demo-in-talk.sagews

Modular Symbols

E = EllipticCurve('11a')
show(E)

$\displaystyle y^2 + y = x^{3} - x^{2} - 10 x - 20$

plot(E)

E.period_lattice().basis()

(1.26920930427955, 0.634604652139777 + 1.45881661693850*I)

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(5077/2015)

-4/5

[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]

︠38915aeb-9620-4f60-9c20-ad1ae6ab91deis︠
%md
# Distribution of values

Faster Cython code...

Distribution of values

Faster Cython code...

%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 = ms(EllipticCurve('11a'))
M = 1000000
%time stats.TimeSeries([s(a, M) for a in range(M)]).plot_histogram()

CPU time: 3.32 s, Wall time: 3.42 s

M = 2000000  # one million
%time v = stats.TimeSeries([s(a, M) for a in range(M)])

CPU time: 6.02 s, Wall time: 6.02 s

len(set(v))

41

v.plot_histogram(bins=len(set(v)))

︠f71ff44d-d18a-4c65-8c72-039e41a553a3is︠
%md
# (Not so) Random Walks from Modular Symbols

(Not so) Random Walks from Modular Symbols

s = EllipticCurve('11a').modular_symbol()
M = 20
v = [s(a/M) for a in range(M)]
w = stats.TimeSeries(v).sums()
w.plot() + points(enumerate(w), pointsize=30, color='black')

s = EllipticCurve('11a').modular_symbol()
M = next_prime(10000)
v = [s(a/M) for a in range(M)]
w = stats.TimeSeries(v).sums()
w.plot() + points(enumerate(w), pointsize=7, color='grey')

︠94ac54d3-ea39-40e1-9fe5-3203464804afis︠
%md
## Modular symbols random walk versus conjecture
$$
\frac{1}{2\pi \omega^+} \cdot
\sum_{n=1}^{\infty} \frac{a_n \sin(2\pi n x)}{n^2}.$$

Modular symbols random walk versus conjecture

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

def f(ms, m):
    m = int(m)
    return stats.TimeSeries([ms(a,m) for a in range(1,m+1)]).scale(1.0/m)
def norm_plot(t, **kwds):
    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)
from sage.stats.intlist cimport IntList   # tricks to get 100x speedup...
cdef float PI = 3.1415926535897932384626433833
def conj(E, int terms):
    cdef IntList v = IntList(E.anlist(terms))
    cdef float D = float( 1 / (2*PI*E.period_lattice().omega()) )
    if E.discriminant()>0: D *= 2
    def f(float x):
        cdef float s = 0
        cdef long an, n = 1
        for n in range(1, terms):
            s += v._values[n] * sin(2*PI*n*x) / (n*n)
        return D * s
    return f

Auto-generated code...  

E = EllipticCurve('11a')
M = 100
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')

E = EllipticCurve('11a')
M = 1000
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')

E = EllipticCurve('11a')
M = 10000
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')

E = EllipticCurve('11a')
M = 100000
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')

E = EllipticCurve('11a')
M = 1000000
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')

E = EllipticCurve('37a')
M = 1000000
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')
︠af1cf606-baef-414d-bec7-bea5fa0d74b8s︠
E = EllipticCurve('389a')
M = 1000000
g = conj(E, 10000)
plot_ms_sum(E, M) + plot(g, 0, 1, color='red')
︠324a764a-0d02-4323-81a0-8de1cc0acdbe︠