Ciro Santilli @cirosantilli 34

 Incoming links: Finite field

Birch and Swinnerton-Dyer conjecture  Updated 2024-12-15  +Created 1970-01-01

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 1. "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 lo g (x)^{r}

(1)

Equation 1.

BSD conjecture

. 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 $E (F_{p}) m o d p$ . $N_{p}$ 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 Wikipedia page of the this conecture is the perfect example of why it is not possible to teach natural sciences on Wikipedia. A million dollar problem, and the page is thoroughly incomprehensible unless you already know everything!

Video 1.

Birch and Swinnerton-Dyer conjecture by Kinertia (2020)

Source.

Video 2.

The $1,000,000 Birch and Swinnerton-Dyer conjecture by Absolutely Uniformly Confused (2022)

Source. A respectable 1 minute attempt. But will be too fast for most people. The sweet spot is likely 2 minutes.

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Deletionism on Wikipedia  Updated 2024-12-15  +Created 1970-01-01

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en.wikipedia.org/wiki/Deletionism_and_inclusionism_in_Wikipedia

Some examples by Ciro Santilli follow.

Of the tutorial-subjectivity type:

This edit perfectly summarizes how Ciro feels about Wikipedia (no particular hate towards that user, he was a teacher at the prestigious Pierre and Marie Curie University and actually as a wiki page about him):
rm a cryptic diagram (not understandable by a professional mathematician, without further explanations
which removed the only diagram that was actually understandable to non-Mathematicians, which Ciro Santilli had created, and received many upvotes at: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426. The removal does not generate any notifications to you unless you follow the page which would lead to infinite noise, and is extremely difficult to find out how to contact the other person. The removal justification is even somewhat ad hominem: how does he know Ciro Santilli is also not a professional Mathematician? :-) Maybe it is obvious because Ciro explains in a way that is understandable. Also removal makes no effort to contact original author. Of course, this is caused by the fact that there must also have been a bunch of useless edits not done by Ciro, and there is no reputation system to see if you should ignore a person or not immediately, so removal author has no patience anymore. This is what makes it impossible to contribute to Wikipedia: your stuff gets deleted at any time, and you don't know how to appeal it. Ciro is going to regret having written this rant after Daniel replies and shows the diagram is crap. But that would be better than not getting a reply and not learning that the diagram is crap.
en.wikipedia.org/w/index.php?title=Finite_field&type=revision&diff=1044934168&oldid=1044905041 on finite fields with edit comment "Obviously: X ≡ α". Discussion at en.wikipedia.org/wiki/Talk:Finite_field#Concrete_simple_worked_out_example Some people simply don't know how to explain things to beginners, or don't think Wikipedia is where it should be done. One simply can't waste time fighting off those people, writing good tutorials is hard enough in itself without that fight.
en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&diff=1193622235&oldid=1193529573 by user Bob K. removed Ciro Santilli's awesome simple image of the Discrete Fourier transform as seen at en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&oldid=1176616763:
Hello. I am a retired electrical engineer, living near Washington,DC. Most of my contributions are in the area of DSP, where I have about 40 years of experience in applications on many different processors and architectures.
with message:
remove non-helpful image
Thank you so much!!
Maybe it is a common thread that these old "experts" keep removing anything that is actually intelligible by beginners? Section "There is value in tutorials written by beginners"
Also ranted at: x.com/cirosantilli/status/1808862417566290252
Figure 1.
Ciro Santilli's awesome graph removed by Bob K. from the Discrete Fourier transform page.
Source at: numpy/fft_plot.py.
when Ciro Santilli created Scott Hassan's page, he originally included mentions of his saucy divorce: en.wikipedia.org/w/index.php?title=Scott_Hassan&oldid=1091706391 These were reverted by Scott's puppets three times, and Ciro and two other editors fought back. Finally, Ciro understood that Hassan's puppets were likely right about the removal because you can't talk about private matters of someone who is low profile:
- en.wikipedia.org/wiki/Wikipedia:Biographies_of_living_persons
- en.wikipedia.org/wiki/Wikipedia:Who_is_a_low-profile_individual
even if it is published in well known and reliable publications like the bloody New York Times. In this case, it is clear that most people wanted to see this information summarized on Wikipedia since others fought back Hassan's puppet. This is therefore a failure of Wikipedia to show what the people actually want to read about.
This case is similar to the PsiQuantum one. Something is extremely well known in an important niche, and many people want to read about it. But because the average person does not know about this important subject, and you are limited about what you can write about it or not, thus hurting the people who want to know about it.

Notability constraints, which are are way too strict:

even information about important companies can be disputed. E.g. once Ciro Santilli tried to create a page for PsiQuantum, a startup with $650m in funding, and there was a deletion proposal because it did not contain verifiable sources not linked directly to information provided by the company itself: en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/PsiQuantum Although this argument is correct, it is also true about 90% of everything that is on Wikipedia about any company. Where else can you get any information about a B2B company? Their clients are not going to say anything. Lawsuits and scandals are kind of the only possible source... In that case, the page was deleted with 2 votes against vs 3 votes for deletion.
should we delete this extremely likely useful/correct content or not according to this extremely complex system of guidelines"
is very similar to Stack Exchange's own Stack Overflow content deletion issues. Ain't Nobody Got Time For That. "Ain't Nobody Got Time for That" actually has a Wiki page: en.wikipedia.org/wiki/Ain%27t_Nobody_Got_Time_for_That. That's notable. Unlike a $600M+ company of course.
In December 2023 the page was re-created, and seemed to stick: en.wikipedia.org/wiki/Talk:PsiQuantum#Secondary_sources It's just a random going back and forth. Author Ctjk has an interesting background:
I am a legal official at a major government antitrust agency. The only plausible connection is we regulate tech firms

There are even a Wikis that were created to remove notability constraints: Wiki without notability requirements.

For these reasons reason why Ciro basically only contributes images to Wikipedia: because they are either all in or all out, and you can determine which one of them it is. And this allows images to be more attributable, so people can actually see that it was Ciro that created a given amazing image, thus overcoming Wikipedia's lack of reputation system a little bit as well.

Wikipedia is perfect for things like biographies, geography, or history, which have a much more defined and subjective expository order. But when it comes to "tutorials of how to actually do stuff", which is what mathematics and physics are basically about, Wikipedia has a very hard time to go beyond dry definitions which are only useful for people who already half know the stuff. But to learn from zero, newbies need tutorials with intuition and examples.

Bibliography:

gwern.net/inclusionism from gwern.net:
Iron Law of Bureaucracy: the downwards deletionism spiral discourages contribution and is how Wikipedia will die.
Quote "Golden wiki vs Deletionism on Wikipedia"

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Elliptic curve  Updated 2024-12-15  +Created 1970-01-01

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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 1. "Definition of the elliptic curves"

y^{2} = x^{3} + a x + b

(1)

Equation 1.

Definition of the elliptic curves

Figure 1.
Plots of real elliptic curves for various values of $a$ and $b$
. Source.

Equation 1. "Definition of the elliptic curves" definies elliptic curves over any field, 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.

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Finite algebraic structure  Updated 2024-12-15  +Created 1970-01-01

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Examples:

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Finite general linear group  Updated 2024-12-15  +Created 1970-01-01

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general linear group over a finite field of order

m

. Remember that due to the classification of finite fields, there is one single field for each prime power

m

Exactly as over the real numbers, you just put the finite field elements into a

n \times n

matrix, and then take the invertible ones.

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Polynomial over a field  Updated 2024-12-15  +Created 1970-01-01

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By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

For example, given the finite field of order 9,

G P (3)

and with elements

{0, 1, 2}

, we can denote polynomials over that ring as

G P (3) [x]

(1)

where

x

is the variable name.

For example, one such polynomial could be:

P (x) = 2 x^{4} + x^{2} + 2

(2)

and another one:

Q (X) = x^{3} + 2 x^{2} + 2

(3)

Note how all the coefficients are members of the finite field we chose.

Given this, we could evaluate the polynomial for any element of the field, e.g.:

P (0) = 2 (0 \times 0 \times 0 \times 0) + (0 \times 0) + 2 = 2 P (1) = 2 (1 \times 1 \times 1 \times 1) + (1 \times 1) + 2 = 2 (1) + 1 + 2 = 2 P (2) = 2 (2 \times 2 \times 2 \times 2) + (2 \times 2) + 2 = 2 (16

(4)

and so on.

We can also add polynomials as usual over the field:

P (x) + Q (x) = 2 x^{4} + x^{3} + (1 + 2) x^{2} + (2 + 2) = 2 x^{4} + x^{3} + (0) x^{2} + 1 = 2 x^{4} + x^{3} + 1

(5)

and multiplication works analogously.

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Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p  Updated 2024-12-15  +Created 1970-01-01

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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} m o d 11 = 3 \times 3 m o d 11 = 9 m o d 11

(1)

where

4^{- 1} = 3 m o d 11

because

4 \times 3 = 1 m o d 11

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Underlying field of a vector space  Updated 2024-12-15  +Created 1970-01-01

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Every vector space is defined over a field.

E.g. in

R^{3}

, the underlying field is

R

, the real numbers. And in

C^{2}

the underlying field is

C

, the complex numbers.

Any field can be used, including finite field. But the underlying thing has to be a field, because the definitions of a vector need all field properties to hold to make sense.

Elements of the underlying field of a vector space are known as scalar.

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