This is an expanded note prepared for a 40-minute elementary introduction to the Birch and Swinnerton-Dyer conjecture presented at the farewell party for Chen-Yu Chi, who was leaving Harvard after his 8 years as a graduate student and a junior fellow here. As majority of the audience are in fields orthogonal to number-theoretic studies, such a talk can easily succeed in convincing the audience that there is a dry, ridiculous, but famous conjecture coming out of nothing; crazy number theorists have wasted their life to contribute to the list of partial results, which cannot even be claimed to be a long list.

Hopefully I have failed in that way.

Our main sources are [1], [2] and [3].

TopDiophantine equations

A Diophantine equation is a polynomial equation with integer unknowns, the study of which dates back to the ancient Greeks. The word "Diophantine" comes from the Greek mathematician Diophantus of Alexandria of the 3rd century, the author of a series of books titled Arithmetica. In 1637, Pierre de Fermat scribbled the famous Fermat's last theorem on the margin of his own copy of Arithmetica. The efforts toward its proof has been widely regarded as one of the most glorious stories of mathematics of all time.

Diophantus studied many problems which are essentially quadratic equations from the modern point of view. For example, which integer triples $(x,y,z)$ are the lengths of a right triangle? In other words, what are the rational solutions to the equation $x^2+y^2=1$? Notice that the latter equation defines a smooth curve $C$ of genus 0 and we are concerned with the set of rational points $C(\mathbb{Q})$.

Theorem 1 Let $C$ be a smooth projective curve of genus 0 defined over a field $k $. Then $C$ is isomorphic to $\mathbb{P}^1$ over $k $ if and only if $C(k)\ne\varnothing$.

So the problem of finding all rational points of a genus 0 curve reduces to finding one rational point. Because each genus 0 curve can be embedded into $\mathbb{P}^2$ as a plane conic, over a number field, this is achieved by a special case of the Hasse principle for quadratic forms. For example, when $k=\mathbb{Q}$, $C$ has a point over $\mathbb{Q}$ if and only if it has a point over $\mathbb{R}$ and over every local field $\mathbb{Q}_p$. The latter can be determined handily using Hilbert symbols.

Question For a smooth curve $C$ of genus $g$ defined over $k $, how do we understand $C(k)$?

The following theorem of Faltings is rather remarkable.

Theorem 2 (Faltings, 1983) Let $C$ be a smooth curve of genus $g\ge2$ over a number field $k $, then $C(k)$ is finite.

In particular, we know that for any $n\ge4$, the Fermat equation $x^n+y^n=z^n$ has at most finitely many solutions, which is far beyond the scope of any elementary methods. Faltings' 18-page proof uses a wide range of techniques from algebraic number theory and algebraic geometry: at least to me, a whole semester seminar on going through this masterpiece is a truly amusing experience.

Finding these finitely many points is another interesting unsolved problem which we will not go into here. Let us concentrate on the case of genus 1 curves. Let $E$ be a smooth curve of genus 1 with a rational point $O\in C(k)$ (such a curve is called an elliptic curve), then $E(k)$ admits a group law with the identity element $O$. In particular, $E(k)$ is an abelian group. The old Mordell-Weil theorem asserts that this group cannot be enormous.

Theorem 3 (Mordell-Weil, 1928) Let $E$ be an elliptic curve over a number field $k $, then $E(k)$ is finitely generated.

In other words, as an abstract group, $E(k)\cong \mathbb{Z}^r\oplus E(k)_\mathrm{tor}$ can be decomposed as a direct some of finitely many copies of $\mathbb{Z}$ and a finite abelian group. Here the integer $r$ is called the rank of $E(k)$. Notice that $E(k)$ can be both finite (when $r=0$) and infinite (when $r>0$). This somehow reflects the tension between the infiniteness on curves of genus 0 and finiteness on curves of higher genus, making the case of elliptic curves most interesting and intricate.

Example 1 Every elliptic curve over $\mathbb{Q}$ can be realized as a plane cubic using the Weierstrass equation $y^2=f(x)$, where $f(x)\in \mathbb{Q}[x]$ is a cubic polynomial, with the point $\infty$ serving as the identity element $O$ here. The group law can be realized as the secant-tangent process. For example, let $E: y^2=x^3-12x^2+20x=x(x-2)(x-10)$. We easily find the rational points $(0,0)$, $(2,0)$, $(10,0)$ and $\infty$. These four are torsion points, all having order 2. The solution $(1,3)$ can also be inspected and it generates infinitely many solutions (e.g., twice of it is $(\frac{361}{36},\frac{-323}{216})$, which is hard to observe directly). As an abstract group, $E(\mathbb{Q})\cong \mathbb{Z}\oplus (\mathbb{Z}/2 \mathbb{Z})^2$.

The arithmetics of elliptic curves is also a significant node of the giant web of arithmetic problems. Some examples are in order.

Example 2 The area of a rational right triangle is called a congruent number, i.e., a congruent number $n$ is equal to $ab/2$ for some positive rational triple $(a,b,c)$ satisfying $a^2+b^2=c^2$. A bit of elementary calculation shows that the elliptic curve $E_n:y^2=x^3-n^2x$ has a rational point $P=\left(\frac{n(a+c)}{b}, \frac{2n^2(a+c)}{b^2}\right)$. Moreover, it can be shown that $P$ must be of infinite order. Conversely, such a point of infinite order gives us back a rational triple $(a,b,c)$. Thus showing that $n$ is a congruent number is equivalent to showing that the elliptic curve $E_n$ has positive rank.
Example 3 A key step of Wiles' proof of Fermat's last theorem is the usage of Frey's curve (1984): if $(a,b,c)$ is a nontrivial solution to $x^n+y^n=z^n$, then the elliptic curve $E: y^2=x(x-a^n)(x+b^n)$ is not modular (this is proved by Ribet (1990) by proving the epsilon conjecture). Wiles (1995) showed that all semistable elliptic curves over $\mathbb{Q}$ (e.g., Frey's curve) are modular, thereby finishing the proof of Fermat's last theorem. In 2001, Breuil-Conrad-Diamond-Taylor extended the modularity to all elliptic curves over $\mathbb{Q}$.
Example 4 Let $K $ be a number field. Poonen (2002) proved that if there is an elliptic curve $E$ over $\mathbb{Q}$ such that $\rank(E(\mathbb{Q}))=\rank(E(K))=1$, then Hilbert's 10th problem has a negative answer over the ring of integers $\mathcal{O}_K$: there does not exist an algorithm to decide whether a Diophantine equation with $\mathcal{O}_K$-coefficients has an $\mathcal{O}_K$-solution or not.

TopThe Birch and Swinnerton-Dyer conjecture

The following statements and conjectures apply for any elliptic curves $E$ over a global field $k $. For simplicity, we consider the case $k=\mathbb{Q}$.

Question How do we understand the structure of $E(\mathbb{Q})$?

Mazur's torsion theorem classifies all possible torsion part of $E(\mathbb{Q})$. His proof is a paradigm of using scheme-theoretic methods to draw concrete arithmetic consequences.

Theorem 4 (Mazur, 1978) Let $E$ be an elliptic curve over $\mathbb{Q}$, then $E(\mathbb{Q})_\mathrm{tor}$ is one of the following groups: $\mathbb{Z}/ n \mathbb{Z}$ for $n=1,\ldots, 10$ and $n=12$ or $\mathbb{Z}/2 \mathbb{Z}\times \mathbb{Z}/2n \mathbb{Z}$ for $n=1,2,3,4$.

On the contrary, we understand far less about the rank $r$ of $E(\mathbb{Q})$. For example, it is not known which integer values $r$ can take; it is even not known whether $r$ can take arbitrarily large values. People intend to believe the following conjectures about the rank $r$.

Conjecture 1 (Rank) $r$ can take arbitrarily large values.

Tate-Shafarevich (1967) proved the unboundedness of the rank over functions fields. The unboundedness in the number field case is still unknown: the current record over $\mathbb{Q}$ is kept by Elkies (2006), he found an elliptic curve with at least 28 independent $\mathbb{Q}$-points via the theory of Mordell-Weil lattices of elliptic K3 surfaces and searching techniques.

Conjecture 2 (Rank distribution) $r=0$: 50%; $r=1$: 50%; $r\ge2$: 0%. The average rank is $1/2$.

It is not even known whether the average rank has an upper bound until the recent seminal work of Bhargava-Shankar (2010). They used methods from the geometry of numbers to count certain integral orbits of the space of binary quartic forms to show that the average rank, if it exists, is less than $0.99$.

Miraculously, the rank of an elliptic curve, which we do not have understand well, is related to the analytic properties of its $L$-function $L(E,s)$. To motivate this, recall that associated to each algebraic variety $X$ over a finite field $\mathbb{F}_p$, its zeta function $$\zeta(X,T)=\exp\left(\sum_{m=1}^\infty\frac{|X(\mathbb{F}_{p^m})|}{m}T^m\right)$$ is a rational function of $T$ (the Weil conjecture). In our case, taking the reduction mod $p$ of an elliptic curve $E$ over $\mathbb{Q}$ gives an elliptic curve $\tilde E_p$ over $\mathbb{F}_p$ (here we ignore what happens for the bad primes). Its zeta function is $$\zeta(\tilde E_p, T)=\exp \left(\sum_{m=1}^\infty\frac{|\tilde E_p(\mathbb{F}_p)|}{m}T^m\right)=\frac{P_1(T)}{P_0(T)P_2(T)}=\frac{1-aT+pT^2}{(1-T)(1-pT)},$$ where $a=p+1-|\tilde E_p(\mathbb{F}_p)|$. Analogous to the Euler factors of the Riemann zeta function, we define the local $L$-factor of $E$ to be $$L_p(E, s)=\frac{1}{P_1(p^{-s})}=\frac{1}{1-ap^{-s}+p^{1-2s}}.$$ When evaluating its value at $s=1$, we retrieve the arithmetic information at $p$, $$L_p(E,1)=\frac{p}{p+1-a}=\frac{p}{|\tilde E_p(\mathbb{F}_p)|}.$$ Notice that each point in $E(\mathbb{Q})$ reduces to a point in $\tilde E_p(\mathbb{F}_p)$. So when $E(\mathbb{Q})$ has high rank, then $L_p(E,1)$ tends to be small. Birch and Swinnerton-Dyer did numerical experiments and suggested the heuristic $$\prod_{p<X}\frac{\tilde E_p(\mathbb{F}_p)}{p}\sim c_E\cdot (\log X)^r.$$

The $L$-function of $E$ is defined to be the product of all local $L$-factors, $$L(E,s)=\prod_p L_p(E,s).$$ Formally evaluating the value at $s=1$ gives $$L(E,1) So intuitively the rank of $E(\mathbb{Q})$ will correspond to the value of $L(E,s)$ at 1: the larger $r$ is, the "smaller" $L(E,1)$ is. However, the value of $L(E,s)$ at $s=1$ does not make sense since the product of $L_p(E,s)$ only converges when $\Re s>3/2$. Nevertheless, if $L(E,s)$ can be continued to an analytic function on the whole of $\mathbb{C}$, it may be reasonable to believe that the behavior of $L(E,s)$ at $s=1$ contains the arithmetic information of the rank of $E(\mathbb{Q})$. The famous Birch and Swinnerton-Dyer conjecture asserts that

Conjecture 3 (Birch and Swinnerton-Dyer, 1960s) Let $E$ be an elliptic curve over a global field $k $, then the order of vanishing of $L(E,s)$ at $s=1$ is equal to the rank of $E(k)$.

For this reason, we call the number $\ord_{s=1}L(E,s)$ the analytic rank of $E$. Even explicit constants are conjecturally described: $$L(E,s)\sim |\Sha(E)| R(E) P(E)\cdot (s-1)^r,\quad s\rightarrow 1.$$ Here $\Sha(E)$ is the Tate-Shafarevich group, measuring the failure of the Hasse principle for $E$-torsors; $R(E)$ is the regulator with respect to the Neron-Tate height pairing on $E(k)$; $P(E)$ is the period, a product of local periods obtained by integrating the Neron differential on $E$.

Remark 1 Because isogenous elliptic curves have the same $L$-functions, the BSD conjecture can be true only if the product $|\Sha(E)| R(E) P(E)$ is isogeny-invariant. This is proved by Cassels (1965) assuming the finiteness of $\Sha(E)$, even though each individual factor may change under isogeny!

This remarkable conjecture has been chosen as one of the seven Millennium Prize problems by the Clay Institute, with a million-dollar prize for its solution. It is even more remarkable when we notice that it is so "meaningless": the left-hand-side has no meaning because we did not even know whether $L(E,s)$ was defined round $s=1$ at the time when conjecture was formulated (except for elliptic curves with complex multiplication due to the work of Deuring and Weil in the 1950s), and the right-hand-side has no meaning either because we do not know in general that $\Sha(E)$ is finite! Nevertheless, this is indeed a good situation for mathematicians: we have more flexibility to figure out what the truth is.

Here are some consequences of the BSD conjecture.

Example 5 Tunnell (1983) proved that if $n$ is a congruent number, then 
\begin{multline*}
\#\{(x,y,z)\in\mathbb{Z}^3: 2x^2+y^2+8z^2=n, z\text{ is odd}\}\\=\#\{(x,y,z)\in\mathbb{Z}^3: 2x^2+y^2+8z^2=n, z\text{ is even}\}.
\end{multline*}
Moreover, the converse is also true if we further assume the BSD conjecture. Therefore the proof of the BSD conjecture will lead to a solution to the congruent number problem using a finite amount of computation.
Example 6 Mazur-Rubin (2010) proved that if the BSD conjecture holds for all elliptic curves over all number fields, then Hilbert's 10th problem has a negative answer over $\mathcal{O}_K$ for any number field $K $.

TopProgress to date

As we mentioned, the formulation of the BSD conjecture relies on the following two conjectures.

Conjecture 4 $L(E,s)$ has an analytic continuation to the whole of $\mathbb{C}$.
Conjecture 5 $\Sha(E)$ is finite.

Thanks to the modularity theorem of Wiles and others, we now know $L(E,s)$ can be always continued analytically to the whole of $\mathbb{C}$, but the finiteness of $\Sha(E,k)$ is still largely open: in each case that the finiteness of $\Sha(E,k)$ is known, the BSD conjecture of $E$ is proved along with it too.

There is much evidence in favor of the BSD conjecture, we now list a few of them. The function field case is of better shape than the number field case due to

Theorem 5 (Artin-Tate, 1960s) The BSD conjecture holds for an elliptic curve $E$ over a function field $k $ if and only if $\Sha(E)$ is finite.

The best general result to date for the BSD conjecture over number fields is due to the groundbreaking Gross-Zagier formula relating the central derivative $L'(E,1)$ and the heights of Heegner points on $E$ defined over an imaginary quadratic field. The Gross-Zagier formula has the following implication.

Theorem 6 (Gross-Zagier, 1986) If an elliptic curve $E$ over $\mathbb{Q}$ satisfies $\ord_{s=1}L(E,s)=1$, then $E(\mathbb{Q})$ has rank $\ge1$.

Building on the Gross-Zagier formula and his theory of Euler systems, Kolyvagin proved that

Theorem 7 (Kolyvagin, 1989) The BSD conjecture holds for all elliptic curves $E$ over $\mathbb{Q}$ with $\ord_{s=1}L(E,s)\le1$. In particular, $\Sha(E)$ is finite for all such elliptic curves.

There is a weaker version of BSD concerning the parity of ranks.

Conjecture 6 (Parity conjecute) $\ord_{s=1}L(E,s)\equiv \rank(E(\mathbb{Q})\pmod{2}$.

Notice that the parity of the analytic rank is directly related to the root number, the sign of the functional equation of $L(E,s)$. The parity conjecture is more tractable because the root number can be defined independent on any conjectures using local Galois actions on the Tate module.

Theorem 8 (Monsky, 1996) The parity conjecture holds true for an elliptic curve $E$ over $\mathbb{Q}$ if $\Sha(E)$ is finite.

Nekovar (2006) extended the parity conjecture to totally real fields. Dokchitser-Dokchitser (2009) proved the $p$-parity conjecture, which allows us to weaker the condition of finiteness of $\Sha(E)$ to the finiteness of the $p$-primary part $\Sha(E)[p^\infty]$ for a prime $p$.

Remark 2 The BSD conjecture has been verified for any elliptic curve $E$ over $\mathbb{Q}$ with $\ord_{s=1}L(E,s)\le1$ and conductor $\le5000$ by Stein and others. There is no elliptic curve $E$ over $\mathbb{Q}$ with analytic rank $\ge2$ for which $\Sha(E)$ can be shown to be finite, nevertheless, there is numerical evidence (up to a rational factor) for many cases with higher analytic rank.
Remark 3 Computing the rank $r$ (when $r$ is large) is still by far a demanding computational problem: there are useful algorithms in practice, but none of them can always guarantee the correct output. The main difficulty again lies in computing the mysterious group $\Sha(E)$. Assuming the BSD conjecture, we can instead compute the analytic rank. However, because we do not have a Gross-Zagier formula for higher derivatives of $L(E,s)$, the analytic rank is hard to compute provably correctly (is $10^{-50}$ zero or a tiny rational number?). It is still an open problem to find an elliptic curve over $\mathbb{Q}$ of provably correct analytic rank $\ge4$.

References

[1]Benedict Gross, Lectures on the conjecture of Birch and Swinnerton-Dyer, Arithmetic of L-functions, AMS PCMI publications, 2011, 169-210.

[2]Alice Silverberg, Ranks "cheat Sheet", 2011, http://math.uci.edu/~asilverb/connectionstalk.pdf.

[3]Joseph H. Silverman, The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics), Springer, 2010.