Notes on Elliptic Curves (II) by BSD by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
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.
Elliptic curve point addition is the group operation of an elliptic curve group, i.e. it is a function that takes two points of an elliptic curve as input, and returns a third point of the elliptic curve as its output, while obeying the group axioms.
The operation is defined e.g. at en.wikipedia.org/w/index.php?title=Elliptic_curve_point_multiplication&oldid=1168754060#Point_operations. For example, consider the most common case for two different points different. If the two points are given in coordinates:then the addition is defined in the general case as:with some slightly different definitions for point doubling and the identity point.
This definition relies only on operations that we know how to do on arbitrary fields:and it therefore works for elliptic curves defined over any field.
Just remember that:means:and that always exists because it is the inverse element, which is guaranteed to exist for multiplication due to the group axioms it obeys.
The group function is usually called elliptic curve point addition, and repeated addition as done for DHKE is called elliptic curve point multiplication.
Elliptic curve point multiplication by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
Number of elements of an elliptic curve over a finite field by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
Number of elements of an elliptic curve over the rational numbers by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
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 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 by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
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 and
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.
Quantum computing university course by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
Waring problem with negative numbers allowed by
Ciro Santilli 37 Updated 2025-07-14 +Created 1970-01-01
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