Clarification on the weak BSD conjecture

Question

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function $$ f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p} $$ as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $E$ has good reduction at $p$. Namely, this function should grow at the order of $$ \log(x)^r $$ when $x$ tends to $+\infty$, where $r$ is the (algebraic) rank of $E$.

Question 1. Why is it natural to look at these kind of products?

Nowadays, people usually state the BSD conjecture as the equality $$ r = \text{ord}_{s=1}L(E,s)\text{.} $$

Question 2. Are these two statements equivalent?

Answer 1

In regards to question 2, in 1982 Goldfeld proved that if $f_{E}(x) \sim C (\log x)^{r}$, then (i) $L(E,s)$ has no zeroes with ${\rm Re}(s) > 1$, and (ii) the order of vanishing at $L(E,s)$ is equal to $r$. I do not know if the converse is true (even assuming GRH for $L(E,s)$), as I don't have a copy of Goldfeld's paper.

Strangely, in the case that $r = 0$, Goldfeld shows that $C = \sqrt{2}/L(E,1)$.

Answer 2

I believe that the original experimental observation was that the product seems to converge to $\infty$ if $E(\mathbb Q)$ is infinite, and to a finite value if $E(\mathbb Q)$ is finite. Also, I may be mis-remembering, but I thought they looked at $$\prod_p \frac{p}{\#E(\mathbb F_p)},$$ with the limit conjecturally being $0$ if and only if $\#E(\mathbb Q)=\infty$.

Here's an intuitive reason to study this quantity. We know from Hasse that $\#E(\mathbb F_p)=p+1-a_p$ with $|a_p|\le2\sqrt p$. So $$\log \left(\prod_p \frac{\#E(\mathbb F_p)}{p} \right)\approx -\sum_p \frac{a_p}{p}.$$ (This is just a heuristic, of course.) Anyway, this quantity diverges to $+\infty$ if the $a_p$ tend to be negative more often than they are positive. So one expects the product to equal $\infty$ if and only if $\#E(\mathbb F_p)$ is biased towards being larger than its "expected" value of $p+1$. On the other hand, if $E(\mathbb Q)=\infty$, then one might hope (guess?) that the image of $E(\mathbb Q)$ in $E(\mathbb F_p)$ tends to make $E(\mathbb F_p)$ somewhat larger than would be expected at random. Ergo, one expects that the product is large if and only if $E(\mathbb Q)$ is large. Then one does experiments to see whether "large" means "infinite".

Answer 3

I recommend you to look at Birch and Swinnerton-Dyer's second paper (Notes on elliptic curves II). They explain beautifully the background for their conjecture.

The basic idea is that if you can prove that $L_E$ has an analytic continuation beyond $\Re(s)=3/2$ (Hasse-Weil conjecture), then

$$L_E(1)=\prod_p\bigg(\frac{|E(\mathbb{F}_p)|}{p}\bigg)^{-1}$$

In their paper they also mention:

After the work of Siegel [19] on quadratic forms, it is natural to look at the product $\prod N_p/p$ where $N_p$ is the number of rational points on the curve over the finite field of $p$ elements.

[19] C. L. Siegel, Über die analytische Theorie der quadratischen Formen. I, II, III. Ann. of Math. 36 (1935), 37 (1936), 38 (1937).

This is explained in more detail at the beginning of their first paper.

Siegel has shown that the density of rational points on a quadric surface can be expressed in terms of the densities of $p$-adic points; which for almost all primes $p$ depends directly on the number of solutions of the corresponding equation in the finite field with $p$ elements.

It is natural to hope that something similar will happen for the elliptic curve $$\Gamma:y^2=x^3-Ax-B$$where $A,B$ are rational numbers.

Answer 4

Question 1: The $L$-functon $L(E/\mathbf{Q},s)$ is a product over $L_p(E/\mathbf{Q},s) := 1/(1-a_pp^{-s}+p^{1-2s})$. Now plug in $s=1$ and use $p+1 - |E(\mathbf{F}_p)| = a_p$. (This is only a heuristic.)

Answer 5

If $E$ is an elliptic curve over $\mathbf{Q}$, define $L_p(E,s) = \frac{1}{1-ap^{-s} + p^{1-2s}}$, where $a$ is $p-(|E_p(\mathbf{F}_p)| - 1)$. 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”): $$ L(E,s) = \prod_{p} L_p(E,s)$$ Since we wish to look at $L(E,1)$, we notice that $\displaystyle L_p(E,1) = \frac{p}{p+1-a}$. From the definition of $a$, we see that $p+1-a=|E_p(\mathbf{F}_p)|$, and so $\displaystyle L_p(E,1) = \frac{p}{|E_p(\mathbf{F}_p)|}$. Taking the product over all good primes $p$ shows that $$L(E,1) = \prod_{p}\frac{p}{|E_p(\mathbf{F}_p)|}.$$