Mathematical background: elliptic curves and L-functions

Day 1 of my Google Summer of Code project! I will try post updates at least once a week; I thought a good starting point would be to give an introduction to the mathematical objects that this project revolves around. The next post will then give a more detailed description of the project itself, and the structure of the code that I'm going to adapt, write and include in Sage.

To that end, for this post I will assume some  knowledge of elementary number theory, algebra and  complex analysis, but nothing more complicated than that.

Let $E$ be an elliptic curve over the rational numbers. We can think of $E$ as the set of rational solutions $(x,y)$ to a two-variable cubic equation in the form:
$$ E: y^2 = x^3 + Ax + B $$
for some integers $A$ and $B$, along with an extra "point at infinity". An important criterion is that the $E$ be a smooth curve; this translates to the requirement that the discriminant of the curve, given by $-16(4A^3+27B^2)$, is not zero.

One of the natural questions to ask when considering an elliptic curve is "how many rational solutions are there?" It turns out elliptic curves fall in that sweet spot where the answer could be zero, finitely many or infinitely many - and figuring out which is the case is a deeply non-trivial problem.

The rational solutions form an abelian group with a well-defined group operation that can be easily computed. By a theorem of Mordell, the group of rational points on an elliptic curve $E(\mathbb{Q})$ is finitely generated; we can therefore write
$$ E(\mathbb{Q}) \approx T \times \mathbb{Z}^r,$$
where $T$ is a finite group (called the torsion subgroup of $E$), and $r$ is denoted the algebraic rank of $E$.

The elliptic curve group law explained visually: three points in a straight line add to zero; because the point at infinity is the identity element, this means that the sum R of two points P and Q is the reflection about the real axis of the other point on the curve on the straight line connecting P and Q.

Determining the torsion subgroup of $E$ is a relatively straightforward endeavor. By a theorem of Mazur, rational elliptic curves have torsion subgroups that are (non-canonically) isomorphic to one of precisely fifteen possibilities: $\mathbb{Z}/n\mathbb{Z}$ for $n = 1$ through $10$ or $12$; or $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2n\mathbb{Z}$ for $n = 1$ though $4$. Computing the rank $r$ - the number of independent rational points on $E$ - is the hard part, and it is towards this end that this project hopes to contribute.

Perhaps surprisingly, we can translate the algebraic problem of finding rational solutions to $E$ to an analytic one - at least conjecturally. To understand this we'll need to know what an elliptic curve $L$-function is. These are holomorphic functions defined on the whole complex plane that somehow encode a great deal of information about the elliptic curve they're attached to.

The definition goes as follows: for each prime $p$, count the number of solutions to the elliptic curve equation modulo $p$; we'll call this number $N_p(E)$. Then define the number $a_p(E)$ by
$$ a_p(E) = p+1 - N_p(E). $$
Hasse's Theorem states that $a_p(E)$ is always less that $2\sqrt{p}$ in magnitude for any $p$, and the Sato-Tate conjecure (recently proven by Taylor et al) states that for a fixed elliptic curve, the $a_p$ (suitably transformed) are asymptotically distributed in a semi-circular distribution about zero.

Next, for a given $p$ define the local factor $L_p(s)$ to be the function of the complex variable $s$ as follows:
$$ L_p(s) = \left(1-a_p(E)p^{-s} + \epsilon(p)p^{-2s}\right)^{-1}, $$
where $\epsilon(p)$ is 0 if $p$ divides the conductor of $E$, and 1 otherwise.

The conductor of $E$ is a positive integer that encodes the bad reduction of $E$: for a finite list of primes $p$, when we reduce the curve modulo $p$, we don't get a smooth curve but rather a singular one instead. The conductor is just the product of these primes to the first or second power (or in the cases $p=2$ or $3$, up to the eighth and fifth powers respectively). If you're unfamiliar with elliptic curves, the thing to note is that the conductor always divides the discriminant of $E$, namely the quantity $-16(4A^3+27B^2)$ mentioned previously.

Finally we can define the $L$-function attached to $E$:
$$ L_E(s) = \prod_p L_p(s) = \prod_p \left(1-a_p(E)p^{-s} + \epsilon(p)p^{-2s}\right)^{-1}. $$
The above representation of $L_E(s)$ is called the Euler product form of the $L$-function. If we multiply out the terms and use power series inversion we can also write $L_E(s)$ as a Dirichlet series:
$$ L_E(s) = \sum_{n=1}^{\infty} a_n(E) n^{-s}, $$
where for non-prime $n$ the coefficients $a_n$ are defined to be exactly the integers you get when you multiply out the Euler expansion.

If you do some analysis, using Hasse's bound on the size of the $a_p(E)$ and their distribution according to Sato-Tate, one can show that the above to representations only converge absolutely when the real part of $s$ is greater than $\frac{3}{2}$, and conditionally for $\Re(s)>\frac{1}{2}$. However, the modularity theorem states that these elliptic curve $L$-functions can actually be analytically continued to the entire complex plane. That is, for every elliptic curve $L$-function $L_E(s)$ as defined above, there is an entire function on $\mathbb{C}$ which agrees with the Euler product/Dirichlet series definition for $\Re(s)>1$, but is also defined - and readily computable - for all other complex values of $s$. This entire function is what we actually call the $L$-function attached to $E$.

The way we analytically continue $L_E(s)$ yields that the function is highly symmetric about the line $\Re(s)=1$; moreover, because the function is defined by real coefficients $L_E(s)$ also obeys a reflection symmetry along the real axis. The point $s=1$ is in a very real sense therefore the central value for the $L$-function. It thus makes sense to investigate the behaviour of the function around this point.

Because $L_E(s)$ is entire, it has a Taylor expansion at any given point. We can ask what the Taylor expansion of $L_E(s)$ about the point $s=1$ is, for example. One of the central unproven conjectures in modern-day number theory is the Birch and Swinnerton-Dyer Conjecture: that the order of vanishing of the Taylor expansions of $L_E(s)$ about the point $s=1$ is precisely $r$, the algebraic rank of the elliptic curve $E$. That is, if we let $z=s-1$ so that
$$ L_E(s) = c_0 + c_1 z + c_2 z^2 + \ldots $$
is the expansion of $L_E(s)$ about the central point, the BSD conjecture holds that $c_0$ through $c_{r-1}$ are all zero, and $c_r$ is not zero.

The values of three elliptic curve L-functions along the critical line $1+it$ for $-10<=1<=10$. Blue corresponds to the curve $y^2 = x^3 - 13392x - 1080432$, a rank 0 curve, red is that of $y^2 = x^3 - 16x + 16$, a rank 1 curve, and green is $y^2 = x^3 - 3024x + 46224$, a rank 2 curve. Note that close to the origin the graphs look like non-zero constant function, a straight line through the origin and a parabola respectively.

Thus if we can compute the order of vanishing of the elliptic curve $L$-function at the central point, we can at least conjecturally compute the rank of the curve. This converts an algebraic problem into a numerical one, which is perhaps more tractible.

The techniques we'll use to attempt to do this will be the subject of the next blog post. Unfortunately there are still plenty of challenges to this approach - the least of which boils down to: how do you numerically determine if the value of a function is zero, or just really, really close to zero? The short answer is that without theorems to back you up, you can't -- but we can still make considerable progress toward the problem of computing an elliptic curve's rank.