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).

Perhaps someone can expand on the specific relation to quadratic forms.

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.

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

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

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).

Perhaps someone can expand on the specific relation to quadratic forms.

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).

Perhaps someone can expand on the specific relation to quadratic forms.