I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with an elliptic curve over rationals. A google search do not gives important answers.
$\begingroup$
$\endgroup$
3
-
1$\begingroup$ Where do you want the series to converge rapidly? If it's the whole complex plane, such a series would be a proof of analytic continuation, which is as hard as modularity. Thus, you probably want to find the associated modular form and look for a rapidly converging series for the L-function of that modular form? $\endgroup$– Will SawinCommented Nov 20, 2012 at 17:22
-
$\begingroup$ for any complex number s. I remamber that there is a rapidly converging series obtained by using Lavrik method, but I can't find it. $\endgroup$– SafwaneCommented Nov 20, 2012 at 17:26
-
2$\begingroup$ Yes, there is a rapidly exponentially converging series for L(E,s) which relies on the functional equation and thus on modularity of E. This has been implemented, see e.g. arxiv.org/abs/math/0207280 If you're more interested in values of s with high imaginary part, then you might need other methods, like the double exponential method. See Pascal Molin's PhD thesis math.jussieu.fr/~molinp/files/these.pdf $\endgroup$– François BrunaultCommented Nov 20, 2012 at 17:48
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
Possibly the answer would be the so-called "approximate functional equation" for the $L$-function. This of course takes as input the modularity of the Hasse-Weil zeta function, and gives rapidly convergent series representing it at any point. I would expect Cremona's book on algorithms for modular elliptic curves to contain a description. Software like Pari/GP implements such algorithms (see the command elllseries).
answered Nov 20, 2012 at 17:39
-
$\begingroup$ Thank you Denis. I think that there is a rapidly converging series obtained by using Lavrik method, but I can't find it in any reference. $\endgroup$– SafwaneCommented Nov 20, 2012 at 17:42
-
3$\begingroup$ See the book Analytic Number Theory by Iwaniec and Kowalski $\endgroup$– StoppleCommented Nov 20, 2012 at 18:29
$\begingroup$
$\endgroup$
Also look at my paper: Computational methods and experiments in analytic number theory, in Recent Perspectives in Random Matrix Theory and Number Theory. Available here: http://arxiv.org/pdf/math/0412181v1.pdf See section 3.4, example 5 which deals with elliptic curves.
Another reasonable reference is Akiyama and Tanigawa:http://www.jstor.org/discover/10.2307/2584959?uid=3739448&uid=2129&uid=2&uid=70&uid=3737720&uid=4&sid=21102659822167
