= Algebra
{wiki}
= Algebraic
{synonym}
Not to be confused with , which is a particular studied within algebra.
= Abstract algebra
{parent=Algebra}
{wiki}
We just use "Abstract algebra" as a synonym for .
= Algebraic structure
{parent=Algebra}
{wiki}
A $S$ plus any number of functions $f_i : S \times S \to S$, such that each $f_i$ satisfies some properties of choice.
Key examples:
* : one function
* : two functions
* : also two functions, but with less restrictive properties
= Commutator
{parent=Algebraic structure}
{wiki}
= Identity element
{parent=Algebraic structure}
{wiki}
= Inverse element
{parent=Identity element}
{wiki}
= Inverse
{synonym}
Some specific examples:
*
= Invertible
{parent=Inverse element}
= Order
{disambiguate=algebra}
{parent=Algebraic structure}
{wiki}
The order of a is just its .
Sometimes, especially in the case of structures with an number of elements, it is often more convenient to talk in terms of some parameter that characterizes the structure, and that parameter is usually called the .
= Degree
{disambiguate=algebra}
{parent=Order (algebra)}
{wiki}
The degree of some is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.
This is particularly useful when talking about structures with an number of elements, but it is sometimes also used for finite structures.
Examples:
* the of degree n acts on n elements, and has order 2n
* the parameter $n$ that characterizes the size of the $GL(n)$ is called the degree of that group, i.e. the dimension of the underlying matrices
= Finite algebraic structure
{parent=Order (algebra)}
Examples:
*
*
\Include[linear-algebra]{parent=algebra}
\Include[group]{parent=algebra}
= Associative property
{parent=Algebra}
{wiki}
= Associative
{synonym}
= Algebraic geometry
{parent=Algebra}
{wiki}
= The beauty of alebraic geometry
{parent=Algebraic geometry}
{tag=The beauty of mathematics}
https://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates Interesting results in algebraic geometry accessible to 3rd year undergraduates
= Algebraic curve
{parent=Algebraic geometry}
{wiki}
= Elliptic curve
{parent=Algebraic geometry}
{wiki}
An elliptic curve is defined by numbers $a$ and $b$. The curve is the set of all points $(x, y)$ of the that satisfy the {full}
$$
y^2 = x^3 + ax + b
$$
{title=Definition of the }
\Image[https://upload.wikimedia.org/wikipedia/commons/thumb/d/db/EllipticCurveCatalog.svg/795px-EllipticCurveCatalog.svg.png]
{title=Plots of real elliptic curves for various values of $a$ and $b$}
{height=800}
definies over any , it doesn't have to the . Notably, the definition also works for , leading to , which are the ones used in cyprotgraphy.
= Elliptic curve group
{parent=Elliptic curve}
The of an is a group in which the elements of the group are points on an .
The is called .
Bibliography:
* https://mathoverflow.net/questions/6870/why-is-an-elliptic-curve-a-group
= Elliptic curve point addition
{parent=Elliptic curve group}
is the of an , i.e. it is a that takes two points of an as input, and returns a third point of the as its output, while obeying the .
The operation is defined e.g. at https://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:
$$
\begin{aligned}
P &+ Q &= R \\
(x_p, y_p) &+ (x_q, y_q) &= (x_r, y_r) \\
\end{aligned}
$$
then the addition is defined in the general case as:
$$
\begin{aligned}
\lambda &= \frac{y_q - y_p}{x_q - x_p} \\
x_r &= \lambda^2 - x_p - x_q \\
y_r &= \lambda(x_p - x_r) - y_p \\
\end{aligned}
$$
with some slightly different definitions for point doubling $P + P$ and the identity point.
This definition relies only on operations that we know how to do on arbitrary [fields]:
* $+$
* $\times$
and it therefore works for defined over any field.
Just remember that:
$$
x/y
$$
means:
$$
x \times y^{-1}
$$
and that $y^{-1}$ always exists because it is the , which is guaranteed to exist for multiplication due to the it obeys.
The group function is usually called , and repeated addition as done for is called .
\Image[https://upload.wikimedia.org/wikipedia/commons/a/ae/ECClines-2.svg]
{title=Visualisation of }
= Elliptic curve point multiplication
{parent=Elliptic curve group}
{wiki}
= Domain of an elliptic curve
{parent=Elliptic curve}
= Not every $x$ belongs to the elliptic curve over a non quadratically closed field
{parent=Domain of an elliptic curve}
One major difference between the or the the is that not every possible $x$ generates a member of the curve.
This is because on the we see that given an $x$, we calculate $x^3 + ax + b$, which always produces an element $y^2$.
But then we are not necessarily able to find an $y$ for the $y^2$, because not all [fields] are not .
For example: with $a = 1$ and $b = 1$, taking $x = 1$ gives:
$$
y^2 = 1^3 + 1 \times 1 + 1 = 3
$$
and therefore there is no $y \in \Q$ that satisfies the equation. So $x = 1$ is not on the curve if we consider this .
That $x$ would also not belong to $\F_4$, because doing everything $\mod 4$ we have:
$$
\begin{aligned}
0*0 &= 0 & &\mod 4 \\
1*1 &= 1 & &\mod 4 \\
2*2 &= 4 &= 0 &\mod 4 \\
3*3 &= 9 &= 1 &\mod 4 \\
\end{aligned}
$$
Therefore, there is no element $y \in \F_4$ such that $y \times y = 2$ or $y \times y = 3$, i.e. $2$ and $3$ don't have a .
For the , it would work however, because the are a , and $\sqrt{3} \in \R$.
For this reason, it is not necessarily trivial to determine the .
= Number of elements of an elliptic curve
{parent=Not every $x$ belongs to the elliptic curve over a non quadratically closed field}
= Elliptic curve over the real numbers
{parent=Domain of an elliptic curve}
{tag=Real number}
{title2=$E(\F)$}
= Elliptic curve over the rational numbers
{parent=Domain of an elliptic curve}
{tag=Rational number}
{title2=$E(\Q)$}
= Number of elements of an elliptic curve over the rational numbers
{parent=Elliptic curve over the rational numbers}
{tag=Number of elements of an elliptic curve}
Can be finite or infinite! TODO examples. But it is always a .
= Mordell's theorem
{c}
{parent=Number of elements of an elliptic curve over the rational numbers}
{title2=1922}
The of all is always a .
The number of points may be either finite or infinite. But when infinite, it is still a .
For this reason, the is always defined.
TODO example.
= Rank of an elliptic curve over the rational numbers
{parent=Mordell's theorem}
{tag=Rank of a group}
{title2=$r$}
{wiki=Rank_of_an_elliptic_curve}
= Rank of the elliptic curve over the rational numbers
{synonym}
guarantees that [the rank] (number of elements in the ) is always well defined for an . But as of 2023 there is no known algorithm which calculates the rank of any curve!
TODO list of known values and algorithms? The would immediately provide a stupid algorithm for it.
= Largest known ranks of an elliptic curve over the rational numbers
{parent=Rank of an elliptic curve over the rational numbers}
https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:
$$
y^2 + xy + y = x^3 - x^2 - 20067762415575526585033208209338542750930230312178956502 x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429
$$
TODO why this non standard formluation?
= Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p
{parent=Elliptic curve over the rational numbers}
= Reduction of an elliptic curve from $E(\Q)$ to $E(\F_p) \mod p$
{synonym}
{title2}
This construction taks as input:
*
* a prime number $p$
and it produces an of order $p$ as output.
The constructions is used in the .
To do it, we just convert the coefficients $a$ and $b$ from the from to elements of the .
For example, suppose we have $a = 3/4$ and we are using $p = 11$.
For the $4$, we just use the , e.g. supposing we have
$$
\frac{3}{4} \to 3 \times 4^{-1} \mod 11 = 3 \times 3 \mod 11 = 9 \mod 11
$$
where $4^{-1} = 3 \mod 11$ because $4 \times 3 = 1 \mod 11$, related: https://math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p
= Birch and Swinnerton-Dyer conjecture
{c}
{parent=Elliptic curve over the rational numbers}
{title2=1965}
{tag=Millennium Prize Problems}
{wiki}
= BSD Conjecture
{c}
{synonym}
{title2}
The BSD conjecture states that if your name is long enough, it will always count as two letters on a famous conejcture.
Maybe also insert a joke about if you're into that kind of stuff.
The conjecture states that holds for every (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=}
{description=
Where the following numbers are definied for the we are currently considering, defined by its constants $a$ and $b$:
* $N_p$: , where the comes from the . $N_p$ can be calculated algorithmically with in
* $r$: . 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 , since we can calculate the by in . 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 . A [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 $y^2 = x^3 - 5x$}
{description=The curve is known to have [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= by Kinertia (2020)}
\Video[https://www.youtube.com/watch?v=tjnwEGBUOLI]
{title=The \$1,000,000 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.}
= BSD conjecture bibliography
{c}
{parent=Birch and Swinnerton-Dyer conjecture}
= Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli
{c}
{parent=BSD conjecture bibliography}
{title2=2023}
Summary:
* overview of the formula of the
* definition of
* . Prerequisite:
* . Prerequisite:
* lets us define the , which is the $r$. Prerequisite:
* lets us define $N_r$ as the number of elements of the generated finite group
\Video[https://www.youtube.com/watch?v=84ig5cih4kI]
= Notes on Elliptic Curves (II) by BSD
{c}
{parent=BSD conjecture bibliography}
{title2=1965}
The paper that states the .
Likely [paywalled] at: https://www.degruyter.com/document/doi/10.1515/crll.1965.218.79/html[]. One illegal upload at: http://virtualmath1.stanford.edu/~conrad/BSDseminar/refs/BSDorigin.pdf[].
= Elliptic curve over a finite field
{parent=Domain of an elliptic curve}
{tag=Finite field}
{title2=$E(\F_p)$}
= Elliptic curve over the finite field
{synonym}
The and definitions on both hold directly.
= Number of elements of an elliptic curve over a finite field
{parent=Elliptic curve over a finite field}
{tag=Number of elements of an elliptic curve}
= Number of elements of the elliptic curve over the finite field
{synonym}
= Schoof's algorithm
{c}
{parent=Number of elements of an elliptic curve over a finite field}
{tag=Polynomial time algorithm}
{wiki}
= Elliptic curve bibliography
{parent=Elliptic curve}
* https://www.cantorsparadise.com/another-math-problem-that-will-earn-you-a-million-dollars-for-solving-it-95546d4841cc
= Elliptic curve university course
{parent=Elliptic curve bibliography}