algebra.bigb

= Algebra
{wiki}

= Algebraic
{synonym}

Not to be confused with <algebra over a field>, which is a particular <algebraic structure> studied within algebra.

= Abstract algebra
{parent=Algebra}
{wiki}

We just use "Abstract algebra" as a synonym for <algebra>.

= Algebraic structure
{parent=Algebra}
{wiki}

A <set (mathematics)> $S$ plus any number of functions $f_i : S \times S \to S$, such that each $f_i$ satisfies some properties of choice.

Key examples:
* <group>: one function
* <field (mathematics)>: two functions
* <ring (mathematics)>: 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 matrix>

= Invertible
{parent=Inverse element}

= Order
{disambiguate=algebra}
{parent=Algebraic structure}
{wiki}

The order of a <algebraic structure> is just its <cardinality>.

Sometimes, especially in the case of structures with an <infinite> 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 (algebra)>.

= Degree
{disambiguate=algebra}
{parent=Order (algebra)}
{wiki}

The degree of some <algebraic structure> 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 <infinite> number of elements, but it is sometimes also used for finite structures.

Examples:
* the <dihedral group> of degree n acts on n elements, and has order 2n
* the parameter $n$ that characterizes the size of the <general linear group> $GL(n)$ is called the degree of that group, i.e. the dimension of the underlying matrices

= Finite algebraic structure
{parent=Order (algebra)}

Examples:
* <finite group>
* <finite field>

\Include[linear-algebra]{parent=algebra}
\Include[group]{parent=algebra}

= Associative property
{parent=Algebra}
{wiki}

= Associative
{synonym}

= Algebraic geometry
{parent=Algebra}
{wiki}

= The beauty of algebraic geometry
{parent=Algebraic geometry}
{tag=The beauty of mathematics}

= The beauty of alebraic geometry
{synonym}

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 <real plane> that satisfy the <equation definition of the elliptic curves>{full}

$$
y^2 = x^3 + ax + b
$$
{title=Definition of the <elliptic curves>}

\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}

<equation definition of the elliptic curves> definies <elliptic curves> over any <field (mathematics)>, it doesn't have to the <real numbers>. Notably, the definition also works for <finite fields>, leading to <elliptic curve over a finite fields>, which are the ones used in <elliptic-curve Diffie-Hellman> cyprotgraphy.

= Elliptic curve group
{parent=Elliptic curve}

The <elliptic curve group> of an <elliptic curve> is a group in which the elements of the group are points on an <elliptic curve>.

The <group operation> is called <elliptic curve point addition>.

Bibliography:
* https://mathoverflow.net/questions/6870/why-is-an-elliptic-curve-a-group

= Elliptic curve point addition
{parent=Elliptic curve group}

<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 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 <field (mathematics)>[fields]:
* <addition> $+$
* <multiplication> $\times$
and it therefore works for <elliptic curves> defined over any field.

Just remember that:
$$
x/y
$$
means:
$$
x \times y^{-1}
$$
and that $y^{-1}$ 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>.

\Image[https://upload.wikimedia.org/wikipedia/commons/a/ae/ECClines-2.svg]
{title=Visualisation of <elliptic curve point addition>}

= 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 <elliptic curve over a finite field> or the <elliptic curve over the rational numbers> the <elliptic curve over the real numbers> is that not every possible $x$ generates a member of the curve.

This is because on the <equation Definition of the elliptic curves> 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 <field (mathematics)>[fields] are not <quadratically closed fields>.

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 <elliptic curve over the rational numbers>.

That $x$ would also not belong to <Elliptic curve over the finite field> $\F_4$, 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 <multiplicative inverse>.

For the <real numbers>, it would work however, because the <real numbers> are a <quadratically closed field>, and $\sqrt{3} \in \R$.

For this reason, it is not necessarily trivial to determine the <number of elements of an elliptic curve>.

= 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 <finitely generated group>.

= Mordell's theorem
{c}
{parent=Number of elements of an elliptic curve over the rational numbers}
{title2=1922}

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.

= 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}

<Mordell's theorem> guarantees that <rank of a group>[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.

= 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 formulation?

= 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 takes as input:
* <elliptic curve over the rational numbers>
* a prime number $p$
and it produces an <elliptic curve over a finite field> of order $p$ as output.

The constructions is used in the <Birch and Swinnerton-Dyer conjecture>.

To do it, we just convert the coefficients $a$ and $b$ from the <equation definition of the elliptic curves> from <rational numbers> to elements of the <finite field>.

For example, suppose we have $a = 3/4$ and we are using $p = 11$.

For the <denominator> $4$, we just use the <multiplicative inverse>, 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 conjecture.

Maybe also insert a joke about <BSD Operating Systems> if you're into that kind of stuff.

The conjecture states that <equation BSD conjecture> holds for every <elliptic curve over the rational numbers> (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=<BSD conjecture>}
{description=
Where the following numbers are defined for the <elliptic curve> we are currently considering, defined by its constants $a$ and $b$:
* $N_p$: <number of elements of the elliptic curve over the finite field>, where the <finite field> comes from the <reduction of an elliptic curve from $E(\Q)$ to $E(\F_p) mod p>. $N_p$ can be calculated algorithmically with <Schoof's algorithm> in <polynomial time>
* $r$: <rank of the elliptic curve over the rational numbers>. 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 <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 $N_p$ 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 <it is not possible to teach natural sciences on Wikipedia>. A <Millennium Prize Problems>[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 <elliptic curve> $y^2 = x^3 - 5x$}
{description=The curve is known to have <rank of the elliptic curve over the rational numbers>[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=<Birch and Swinnerton-Dyer conjecture> by Kinertia (2020)}

\Video[https://www.youtube.com/watch?v=tjnwEGBUOLI]
{title=The \$1,000,000 <Birch and Swinnerton-Dyer conjecture> 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 <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 $r$. Prerequisite: <generating set of a group>
* <reduction of an elliptic curve from $E(\Q)$ to $E(\F_p) \mod p$> 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 <BSD conjecture>.

Likely <closed access academic journals are evil>[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 <equation Definition of the elliptic curves> and definitions on <elliptic curve point addition> 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}