Trac #11271: there is a serious bug in the documentation or code for is_surjective for Galois representations attached to elliptic curves

David Zywina pointed out to me that the docstring for the Sage function
clearly states: "Return True if the mod-p representation is surjective
onto Aut(E[p])=GL2(Fp).  False if it is not, or None if we were unable
to determine whether it is or not."
So according to the docstring, if the functions returns either True or
False, then it returns the right answer.  The documentations suggests
that if the function returns None (which is completely different than
True/False in Python), then you have to re-call it with a bigger
parameter.  That said, I just looked through the actual source code, and
this simply isn't true at all!
{{{
sage: E = EllipticCurve('147b1')
sage: rho = E.galois_representation()
sage: rho._is_surjective??
...
    while ell < A:
        ell = arith.next_prime(ell)
        if Np % ell != 0:
            ...
    return False    #, signs
}}}

That "return False" at the end should be "return None" according to the
docstring.  So this is a serious bug in the Sage documentation or in the
function.  The options for a fix are:

1.  Probably the function as is can never ever prove non-surjectivity,
except in various special cases.   It should be changed to a better one
that can prove non-surjectivity maybe following [1], or at least using
division polynomials (since p<=37 anyways).

2. Change the lying docstring a lot to agree with the code.    NOTE: One
should probably also change the docstring for the deprecated
"E.is_surjective()" which is also misleading.

[1] http://www.math.upenn.edu/~zywina/papers/EffectiveModl.pdf

I could imagine doing 2 above quickly, then opening another ticket for
1, or even having 1 as part of #11270 later.

I'm marking this as "critical" since it is a major misleading
mathematical bug, and it is functionality that gets used as a hypothesis
for computational papers a lot (though in practice always in the
direction that is safe, but who knows?).

See also: #11270

URL: http://trac.sagemath.org/11271
Reported by: was
Ticket author(s): Chris Wuthrich
Reviewer(s): John Cremona
Trac #14858: Type checks in arith.py

Many type checks in `rings/arith.py` use `isinstance` for checking
whether something is an integer, for example. They should use `x in ZZ`
instead, because not all cases are covered, leading to errors like the
following:

{{{
sage: rising_factorial(-4, 2)
12
sage: rising_factorial(-4, SR(2))
0
sage: rising_factorial(SR(-4), SR(2))
RuntimeError: indeterminate expression: 0 * infinity encountered.
}}}

Setting to critical because this is very confusing, and can give silent
errors such as `rising_factorial(-4, 2)` being different from
`rising_factorial(-4, SR(2))`. The fact that `rising_factorial(-4,
SR(2))` gives 0 is a separate error, ticket #14857.

URL: http://trac.sagemath.org/14858
Reported by: eviatarbach
Ticket author(s): Eviatar Bach
Reviewer(s): Nathann Cohen
Trac #15587: Package the git trac subcommand

Packaged version of https://github.com/sagemath/git-trac-command

Tarball: http://boxen.math.washington.edu/home/release/sage-6.1.beta3/up
stream/git_trac-1.0.tar.bz2

URL: http://trac.sagemath.org/15587
Reported by: vbraun
Ticket author(s): Volker Braun
Reviewer(s): R. Andrew Ohana
Trac #15556: Convert Stein-Watkins database to git spkg format

There are two optional spkgs containing small and full version of the
Stein-Watkins database of elliptic curves, namely
database_stein_watkins_mini.p0.spkg (the small version, approximately
20mb) and stein-watkins-ecdb.p0.spkg (the huge version, approximately
2.7gb).

Both should be converted to the new git format for spkgs, at the same
time making their names more consistent and completing their SPKG.txt
information.  The interface in sage/databases/stein-watkins.py will need
to be tested.

New '''upstream''' tarballs:
 1. [http://boxen.math.washington.edu/home/cremona/database_stein_watkin
s_mini-20030423.tar.bz2]
 1. [http://boxen.math.washington.edu/home/cremona/database_stein_watkin
s-20030423.tar.bz2]

URL: http://trac.sagemath.org/15556
Reported by: cremona
Ticket author(s): John Cremona, Jeroen Demeyer
Reviewer(s): Jeroen Demeyer, John Cremona