Just to be specific, I deal only with the elliptic curves $E$ over $\mathbb{Q}$, and most of the explaination here are obtained from the paper: Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory.
I understand that Manin gave an algorithm to compute rank $r$ if we assume the truth of BSD. Finding non-torsion points gives a lower bound $r_l\rightarrow r$ and finding $L^{(k)}(E,1)$ to a high enough precision gives an upper bound $k\rightarrow r_{an}$. BSD gives $r=r_{an}$ and so we know when to stop.
From my understanding of $p$-adic BSD, for simplicity, I restrict to dealing with good and ordinary primes here.
There is the work of Kato, Schneider, Perrin-Riou in Iwasawa Theory for Elliptic Curves and the corresponding Main Conjecture. I understand that there is a very similar algorithm to the above to compute the rank $r$ and using $\mathcal{L}_p(E,T)$ in place of $L^{(k)}(E,1)$ in finding this upper bound.
It seems that by the method in the $p$-adic BSD case, we get an unconditional upper bound as opposed to the standard BSD case. My question is regarding this.
What is the upper bound of the standard BSD case conditional upon that the $p$-adic BSD is not? It seems to me that both are dependent on the precision of their corresponding $L$-functions.
