Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at most $1$

Question

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).

Answer 1

I think that the answer to your questions depends in subtle ways on whether $r=0$ or $r=1$.

In full generality, I believe you are right that none of the properties you state are known for all elliptic curves. For instance, I don't think it is currently known how to prove them for $p$ a prime of very bad reduction of $E$ if in addition $E[p]$ is a reducible Galois representations and I am almost sure it is not known how to prove them at $p$ a prime if the conductor of $E$ is $N=p^2$ (this can happen). You can find infinite families of elliptic curves for which all the properties you state are known for a given very small prime $p$ but the exercise is not very illuminating.

On the other hand, if you pick a specific elliptic curve with $r=0$ and smallish conductor $N$ (say $N<5000$) it is nowadays typically possible to check that all properties you state are indeed verified (though that may require among the most recent results on the Iwasawa theory of elliptic curves to treat some of the bad primes, which are the primes where $E[p]$ is reducible and the primes where $E/\mathbb{Q}_p$ has very bad reduction).

A good starting points for references are to look at recent publications of C. Skinner, F. Castella and X. Wan.