MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $4$?

share|cite|improve this question
up vote 2 down vote accepted

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

share|cite|improve this answer
I'm in case 1. I'm just curious where to find such curves E, and how to turn them in to $L_E(s)$. – paarshad May 16 '10 at 7:32
In that case you're probably best off looking at Cremona's very detailed tables of elliptic curves of conductor < 130000: 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: – user1594 May 16 '10 at 17:59
This is what I was looking for. Thank you. – paarshad May 17 '10 at 1:21
"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? – Idoneal May 19 '10 at 5:58
Sorry. That was sheer nonsense. – Idoneal May 19 '10 at 6:05

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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