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

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as

J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analogue, Sem.
Bonrbaki 18 (1966)

(please don’t ask me to mention the reference of the above article, I have missed the link, I have only soft-copy with me)
after a deep internet search, but the article was very technical and very sophisticated, rather after going through it dozens of times, I understood that Prof. J. Tate was trying to convey that one must relate the $L$-function to some Galois-groups, that was the rough picture in my mind! .

But later luckily to my surprise it was a coincidence that in an work of John Coates(John Coates, The Arithmetic of Elliptic Curves with Complex
Multiplication, Proceedings of the International Congress of Mathematicians
Helsinki, 1978
), I encountered the same words where Coates describe that

“…..They also gave a conjectural formulathe coefficient of $\large(s-1)^{g_{Q}}$($g_{Q}$ is rank) in the expansion of $L(E,s)$ about $s=1$ but we shall not discuss this here.

Now Tate’s work on the geometric analogue suggests that, in order to attack this conjecture, one must relate $L(E,s)$ to the
characteristic polynomial of some canonical element in a
representation of a certain Galois group.

Now I request anyone who is currently working in that area/ know that area, answer me what does the above statement mean.

i.e. what was the intuition behind linking the $L(E,s)$ to the characteristic polynomial of some canonical element in a representation of a certain Galois group ? .

And why does one need to look at characteristic polynomial of some element in Galois group in order to proceed with the Birch and Swinnerton Dyer conjecture ? .

Can anyone give a detailed summary of what was the Tate’s idea and in which sense Coates refer to that sentence ? .

Please frame your answer not in a high-technical manner, but in the way a beginner can understand, but please answer me in a detailed manner.

I hope this question doesn’t go unanswered, and I get the answer in a detailed form(describing to the maximum extent).
Please help me.

Thanking you all.

Solutions Collecting From Web of "Reading the mind of Prof. John Coates (motive behind his statement)"

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.