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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 $4$?

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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

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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: 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 –  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
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