4 ms·

To put a little color on the BSD conjecture, it states that the rank (0, 1, 2, 3, etc.) of rational points on an elliptic curve is related to the residue (coeff

To put a little color on the BSD conjecture, it states that the rank (0, 1, 2, 3, etc.) of rational points on an elliptic curve is related to the residue (coefficient of 1/q) of the L-function for the curve. There are some additional multiplicative factors, in particular the size of the Tate-Shafarevich group. No one knows how to compute the size of that group in general (in fact no one has proved that it's finite!). Computing the rank of a curve via non-analytic means is more akin to a bespoke proof than a straightforward computation (see Noam Elkies' work). So saying you're going to disprove BSD with blind computation is rather naive unless you're sitting on several career-defining proofs and not sharing them.

williamstein 2y ago
If the BSD rank conjecture were false, then the simplest counterexample might be an elliptic curve with algebraic rank 4 and analytic rank 2. This could be established for a specific curve by rigorously numerically computing the second derivative of the L-series at 1 to some number of digits and getting something nonzero (which is possible because elliptic curves are modular - see work of Dikchitser). This is a straightforward thing to do computations about and there are large tables of rank 4 curves. This is also exactly the problem I suggested to the OP in grad school. :-) In number theory doing these sorts of “obvious computational investigations” is well worth doing and led to many of the papers I have written. I remember doing one in grad school and being shocked when we found a really interesting example in minutes, which led to a paper.
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