Topics (203k) Articles (205k) Users (298) Discussions (237) Comments (383) Files (715) New article
Their status is a mess as of 2020s, with several systems ongoing. Long live the "original" collegiate university!
math.mit.edu/classes/18.783, Wow, good slides! Well organized site! This is a good professor! And brutal course. 25 lectures, and lecture one ends in BSD conjecture!
Some points from math.mit.edu/classes/18.783/2022/LectureSlides1.pdf:
Mordell's theorem guarantees that the rank (number of elements in the generating set of the group) is always well defined for an elliptic curve over the rational numbers. But as of 2023 there is no known algorithm which calculates the rank of any curve!
It is not even known if there are elliptic curves of every rank or not: Largest known ranks of an elliptic curve over the rational numbers, and it has proven extremely hard to find new ones over time.
TODO list of known values and algorithms? The Birch and Swinnerton-Dyer conjecture would immediately provide a stupid algorithm for it.
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.
3 is not enough by Legendre's three-square theorem.
The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem.
Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16This construction takes as input:and it produces an elliptic curve over a finite field of order as output.
The constructions is used in the Birch and Swinnerton-Dyer conjecture.
To do it, we just convert the coefficients and from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.
For the denominator , we just use the multiplicative inverse, e.g. supposing we havewhere because , related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p
Number of elements of an elliptic curve over the rational numbers by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16 Number of elements of an elliptic curve over a finite field by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16 Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact