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 1. "BSD conjecture" holds for every elliptic curve over the rational numbers (which is defined by its constants and )
Equation 1. . Where the following numbers are defined for the elliptic curve we are currently considering, defined by its constants and :
- : number of elements of the elliptic curve over the finite field, where the finite field comes from the reduction of an elliptic curve from to . can be calculated algorithmically with Schoof's algorithm in polynomial time
- : rank of the elliptic curve over the rational numbers. We don't really have a good general way to calculate this besides this conjecture (?).
- : 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 like that up to some point at which we are quite certain about .
The Wikipedia page of the this conecture is the perfect example of why it is not possible to teach natural sciences on Wikipedia. A million dollar problem, and the page is thoroughly incomprehensible unless you already know everything!
Birch and Swinnerton-Dyer conjecture by Kinertia (2020)
Source. The $1,000,000 Birch and Swinnerton-Dyer conjecture by Absolutely Uniformly Confused (2022)
Source. A respectable 1 minute attempt. But will be too fast for most people. The sweet spot is likely 2 minutes.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 . Prerequisite: generating set of a group
- reduction of an elliptic curve from to lets us define as the number of elements of the generated finite group
The paper that states the BSD conjecture.
Likely paywalled at: www.degruyter.com/document/doi/10.1515/crll.1965.218.79/html. One illegal upload at: virtualmath1.stanford.edu/~conrad/BSDseminar/refs/BSDorigin.pdf.
