#10546: implement a custom cusps() method for principal congruence subgroups
Gamma(N)
-----------------------------+----------------------------------------------
Reporter: rje | Owner: John Cremona
Type: enhancement | Status: new
Priority: minor | Milestone: sage-4.6.2
Component: modular forms | Keywords: inequivalent cusps, principal
congruence subgroup, Gamma(n), cusps()
Author: rje | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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The command "Gamma(n).cusps()" computes a complete list of inequivalent
cusps for the congruence subgroup
Gamma(n), but it is very slow.
For example, look at the time required just for n=8:
----------------------------------------------------
sage: time Gamma(8).cusps()
CPU times: user 1086.11 s, sys: 5.60 s, total: 1091.71 s
Wall time: 1092.30 s
[Infinity, 0, 1, 2, 3, 4, 5, 6, 7, -1/2, -1/3, -1/4, -1/5, -1/6, 2/3, 3/4,
4/5, 5/6, 5/3, 11/6, 8/3, 11/3, 14/3, -3/8]
---------------------------------------------------------
I've defined a function f(n) below that returns a complete list of
inequivalent cusps for the group Gamma(n) (n>2). The elements of the list
are in so-called "reduced_cusp form". The cusp 1/n in the list is the one
equivalent to Infinity. Please use f(n) to make a patch for
Gamma(n).cusps(), n>2.
------------------------------------------------------------
def f(n):
C=[0..n-1]
n1=integer_floor((n-1)/2)
n0=integer_floor(n/2)
for r in [1..n1]:
if gcd(r,n)==1:
C.append(r/n)
if n0==n/2 and gcd(r,n0)==1:
C.append(r/n0)
for s in [2..n1]:
for r in [1..n]:
if gcd([s,r,n])==1:
m=r
while gcd(m,s)>1:
m=m+n
C.append(m/s)
return(C)
--------------------------------------------------
Here is an example for n=8:
--------------------------------------------------
sage: time f(8)
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.00 s
[0, 1, 2, 3, 4, 5, 6, 7, 1/8, 1/4, 3/8, 3/4, 1/2, 3/2, 5/2, 7/2, 1/3, 2/3,
11/3, 4/3, 5/3, 14/3, 7/3, 8/3]
--------------------------------------------------
The following gives an idea of the speed of f(n):
--------------------------------------------------
sage: time len(f(1260))
CPU times: user 16.66 s, sys: 0.41 s, total: 17.07 s
Wall time: 17.07 s
497664
---------------------------------------------------
The following checks the length 497664 of the list of inequivalent cusps:
---------------------------------------------------
sage: prod([p**(2*e) - p**(2*e-2) for (p,e) in 1260.factor()])//2
497664
--------------------------------------------------
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10546>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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