talk.sagews

E = EllipticCurve('11a')

s = E.modular_symbol(); s

Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field

s(17/13)

-4/5

s(0)

1/5

S = E.modular_symbol_space(sign=1)

phi = S.integral_period_mapping()
phi([oo, 0])
phi([oo, 17/13])

(2/5) (-8/5)

[s(n/13) for n in [-13..13]]

[1/5, -4/5, 17/10, 17/10, -4/5, -4/5, -4/5, -4/5, -4/5, -4/5, 17/10, 17/10, -4/5, 1/5, -4/5, 17/10, 17/10, -4/5, -4/5, -4/5, -4/5, -4/5, -4/5, 17/10, 17/10, -4/5, 1/5]

E = EllipticCurve('11a')
s = E.modular_symbol()
M = 13
print [s(a/M) for a in range(M)]
pl = stats.TimeSeries([s(a/M) for a in range(M)]).plot_histogram()
pl
pl.save('plots/11a-Z%s.pdf'%M)

[1/5, -4/5, 17/10, 17/10, -4/5, -4/5, -4/5, -4/5, -4/5, -4/5, 17/10, 17/10, -4/5]

E = EllipticCurve('11a')
s = E.modular_symbol()
M = 100
v = [s(a/M) for a in range(M)]; print(v)
pl = stats.TimeSeries(v).plot_histogram()
pl
pl.save('plots/11a-Z%s.pdf'%M)

[1/5, 1/5, 6/5, 1/5, -3/10, -4/5, 6/5, 1/5, -3/10, 1/5, 1/5, 1/5, -3/10, 1/5, 6/5, 17/10, 11/5, 27/10, 6/5, 1/5, 6/5, 27/10, 6/5, 27/10, -3/10, 7/10, 6/5, 1/5, -3/10, 27/10, 1/5, -23/10, -3/10, 1/5, -13/10, -4/5, -3/10, -23/10, 6/5, -23/10, -13/10, -23/10, -19/5, -23/10, -3/10, -4/5, -13/10, -23/10, -3/10, -23/10, -4/5, -23/10, -3/10, -23/10, -13/10, -4/5, -3/10, -23/10, -19/5, -23/10, -13/10, -23/10, 6/5, -23/10, -3/10, -4/5, -13/10, 1/5, -3/10, -23/10, 1/5, 27/10, -3/10, 1/5, 6/5, 7/10, -3/10, 27/10, 6/5, 27/10, 6/5, 1/5, 6/5, 27/10, 11/5, 17/10, 6/5, 1/5, -3/10, 1/5, 1/5, 1/5, -3/10, 1/5, 6/5, -4/5, -3/10, 1/5, 6/5, 1/5]

E = EllipticCurve('11a')
s = E.modular_symbol()
M = 1000
pl = stats.TimeSeries([s(a/M) for a in range(M)]).plot_histogram()
pl
pl.save('plots/11a-Z%s.pdf'%M)

stats.TimeSeries([s(a/M) for a in range(M)]).plot_histogram().save('plots/11a-Z1000.pdf')

M = 10000
stats.TimeSeries([s(a/M) for a in range(M)]).plot_histogram().save('11a-Z10000.pdf')

%load modular_symbol_map.pyx
def ms(E, sign=1):
    g = E.modular_symbol(sign=sign)
    h = ModularSymbolMap(g)
    d = float(h.denom)  # otherwise get int division!
    return lambda a,b: h._eval1(a,b)[0]/d

Compiling ./modular_symbol_map.pyx...

s = ms(EllipticCurve('11a'))

%time
M = 100000
i = 7
g = stats.TimeSeries([s(a, M) for a in range(M)]).plot_histogram()
g
g.save('plots/11a-Z%s.pdf'%M)

CPU time: 1.15 s, Wall time: 1.20 s

M = 1000000
ell = 7
for i in range(7):
    w = stats.TimeSeries([s(a, M) for a in range(M) if a%ell==i])
    w.plot_histogram()
︠9b0204be-c869-4ce6-b29c-9e5c4f30415f︠
%time
M = 1000000
v = stats.TimeSeries([s(a, M) for a in range(M)])
len(set(v))
stats.TimeSeries(v).plot_histogram().save('plots/11a-Z%s.pdf'%M)

38 CPU time: 3.55 s, Wall time: 3.55 s

M = 1500000
v = stats.TimeSeries([s(a, M) for a in range(M)])
len(set(v))

40

v

[0.2000, 0.7000, -0.8000, 1.2000, 0.2000 ... -0.3000, 0.2000, 1.2000, -0.8000, 0.7000]

stats.TimeSeries(v).plot_histogram()

E = EllipticCurve('11a')
s = E.modular_symbol()
M = 13
v = [s(a/M) for a in range(M)]; print(v)
w = stats.TimeSeries(v).sums()
pl = w.plot() + points(enumerate(w), pointsize=60, color='black')
pl
pl.save('plots/walk-11a-Z%s.pdf'%M)

[1/5, -4/5, 17/10, 17/10, -4/5, -4/5, -4/5, -4/5, -4/5, -4/5, 17/10, 17/10, -4/5]

stats.TimeSeries(v).sums().plot().save('plots/walk-11a-Z13.pdf')

%load modular_symbol_map.pyx
def ms(E, sign=1):
    g = E.modular_symbol(sign=sign)
    h = ModularSymbolMap(g)
    d = float(h.denom)  # otherwise get int division!
    return lambda a,b: h._eval1(a,b)[0]/d

s = EllipticCurve('11a').modular_symbol()
M = 20; v = [s(a/M) for a in range(M)]; print(v)
w = stats.TimeSeries(v).sums()
pl = w.plot() + points(enumerate(w), pointsize=60, color='black')
pl
pl.save('plots/walk-11a-Z%s.pdf'%M)

[1/5, -4/5, 1/5, 17/10, 6/5, 7/10, 1/5, -4/5, -13/10, -4/5, -4/5, -4/5, -13/10, -4/5, 1/5, 7/10, 6/5, 17/10, 1/5, -4/5]

s = EllipticCurve('11a').modular_symbol()
M = 50; v = [s(a/M) for a in range(M)]; print(v)
w = stats.TimeSeries(v).sums()
pl = w.plot() + points(enumerate(w), pointsize=20, color='black')
pl
pl.save('plots/walk-11a-Z%s.pdf'%M)

[1/5, 6/5, -3/10, 6/5, -3/10, 1/5, -3/10, 6/5, 11/5, 6/5, 6/5, 6/5, -3/10, 6/5, -3/10, 1/5, -3/10, -13/10, -3/10, 6/5, -13/10, -19/5, -3/10, -13/10, -3/10, -4/5, -3/10, -13/10, -3/10, -19/5, -13/10, 6/5, -3/10, -13/10, -3/10, 1/5, -3/10, 6/5, -3/10, 6/5, 6/5, 6/5, 11/5, 6/5, -3/10, 1/5, -3/10, 6/5, -3/10, 6/5]

s = EllipticCurve('11a').modular_symbol()
M = 100; v = [s(a/M) for a in range(M)]; print(v)
w = stats.TimeSeries(v).sums()
pl = w.plot() + points(enumerate(w), pointsize=10, color='grey')
pl
pl.save('plots/walk-11a-Z%s.pdf'%M)

[1/5, 1/5, 6/5, 1/5, -3/10, -4/5, 6/5, 1/5, -3/10, 1/5, 1/5, 1/5, -3/10, 1/5, 6/5, 17/10, 11/5, 27/10, 6/5, 1/5, 6/5, 27/10, 6/5, 27/10, -3/10, 7/10, 6/5, 1/5, -3/10, 27/10, 1/5, -23/10, -3/10, 1/5, -13/10, -4/5, -3/10, -23/10, 6/5, -23/10, -13/10, -23/10, -19/5, -23/10, -3/10, -4/5, -13/10, -23/10, -3/10, -23/10, -4/5, -23/10, -3/10, -23/10, -13/10, -4/5, -3/10, -23/10, -19/5, -23/10, -13/10, -23/10, 6/5, -23/10, -3/10, -4/5, -13/10, 1/5, -3/10, -23/10, 1/5, 27/10, -3/10, 1/5, 6/5, 7/10, -3/10, 27/10, 6/5, 27/10, 6/5, 1/5, 6/5, 27/10, 11/5, 17/10, 6/5, 1/5, -3/10, 1/5, 1/5, 1/5, -3/10, 1/5, 6/5, -4/5, -3/10, 1/5, 6/5, 1/5]

s = ms(EllipticCurve('11a'))
M = 1000
v = [s(a, M) for a in range(M)]
stats.TimeSeries(v).sums().plot().save('plots/walk-11a-Z%s.pdf'%M)

s = ms(EllipticCurve('11a'))
M = 10000
v = [s(a, M) for a in range(M)]
stats.TimeSeries(v).sums().plot().save('plots/walk-11a-Z%s.pdf'%M)

s = ms(EllipticCurve('11a'))
M = 100000
v = [s(a, M) for a in range(M)]
stats.TimeSeries(v).sums().plot().save('plots/walk-11a-Z%s.pdf'%M)

s = ms(EllipticCurve('11a'))
M = next_prime(100000); M
v = [s(a, M) for a in range(M)]
stats.TimeSeries(v).sums().plot().save('plots/walk-11a-Z%s.pdf'%M)

100003

M = 1000000
for lbl in ['11a', '37a', '389a']:
    name = 'plots/walk-%s-Z%s.pdf'%(lbl,M)
    if os.path.exists(name): continue
    print name
    s = ms(EllipticCurve(lbl))
    v = [s(a, M) for a in range(M)]
    stats.TimeSeries(v).sums().plot().save(name)

Modular symbols sum versus the theoretical conjecture

%load modular_symbol_map.pyx

def ms(E, sign=1):
    g = E.modular_symbol(sign=sign)
    h = ModularSymbolMap(g)
    d = float(h.denom)  # this is *critical*; otherwise get int division below!
    return lambda a,b: h._eval1(a,b)[0]/d

def f(ms, m):
    m = int(m)
    return stats.TimeSeries([ms(a,m) for a in range(1,m+1)]).scale(1/float(m))

def norm_plot(t, **kwds):
    "Normalized plot of a time series, normalized so the x-axis goes from 0 to 1."
    s = float(len(t))
    return line([(i/s, y) for i,y in enumerate(t.list())], **kwds)

def plot_ms_sum(E, M, sign=1):
    g = ms(E, sign=sign)
    return norm_plot(f(g,M).sums())

%cython

cdef extern from "math.h":
    float sin(float)

def conj(float x, list a):
    cdef float PI = 3.1415926535897932384626433833
    cdef float s = 0
    cdef long an, n = 1
    for an in a:
        s += an * sin(2*PI*n*x) / (n*n)
        n += 1
    return s

/projects/c6b42aa2-a189-445d-8bb7-880848c5b633/.sagemathcloud/sage_server.py:920: DeprecationWarning: Importing tmp_dir from here is deprecated. If you need to use it, please import it directly from sage.misc.temporary_file See http://trac.sagemath.org/17460 for details. code = code_decorator(code)

Auto-generated code...  

$$f(x) =_? \frac{1}{2\pi \Omega^+} \cdot \sum_{n=1}^{\infty} \frac{a_n \sin(2\pi n x)}{n^2}.$$

E = EllipticCurve('11a')
D = float( 1 / (2*pi*E.period_lattice().omega()) )
anlist = E.anlist(10^4, python_ints=True)[1:]
%time f_conj = plot(lambda x : conj(x, anlist)*D, (0,1), color='red')

for M in [13, 50, 1000, 10^4, 10^5, 10^6]:
    print M
    g = plot_ms_sum(E, M) + f_conj
    show(g)
    g.save('plots/fM-11a-%s.pdf'%M)

CPU time: 1.54 s, Wall time: 1.54 s 13

50

1000

10000

100000

1000000

E = EllipticCurve('11a')
D = float( 1 / (2*pi*E.period_lattice().omega()) )
anlist = E.anlist(10^4, python_ints=True)[1:]
modsym = ms(E)

for M in [50, 100, 1000, 10000]:
    print M
    tm = walltime()
    z = f(modsym, M)
    dist = [conj(i/M, anlist)*D - z[i] for i in range(M)]
    v = stats.TimeSeries(dist)
    pl = v.plot_histogram(bins=len(v)/10)
    show(pl)
    pl.save('plots/dist-11a-%s.pdf'%M)
    print walltime(tm)

50

0.851037025452 100

0.808167934418 1000

1.7163040638 10000

9.56828713417