3
$\begingroup$

If the group of rational points of $E$, which is finitely generated by the Mordell-Weil Theorem, has $g$ generators of infinite order, then the Birch-Swinnerton-Dyer conjecture gives

$L_E(s)$ has a zero of order $g$ at $s=1$.

Assuming the BSD conjecture, is it possible to (and if so how) to construct such $L_E(s)$? Specifically, if we want $g=3$ or $4$?

$\endgroup$

1 Answer 1

2
$\begingroup$

I'm not entirely sure what you mean by your question. Here are two remarks:

  1. If you assume BSD, then to "construct" $L_E$ you just need to give the curve $E$. There are (many) elliptic curves /$\mathbb{Q}$ whose ranks have been computed, and are (say) equal to 3 or 4.

  2. If one wants an example without assuming BSD, then you are in trouble - for given $E$, you can compute $L^{(n)}(s)$ to any desired degree of accuracy, but proving that it vanishes computationally is impossible.

However, two things help you. If the sign in the functional equation is -1, then you have that the order of vanishing is also odd. The Gross-Zagier formula can be used to check the vanishing of the first derivative. For example, this is used in the following paper to exhibit an elliptic curve $E$ whose $L_E$ provably vanishes to order 3.

On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3 Author(s): Joe P. Buhler, Benedict H. Gross, Don B. Zagier Source: Mathematics of Computation, Vol. 44, No. 170 (Apr., 1985), pp. 473-481

$\endgroup$
5
  • $\begingroup$ I'm in case 1. I'm just curious where to find such curves E, and how to turn them in to $L_E(s)$. $\endgroup$
    – paarshad
    May 16, 2010 at 7:32
  • $\begingroup$ In that case you're probably best off looking at Cremona's very detailed tables of elliptic curves of conductor < 130000: warwick.ac.uk/staff/J.E.Cremona/ftp/data They include the $a_p$, which will allow you to write down the $L$-function by hand. For computing values of the L-function, T. Dokchitser has written a nice program that can do this given the input of the $a_p$ and a functional equation. It has been implemented in sage, and linked up with the elliptic curve functionality - cf. the example here: sagemath.org/doc/reference/sage/lfunctions/dokchitser.html $\endgroup$
    – user1594
    May 16, 2010 at 17:59
  • $\begingroup$ This is what I was looking for. Thank you. $\endgroup$
    – paarshad
    May 17, 2010 at 1:21
  • $\begingroup$ "you can compute L(n)(s) to any desired degree of accuracy, but proving that it vanishes computationally is impossible." Can't one discretize the L-value and check whether it is zero? $\endgroup$
    – Idoneal
    May 19, 2010 at 5:58
  • $\begingroup$ Sorry. That was sheer nonsense. $\endgroup$
    – Idoneal
    May 19, 2010 at 6:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.