For an elliptic curve, $E$, defined over $\mathbb{Q}$ and with the Mordell-Weil group, $E(\mathbb{Q})$, having rank, $r$, the Birch Swinnerton-Dyer (BSD) conjecture states that $$ c_{E} = \lim_{s \rightarrow 1} \frac{L(E,s)}{(s-1)^{r}} = \frac{\Omega(E) \mathrm{Reg}(E) \left( \prod_{p} c_{p} \right) |\mathrm{Sha}(E/\mathbb{Q})|}{| E(\mathbb{Q})_{tors}|^{2}}. $$

For $r=0,1$, it is known from Kolyvagin's work that the analytic rank of $E$ equals the rank of the Mordell-Weil group and that the Tate-Shafarevich group is finite.

My question concerns what is currently proven about the conjectured value for $c_{E}$ above when $r=0,1$.

For brevity, put $$ c_{1}(E) = \frac{\Omega(E) \mathrm{Reg}(E) \left( \prod_{p} c_{p} \right)}{| E(\mathbb{Q})_{tors}|^{2}}. $$

BSD states that $$ c(E)=c_{1}(E) \cdot |\mathrm{Sha}(E/\mathbb{Q})|. $$

(1) What is currently proven about $c(E)/c_{1}(E)$ when $r=0,1$?

From BSD, it should be the square of a non-zero rational integer.

(2) This is still unproven for $r=0,1$, isn't it?

(2-a) If still unproven, is it proven for any families etc of elliptic curves with $r=0,1$? If so, references, please?

(3) Do we have a proof that $c(E)/c_{1}(E)$ is always an integer when $r=0,1$?

(3-a) If so, could someone provide a reference?

(3-b) If not, then similar to question (2-a) above, is it proven for any families, infinite sets,... of such elliptic curves?

(4) What do we know $p$-adically about $c(E)/c_{1}(E)$?

From Theorem 3.3(3), and the paragraph after that theorem, on page 193 of Gross' Lecture 3 in [1] below, we know that $c(E)/c_{1}(E)$ (note that his $R(E/\mathbb{Q}) \cdot P(E/\mathbb{Q})$ is the same as my $c_{1}(E)$) is a non-zero rational number and that except for a specified finite set of primes, $p$, that depends on $E$, we know that the $p$-part of $|\mathrm{Sha}(E/\mathbb{Q})|$ equals the $p$-part of $c(E)/c_{1}(E)$.

(4-a) is there a good reference and explanation for how to determine this finite set of primes, $p$, for a given curve, $E$?

[1] ``Arithmetic of $L$-functions'' (ed. Popescu, Rubin, Silverberg), AMS (2011).

For an elliptic curve, $E$, defined over $\mathbb{Q}$ and with the Mordell-Weil group, $E(\mathbb{Q})$, having rank, $r$, the Birch Swinnerton-Dyer (BSD) conjecture states that $$ c_{E} = \lim_{s \rightarrow 1} \frac{L(E,s)}{(s-1)^{r}} = \frac{\Omega(E) \mathrm{Reg}(E) \left( \prod_{p} c_{p} \right) |\mathrm{Sha}(E/\mathbb{Q})|}{| E(\mathbb{Q})_{tors}|^{2}}. $$

For $r=0,1$, it is known from Kolyvagin's work that the analytic rank of $E$ equals the rank of the Mordell-Weil group and that the Tate-Shafarevich group is finite.

My question concerns what is currently proven about the conjectured value for $c_{E}$ above when $r=0,1$.

For brevity, put $$ c_{1}(E) = \frac{\Omega(E) \mathrm{Reg}(E) \left( \prod_{p} c_{p} \right)}{| E(\mathbb{Q})_{tors}|^{2}}. $$

BSD states that $$ c(E)=c_{1}(E) \cdot |\mathrm{Sha}(E/\mathbb{Q})|. $$

(1) What is currently proven about $c(E)/c_{1}(E)$ when $r=0,1$?

From BSD and the Cassels-Tate pairing, it should be the square of a non-zero rational integer.

(2) This is still unproven for $r=0,1$, isn't it?

(2-a) If still unproven, is it proven for any families etc of elliptic curves with $r=0,1$? If so, references, please?

(3) Do we have a proof that $c(E)/c_{1}(E)$ is always an integer when $r=0,1$?

(3-a) If so, could someone provide a reference?

(3-b) If not, then similar to question (2-a) above, is it proven for any families, infinite sets,... of such elliptic curves?

(4) What do we know $p$-adically about $c(E)/c_{1}(E)$?

From Theorem 3.3(3), and the paragraph after that theorem, on page 193 of Gross' Lecture 3 in [1] below, we know that $c(E)/c_{1}(E)$ (note that his $R(E/\mathbb{Q}) \cdot P(E/\mathbb{Q})$ is the same as my $c_{1}(E)$) is a non-zero rational number and that except for a specified finite set of primes, $p$, that depends on $E$, we know that the $p$-part of $|\mathrm{Sha}(E/\mathbb{Q})|$ equals the $p$-part of $c(E)/c_{1}(E)$.

(4-a) is there a good reference and explanation for how to determine this finite set of primes, $p$, for a given curve, $E$?

[1] ``Arithmetic of $L$-functions'' (ed. Popescu, Rubin, Silverberg), AMS (2011).