Reading the mind of Prof. John Coates (motive behind his statement)

Solution 1:

In the case of an elliptic curve over a global field of positive characteristic (i.e. over the function field of a curve over a finite field), Tate reinterpreted the BSD conjecture in a more geometric way.

Namely, an elliptic curve over the function field of a curve $C$ over a finite field $\mathbb F$ can be "spread out" to form an elliptic surface $S$ over $C$ (i.e. a surface mapping to $C$ whose generic fibre is an elliptic curve).

Giving rational points on the original elliptic curve corresponds to giving sections of the projection $S \to C$. To determine such sections essentially amounts to determining all the curves lying on $S$ that can be defined over $\mathbb F$.

Now to determine these curves, one can look at the cycle class map which takes any curve to its class in the second etale cohomology group of $S$ over $\overline{\mathbb F}$. Since one is considering curves defined over $\mathbb F$, the image of this map lies in the Frobenius invariants of the etale $H^2$, and Tate showed that (the rank part of) BSD is equivalent to the statement that every Frobenius invariant element actually arises from a curve defined over $\mathbb F$. (He was then led to make his general conjecture, known as the Tate conjecture, which I have discussed here.)

There is a general philosophy, known as Iwasawa theory, which tries to take intuition from the Weil conjectures and the Tate conjecture (which are about varieties over finite fields) to formulate analogous statements for varieties over number fields.

The idea is that passage from $\mathbb F$ to $\overline{\mathbb F}$ should be replaced by passage from $\mathbb Q$ to $\mathbb Q(\zeta_{p^{\infty}})$ (i.e. adjoin all the $p$-power roots of $1$, for some prime $p$). At least if $p$ is odd, the Galois group of $\mathbb Q(\zeta_{p^{\infty}})$ over $\mathbb Q$ is pro-cyclic, just as $Gal(\overline{\mathbb F}/\mathbb F)$ is. Unlike in the finite field context (where one has the Frobenius element), it does not admit a canonical generator, but we can just choose a generator; traditionally it is labelled $\gamma$.

Now if $E$ is an elliptic curve over $\mathbb Q$, one can construct a certain Galois cohomology group attached to $E$ over $\mathbb Q(\zeta_{p^{\infty}})$, which will have an action of $\gamma$ on it, which is analogous to the second etale cohomology group of the elliptic surface in the function field case.

It is the action of $\gamma$ on this Galois cohomlogy group that Coates is referring to.

In fact, the Galois cohomology group in question is the Selmer group of $E$ over $\mathbb Q(\zeta_{p^{\infty}})$, and the main conjecture of Iwasawa theory for $E$ over $\mathbb Q$ relates the characteristic polynomial (actually, characteristic power series, but let me not get into that detail here) of $\gamma$ on this Selmer group to the $p$-adic $L$-function of $E$.

There are various caveats (e.g. as I'm describing it here, the conjecture only makes sense if $E$ is ordinary at $p$), but let me say that in broad terms, the two (characteristic power series and $p$-adic $L$-function) are supposed to be equal up to multipication by a unit in the ring of power series; that one divisibility was proved by Kato; and that more recently the other divisibility (and hence the main conjecture itself) was proved by Skinner and Urban.

Knowing the main conjecture does not actually imply BSD, since it relates the Selmer group to the $p$-adic $L$-function (rather than the usual $L$-function), and since it doesn't deal with the problem of proving that Sha is finite. But it is natural to make a $p$-adic BSD conjecture, and the main conjecture is closely related to this. Unfortunately, it doesn't actually imply $p$-adic BSD either (even if one grants the finiteness of Sha), because of possible non-semisimplicity of the action of $\gamma$ on the Selmer group. (This echoes the problem, in etale cohomology, of proving that Frobenius acts semisimply --- an important problem that is open in most situations.) Thus $p$-adic BSD is also currently open (as far as I know).

Finally, although the main conjecture is weaker than $p$-adic BSD, which is in turn different to the usual BSD, there are relations between all three, and in particular, with the main conjecture proved, the only obstruction to the following statement:

• the $L$-function of $E$ vanishes at $s = 1$ iff the Mordell--Weil group of $E$ is infinite

is the finiteness of Sha. (I.e., if we could prove that Sha is finite, we would get the preceding statement.)

For a (much more technical) discussion of the main conjecture and $p$-adic BSD, one could look at the text of Colmez's Bourbaki seminar.