Background: The Birch and Swinnerton-Dyer conjecture states that the Taylor expansion around the point $s=1$ of the L-function of an elliptic curve $E$ has the form $$c(s-1)^r+\text{higher order terms},$$ where $c$ is a constant not equal to zero, and $r$ is the rank of the curve. This implies that the L-function $$L(E,1)=\prod_p F_p(1)^{-1} \sim (k(\log X)^r )^{-1},$$ where $F_p(1)=N_p/p$, $k$ is a constant (without any non-zero restrictions), $N_p$ is the number of solutions $E (mod p)$, and $p$ are the prime numbers. The conjecture states that $L(E,1)=0$ if and only if the number of points on $E$ is infinite (see Birch and Swinnerton-Dyer’s paper “Notes on Elliptic Curves. II”).
The implication is clear that when the number of points is infinite, then it could increase faster than the infinite primes, as if $N_p$ in
$$F_p(1)^{-1}=p/N_p$$
continues to grow faster than $p$, as $p$ approaches infinity, then $L(E,1)=0$. But that’s only half of the conjecture. The reverse implication (“and only if”) is not as clear to me; that is, if $N_p$ is finite, then once passed the last $N_p$ in the list of
$$L(E,1)=\prod_p p/N_p$$
the primes $p$ will still continue to grow to infinity such that $L(E,1)$ too will grow to infinity (an infinite product over a finite product equals an infinite product). This seems contradictory to the conjecture based on the Modularity Theorem that would imply that $L(E,1)$ is finite for all elliptic curves.
Question: If one could prove that $$T_{L(E,1)}=c(s-1)^r+\text{higher order terms},$$ where $T_{L(E,1)}$ is the Taylor expansion of $L(E,1)$, and $c\neq 0$, would it actually prove the Birch and Swinnerton-Dyer conjecture, or would it disprove it?
Discussion: If the answer to the above is “it proves”, then one could prove that the conjecture is false in just one sentence. I will show.
In reading Gross and Zagier’s 1983 paper “Heegner Points and Derivatives of L-Series”, they prove that there are an infinite number of points on rank one modular curves when $L(E,1)=0$ but $L'(E,1)\neq 0$. But I do not see that they proved $L(E,1)=0$ is possible for rank 1 modular curves, only that if $L(E,1)=0$ is possible but $L'(E,1)\neq 0$, then there will exist an infinite number of points on the curve. This is a subtle detail but important. Much (if not all) interpretation on this that I can find suggests exactly the opposite of what I am raising. But similarly, in reading Kolyvagin’s 1989/90 paper “Finiteness of E(Q) and III(E,Q) for a Subclass of Weil Curves,” which is typically thought to prove the reverse implication, Kolyvagin proves for modular curves, when rank equals zero, $L(E,1)\neq 0$, and when rank equals one, $L(E,1)=0$ and $L'(E,1)\neq 0$. This does imply the reverse implication of $L(E,1)=0$ to rank 1, but it does not imply $c\neq 0$. It implies the contrary, so it seems to me.
The leading term of the Taylor expansion of a function is always the function itself at the point expanded around. If the leading term is zero, then all the remaining terms add up to zero. The conjecture predicts that the leading term of the Taylor expansion of the L-function of $E$ is $c(s-1)^0$, where at $s=1$, $(s-1)^0=1$, as any number raised to zero is one (by the order of arithmetic operations, the power is determined before even looking at what’s within the parentheses). Thus, if the first term is zero, then $c(s-1)^0=0$ implies $c=0$ (again because $(s-1)^0=1$), but this result says nothing about the BS-D conjecture itself, as by definition of the conjecture, $c$ cannot be zero.
However, following the Modularity Theorem from the mid-90s, all elliptic curves are now proven to be modular, and therefore all $L(E,s)$ are finite. This allows:
Theorem: If $$T_{L(E,1)}=c(s-1)^r+\text{higher order terms},$$ where $T_{L(E,1)}$ is the Taylor expansion of $L(E,1)$, then $L(E,s)=0$ implies $c=0$, as the leading term of the series is always equal to the function itself at the point expanded around, and $c(s-1)^0=0$ implies $c=0$.
In other words, a contradiction would exist in the conjecture and therefore it is incorrect. That is, unless 1) I am mistaken (please point out where) or 2) the conjecture is incorrect if it can be proven that $$T_{L(E,1)}=c(s-1)^r+\text{higher order terms}$$ (proof by contradiction). But I am only asking if this would only disprove the reverse implication of the conjecture: the idea that no finite curves exist of greater than rank zero. The implication of infinite curves existing for rank greater than zero would still hold.
Lastly, please understand that I am 99% sure I am wrong about all of this above, but I cannot find where my error arises.