if t1 < 0: print N
        if N.is_prime():
            j += 1
            if t1 == 0: i += 1
print("No more examples.")
print("For prime levels, proportion where equality holds: " + str(1.0*i/j))

No more examples. For prime levels, proportion where equality holds: 0.480159609842607

# Test lower bound for 1-multiplicity for N == 7 (mod 8).
i=j=0
for N in srange(1, cutoff, 2):
    (m0, m1) = d2[N]
    if N%8 == 7 and N.is_squarefree():
        t1 = m1 - (hodd(N)-1 + 2*(hmodord2(-N) - 1))
        t1 -= 2^(len(factor(N))-1) - 1
        if t1 < 0: print N
        if N.is_prime():
            j += 1
            if t1 == 0: i += 1
print("No more examples.")
print("For prime levels, proportion where equality holds: " + str(1.0*i/j))

No more examples. For prime levels, proportion where equality holds: 0.847953216374269