Question regarding normalization of modular symbols.

[francisco]

Hello,

I have been computing modular symbols for distinct curves on the Cremona data base.
But, in a few curves, I recived a WARNING messages like this:

Warning : Could not normalize the modular symbols, maybe all further
results will be multiplied by -1, 2 or -2.

Why sage does not give the exact normalization? Is there a theoretical reason?
How sage normalize modular symbols?

[John Cremona]

This may be related to https://trac.sagemath.org/ticket/10256 . The
answer to your last question is that it depends on whether they are
computed using Sage itself (slower) or my code (=eclib).

John Cremona

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[chris wuthrich]

Most of the normalisation of modular symbols was introduced by me so let me comment.

There are currently two ways of computing modular symbols, one using eclib and the other the native sage implementation. Both are only correct up to a scaling factor. That is because their main use was as generators for some Q-vector space to produce modular forms etc. Instead if we start with an elliptic curve then there is a unique normalisation such that the modular symbol [0] takes the value L(E,1)/Omega+ where Omega+ is the least positive real period. In particular this depends on the curve and not just on the isogeny class (i.e. the modular form). So currently sage or eclib produce some Q-valued symbol and then we use the L-value to find the correct rational scaling factor. However if this L-value (and [0]) is zero, then we can't do it like that. Instead we can use a quadratic twist to get a non-zero L-value. For the curves where the warning message comes, sage was not able to find a good quadratic twist. This could be improved.

However there is a ticket waiting that will implement a thrid way of computing the modular symbols (much faster when asked only a few of them): https://trac.sagemath.org/ticket/21046 . Once this is in, I will change the normalisation to used these numerical symbols to get the right scaling factor also for the other two implementations. Then this warning will disappear.

Chris

[francisco]

Chris,
Thank you for your comments, they are very helpful!

On my computations, I am considering elliptic curves with rank 1.  So, the value of the modular symbol [0] is zero, and we cannot normalise using
the value of the L function at 1.
I need the correct normalization because I am working on computer evidence of the Mazur and Tate conjecture. And the two sides on the
conjecture should agree.

Until now, whenever SAGE prompts me a Warning Message, I was checking each of the suggested values, and see which one works (or is the correct one)
for the formula in the conjecture. Of course, the problem is that I pick the value (among the suggested ones), so that the conjecture works!!!
(Which is not so bad neither, since the conjecture gives an equation for each prime (almost all), and the normalisation constant does not depend on the prime)

But now, I will follow your suggestion of using numerical computation of modular symbols to obtain the correct constant!
In fact, before I start computing with SAGE, I was doing something similar on PARI by estimating [a/b] with the partial sum
$\sum_{i=1}^N a_n cos(2\pi n a/b )/n$  for large $N$.

But, I didnt have any theoretical control of the aproximation error, as you have shown in your article in the link, so I was aproximating many values of the form [r/q] with $q$ prime to the conductor, until I figure out the correct constant.

Best,

Francisco Portillo