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{tag=Millennium Prize Problems}
{title2=1965}
{wiki}

= BSD conjecture
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{synonym}
{title2}

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

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>}
{description=
Where the following numbers are defined 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 https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture[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!

\Image[https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/BSD_data_plot_for_elliptic_curve_800h1.svg/640px-BSD_data_plot_for_elliptic_curve_800h1.svg.png]
{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.}
{height=400}

\Video[https://www.youtube.com/watch?v=R9FKN9MIHlE]
{title=<Birch and Swinnerton-Dyer conjecture> by Kinertia (2020)}

\Video[https://www.youtube.com/watch?v=tjnwEGBUOLI]
{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.}


Birch and Swinnerton-Dyer conjecture

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{title2=2023}

Summary:
* overview of the formula of the <BSD conjecture>
* definition of <elliptic curve>
* <domain of an elliptic curve>. Prerequisite: <field>
* <elliptic curve group>. Prerequisite: <group>
* <Mordell's theorem> lets us define the <rank of an elliptic curve over the rational numbers>, which is the $r$. Prerequisite: <generating set of a group>
* <reduction of an elliptic curve from E(\Q) to E(\F_p) \mod p> lets us define $N_r$ as the number of elements of the generated finite group

\Video[https://www.youtube.com/watch?v=84ig5cih4kI]


Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli

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{title2=1922}

The <elliptic curve group> of all <elliptic curve over the rational numbers> is always a <finitely generated group>.

The number of points may be either finite or infinite. But when infinite, it is still a <finitely generated group>.

For this reason, the <rank of an elliptic curve over the rational numbers> is always defined.

TODO example.


Ciro Santilli @cirosantilli 34

 Incoming links: Rank of an elliptic curve over the rational numbers