# Birch-Swinnerton-Dyer over function fields

Last weekend, I talked to Noam Elkies about elliptic curves and a problem that we were working on. We reached the topic of the Birch-Swinnerton-Dyer conjecture (which is a Millenium problem). Just for posterity, let me recall what the conjecture is about. Let $E$ be an elliptic curve over a number field $K$, and let $E(K)$ denote the group of $K$-rational points of $E$.

Theorem (Mordell-Weil): The group $E(K)$ is finitely generated. In other words, $E(K)\cong \mathbf{Z}^r\oplus E(K)_\mathrm{tor}$. The integer $r$ is called the rank of $E$.

For example, a number $n$ is congruent if and only if the rank of the elliptic curve $E_n:y^2 = x^3 - n^2x$ is positive. For simplicity, we will now work over $\mathbf{Q}$. There is a remarkable result by Mazur:

Theorem (Mazur): If $E$ is an elliptic curve over $\mathbf{Q}$, the torsion group $E(\mathbf{Q})_\mathrm{tor}$ is either $\mathbf{Z}/n\mathbf{Z}$ for some $n\in \{1,\cdots,10,12\}$ or $\mathbf{Z}/2\mathbf{Z}\times\mathbf{Z}/2n\mathbf{Z}$ for $n\in\{1,...,4\}$.

(Over a general number field $K$ of dimension $n$ over $\mathbf{Q}$, Merel proved that the size of the torsion group is bounded by some constant $B(n)$.) This is an incredible theorem, and tells us about the structure of $E(\mathbf{Q})$ for an elliptic curve $E$. But what about the rank? Are there any special properties of the rank? There’s a conjecture known as the rank conjecture:

The rank conjecture: The rank of $E$ can be arbitrarily large.

That doesn’t really help us much, though, so we ask: is there a way of determining the rank of a general curve? This is the content of the Birch-Swinnerton-Dyer (BSD) conjecture (still over number fields!).

Recall the zeta function of an elliptic curve $E$ over $\mathbf{Q}$, where $E_p$ denotes the reduction of $E$ mod a prime $p$ (ignoring what happens when $p|2\Delta$ where $\Delta$ is the discriminant of $E$), defined as $\displaystyle\zeta(E_p,T) = \exp\left(\sum^\infty_{k=1}|E_p(\mathbf{F}_{p^k})|\frac{T^k}{k}\right)$. By the Weil conjecture, this is a rational function in $T$, and it turns out that the exact form of $\zeta(E_p,T)$ as a rational function in $T$ is as $\displaystyle \frac{1-aT+pT^2}{(1-pT)(1-T)}$, where $a$ is $p-(|E_p(\mathbf{F}_p)| - 1)$. Now define $\displaystyle L_p(E,s) = \frac{1}{1-ap^{-s} + p^{1-2s}}$. The $L$-function of $E$ is defined as the product over the $L_p(E,s)$ over all primes not dividing $2\Delta$ (called “good primes”); more precisely, $\displaystyle L(E,s) = \prod_{p\not|2\Delta} L_p(E,s)$.

This is a function in $s$, and converges for $\mathrm{Re}(s)>3/2$. Hasse conjectured that this could be analytically continued to the whole complex plane, and this is proved. Let’s look at $L_p(E,s)$ when $s=1$. This gives us $\displaystyle L_p(E,1) = \frac{p}{p+1-a}$. Wait! Recall that $a=p+1-|E_p(\mathbf{F}_p)|$. Thus, $p+1-a$ is $|E_p(\mathbf{F}_p)|$, and $\displaystyle L_p(E,1) = \frac{p}{|E_p(\mathbf{F}_p)|}$. Now, the $L$-function of $E$ is a product over all good primes, so you’d expect $L(E,1)$ to tell us something about the rank of $E(\mathbf{Q})$. Indeed, that’s what the Birch-Swinnerton-Dyer conjecture states:

Conjecture (BSD): The order of vanishing of $L(E,s)$ when $s=1$ is the rank of $E$. In other words, the Taylor expansion of $L(E,s)$ must be $C(s-1)^{\mathrm{rank}(E)} + \text{higher terms}$.

(There’s an explicit formula for $c$ when you factor in the “bad primes” as well.) As mentioned above, all of this makes sense for number fields as well (we just chose $K=\mathbf{Q}$ because in that case one can state the exact classification of the torsion subgroups of $E(\mathbf{Q})$). When you’re asking questions over number fields, a very natural question to ask is what happens to the same question over function fields (which, recall, are finite-dimensional field extensions of $\mathbf{F}(x)$).

Let $C$ be a smooth connected projective curve over $\mathbf{F}_q$, and let $E$ be an elliptic curve over $\mathbf{F}_q(C)$. The conductor $\mathfrak{n}$ of $E$ is the product over all places $v\not|\infty$ of the ideals $(\mathfrak{p}_v)^{f(v)}$ where $f(v)$ is $0$ if $E$ has good reduction at $v$ (so the conductor measures bad reduction), $1$ if $E$ has multiplicative reduction at $v$, and $2+\text{some factor}$ if $E$ has additive reduction at $v$. It turns out that the Mordell-Weil conjecture and rank conjecture hold true in this (i.e., the function field) context! This is really cool, so we can ask: does the analogue of the BSD conjecture hold in the function field case? But what is the $L$-function? The $L$-function in this case is defined as follows, where $q_v$ is the cardinality of the residue field $\mathbf{F}_v$, and $a_v$ is, like before, $q_v + 1 - |E_v(\mathbf{F}_v)|$:

$\displaystyle L(E,s)=\prod_{v\not|\mathfrak{n}} \frac{1}{1-a_v(q_v)^{-s}(q_v)^{1-2s}}\times\prod_{v|\mathfrak{n}}\begin{cases}1 & E\text{ has additive reduction at }v \\ \frac{1}{1 + q_v^{-s}} & E\text{ has nonsplit multiplicative reduction at }v \\ \frac{1}{1-q_v^{-s}} & E\text{ has split multiplicative reduction at }v\end{cases}$

This function converges absolutely for $\mathrm{Re}(s)>3/2$, and can be meromorphically continued onto the complex plane. What we’re going to do next by relating it to the BSD conjecture seems quite obvious; indeed, the BSD over function fields says that the rank of $E(\mathbf{F}_q(C))$ is the order of vanishing of $L(E,s)$ at $s=1$. There’s a remarkable result by Artin and Tate; in order to state this, we need to introduce one more definition.

Recall the Hasse principle basically says that certain equations have solutions in the rationals if they have solutions in all completions of $\mathbf{Q}$ (where, recall, there are only two norms on $\mathbf{Q}$. But this does not work in all cases, and so we need something to measure its failure – this, in the case of elliptic curves (and abelian varieties in general, but we won’t go there), is called the Tate-Shafarevich group. Precisely, let $k$ be a perfect field, and define $H^1(k,E)$ to be the cohomology group $H^1(\mathrm{Gal}(\overline{k}/k),E(\overline{k}))$. Define the Tate-Shafarevich group (which refuses to compile in WordPress LaTeX, so I use the older notation; the more modern notation is “Ш(E/K)”) $\mathrm{TS}(k,E)$ as the kernel of the map $\displaystyle H^1(k,A)\to\prod_{v}H^1(k_v,E)$. We may now state Artin-Tate’s result:

Theorem (Artin-Tate): The BSD conjecture holds for $E$ over $\mathbf{F}_q(C)$ if and only if $\mathrm{TS}(\mathbf{F}_q(C),E)$ is finite.

This is an incredibly awesome result, which hints at us that BSD might be true in the number field case! The way this is proved is very cool; you consider the elliptic surface over $C$ attached to $E$, which is the smooth proper surface with a flat morphism to $C$ with fiber $E$, and then use work of Grothendieck on $L$-functions to complete the proof. In the notes I have of my conversation with Elkies, I have Elkies’ drawing of such an elliptic surface. Now, the Neron-Severi group is isomorphic to the direct sum of $E(\mathbf{F}_q(C))$ and some other groups; based on this, Artin and Tate formulated a more general conjecture relating the rank of the Neron-Severi group of the elliptic surface used in the proof and some other groups telling us algebraic information about the elliptic surface and its zeta function. (I refer you to this link for more information on the Artin-Tate conjecture.)

SKD