Source: /cirosantilli/birch-and-swinnerton-dyer-conjecture

= Birch and Swinnerton-Dyer conjecture
{tag=Millennium Prize Problems}

= BSD Conjecture

The BSD conjecture states that if your name is long enough, it will always count as two letters on a famous conejcture.

Maybe also insert a joke about <BSD Operating Systems> if you're into that kind of stuff.

The conjecture states that <equation BSD conjecture> holds for every <elliptic curve over the rational numbers> (which is defined by its  constants $a$ and $b$)

\lim_{x \to \infty} \prod_{p \leq x} \frac{N_p}{p} = C \log(x)^r
{title=<BSD Conjecture>}
Where the following numbers are definied for the <elliptic curve> we are currently considering, defined by its constants $a$ and $b$:
* $N_p$: <number of elements of the elliptic curve over the finite field>, where the <finite field> comes from the <reduction of an elliptic curve from E(\Q) to E(\F_p) \mod p>. $N_p$ can be calculated algorithmically with <Schoof's algorithm> in <polynomial time>
* $r$: <rank of the elliptic curve over the rational numbers>. We don't really have a good general way to calculate this besides this conjecture (?).
* $C$: a constant

The conjecture, if true, provides a (possibly inefficient) way to calculate the <rank of an elliptic curve over the rational numbers>, since we can calculate the <number of elements of an elliptic curve over a finite field> by <Schoof's algorithm> in <polynomial time>. So it is just a matter of calculating $N_p$ like that up to some point at which we are quite certain about $r$.

The[Wikipedia page of the this conecture] is the perfect example of why <it is not possible to teach natural sciences on Wikipedia>. A <Millennium Prize Problems>[million dollar problem], and the page is thoroughly incomprehensible unless you already know everything!

{title=$\lim_{x \to \infty} \prod_{p \leq x} \frac{N_p}{p}$ as a function of $p$ for the <elliptic curve> $y^2 = x^3 - 5x$}
{description=The curve is known to have <rank of the elliptic curve over the rational numbers>[rank] 1, and the logarithmic plot tends more and more to a line of slope 1 as expected from the conjecture, matching the rank.}

{title=<Birch and Swinnerton-Dyer conjecture> by Kinertia (2020)}

{title=The \$1,000,000 <Birch and Swinnerton-Dyer conjecture> by Absolutely Uniformly Confused (2022)}
{description=A respectable 1 minute attempt. But will be too fast for most people. The sweet spot is likely 2 minutes.}