This 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 example, suppose we have and we are using .
For the denominator , we just use the multiplicative inverse, e.g. supposing we have
where because , related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p
Mordell's theorem by Ciro Santilli 37 Updated 2025-07-16
The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.
TODO example.
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.
18.783 MIT course by Ciro Santilli 37 Updated 2025-07-16
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!
Their status is a mess as of 2020s, with several systems ongoing. Long live the "original" collegiate university!
web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:
TODO why this non standard formulation?

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