Short intro: Birch and Swinnerton-Dyer conjecture

Abstract

In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute. Let me start to describe this conjecture:

why study elliptic curve?

a number theorists like study solution for algebraic variety: we know that $$ ax+b=0 \rightarrow x=\frac{-b}{a} $$ and $$ ax^2+bx+c=0 \rightarrow x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} $$ For the higher degree one variable, Galois solved this question by introducing . Galois theory But at least, Gauss said: $$ a_nx^n+a_{n-1}x^{n-1}+....a_0=0 $$ have n complex solutions. This is called algebraic fundamental theorem. Ref can be seen here Fundamental theorem of algebra

Rational Solution

sometimes, we only care about rational solution: in one varible, this is not so hard: $$ a_nx^n+a_{n-1}x^{n-1}+....a_0=0 $$ Newton method: if there is a solution \( x=\frac{p}{q} : p, q \in \text{Integer}\) $$ q \text{ is a divisor for } a_n, p \text{ is a divisor for } a_0 $$. So it must be finite and can be computed via computers very easily.

Two variables?

The simple two variable can be viewed as: $$ ax^2+bxy+cy^2+dx+ef+g=0 $$ This is called the conic section: More generally:

Genus of a curve

we define the \( g \) $$ g=(d-1)(d-2)/2$$ for the homogenious polynomial.

Faltings's theorem

A non-singular algebraic curve of genus \( g \) over rational fields.

1. \( g=0 \): no rational points points or infinitely many rational points
2. \(g \geq 2 \): only finitely many points
3. \(g=1 \) :no rational points, or rational points forms an finitely generated abelian group

So the for \(g=1\) the number of rational points can be any cases.

Elliptic curve

$$ E: y^2=x^3+ax+b$$ Provided that : $$4a^3+27b^2 \neq 0 $$ We can discuss by restrict a ,b in different fieid: We only discuss the rational field and finite field: First, we need to introduce algebraic structure for an elliptic curve: We defined $$ P+Q $$ like the image. First, draw a line connect \(P, Q \) then it will intersect the second point on the elliptic curve and then reflection it, we can get the \(P+Q \)

(Mordell, 1922)

Let E be an elliptic curve given by an equation $$ E: y^2=x^3+ax+b$$ \( E(\mathbb{Q}) \) forms an abelian group via this addition. \( E(\mathbb{Q}) \) is a finitely generated abelian groups. ie Moreover, the Torsion subgroup has only these possibilities. So it turns out that the torsion subgroup can be computed easily.

Rank

The major question is how can we computed r efficiently? ANs: NO Up to now, we don't have efficient algorithm for finding rank for given general elliptic curve. So how can we do?? Birch and Swinnerton-Dyer in late 60 do the experiment on the elliptic curve on finite field, let us start to discuss it:

Finite field

Finite field denoted \( F_p \) are just like usually addition, multiplication but only have p-1 elements (only count the reminder). For example consider the algebriac equation $$ x^2+1=0 $$ in \(F_5 \) : 2,3 are solutions Given an elliptic curve, we can use the same way: for example: consider $$ y^2=x^3+2 $$ we list solution below \(O )\ means infinity: Solutions for elliptic curve and Hesse Weil bound Hesse Weil bound: number of solutions \(N(p)-p-1 \) are bounded by \( 2\sqrt{p} \).

Key idea for Birch and Swinnerton-Dyer conjecture:

If elliptic curve has higher rank, then the \(N(p) \) should have some relationship with r. So they computed this functions and found that:

primitive bsd conjecture 
  Let means this function grows \(ln(x)^r \) which related to the rank. (good means that the elliptic curve is non singular when it is restricted at \( F_{p} \), this is too detail here).

More subtle: Corresponding L function

Given an elliptic curve over rational. We define $$L(E, s)=\prod_{p \in \text{Primes}}\frac{1}{1-a_pp^{-s}+p^{1-2s}}$$ $$a_p=p+1-N(p) $$ we found that when s=1: so the rank connect the zero of L functions: Up to now, we can state this famous conjecture in wiki and other books (skip the analytic continuation and some details): Actually the L function did not converge at s=1, but we skip the detail here.

Birch and Swinnerton-Dyer conjecture

Given an elliptic curve \( E \) defined on rational field, the rank \(r \) of the groups is equal to the order of L function for the elliptic curve. $$ \text{the rank r for} \mathbb{Q}= \text{the order of s=1 at L function.}$$ write it more explicitly: if $$ E(\mathbb{Q}) =\mathbb{Z}^{r} + \text{ torsion} $$ then the expansion for L function is \(a_0 \neq 0 \): $$ L=0+0+a_0(s-1)^r+ ..\text{higher orders}$$.

Conclusion: Why we are interested in this?

We are interested in particular solutions of algebraic equations, for example, famous Fermat's last theorem said $$ x^n+y^n=z^n, n \in \mathbb{N} , n \geq2$$ has no positive integer \(xyz>0 \) solution: It is very hard to proof, even though the well educated professor in mathematics nowadays. In order to proof this, one has to proof Modularity theorem which connect the number theory and analysis. So if someone can proof Birch and Swinnerton-Dyer conjecture, it means that we can imitate this conjecture in higher dimensional algebraic variety and probably use same techniques to extend this result. For example: Euler(1769) conjectured that $$x^4+y^4+z^4=w^4 $$ has no positive integer solution \(xyzw>0 \). This looks very similar to Fermat last theorem!! It turns out it has a solutions(XD). Elkies found the solution: $$ 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$$. In conclusion, up to now, there are still almost no efficient methods finding such rational or integer points for given an algebraic variety. We hope that solving this conjecture can help us to proceed other problems in the future.

Reference and good source

You can find elliptic curve on WyomingEllipticCurve.pdf Or on wiki or find clay mathematics.