by

Ciro Santilli (@cirosantilli, 34)

Elliptic curve over the rational numbers ( $E (Q)$ )

Table of contents

Number of elements of an elliptic curve over the rational numbers

Can be finite or infinite! TODO examples. But it is always a finitely generated group.

Mordell's theorem (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 ( $r$ )

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!

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

web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:

y^{2} + x y + 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 (Reduction of an elliptic curve from $E (Q)$ to $E (F_{p}) m o d p$ )

This construction taks 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

where

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

because

4 \times 3 = 1 m o d 11

, related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p

Birch and Swinnerton-Dyer conjecture (1965, BSD conjecture)

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 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}

. Where the following numbers are definied 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!

Figure 1.
$lim_{x \to \infty} \prod_{p \leq x} \frac{N _{p}}{p}$ as a function of $p$ for the elliptic curve $y^{2} = x^{3} - 5 x$
. Source. 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.

Birch and Swinnerton-Dyer conjecture by Kinertia (2020)

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.

BSD conjecture bibliography

Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli (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 $E (F_{p}) m o d p$ lets us define $N_{r}$ as the number of elements of the generated finite group

Video 1. Source.

Notes on Elliptic Curves (II) by BSD (1965)

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.

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