Is there is a known relation or expression containing the algebraic rank $r$?

Question

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-function of a holomorphic cusp form for a congruence subgroup and it is entire function and have a holomorphic continuation. The order of vanishing (the analytic rank) of $L(C,s)$ at $s=1$ is denoted by $m$ (the minimal integer $m≥0$ such that $L(C,s)^{(m)}(1)≠0$) and the algebraic rank of $C(ℚ)$ is denoted by $r$.

My question is: Is there is a known relation or expression containing the algebraic rank $r$? I am looking for any kind of relations (equalilities, inequalities, etc...). In particular, I have this one: $$2^{r}=(|Imα||Imα′|)/4$$

for some well defined maps $α$ and $α′$. See: Rational Points on Elliptic Curves by Alexandru Gica (2006).

Answer 1

First, is the reference supposed to be to the book "Rational Points on Elliptic Curves" by John Tate and myself? The formula $$2^r=\frac{\#\alpha(\Gamma)\cdot\#\overline\alpha(\Gamma)}{4}$$ appears at the bottom of page 91 of our book. So maybe Dr. Gica's notes are based on that?

Anyway, this is all a special case of the standard descent on an elliptic curve. For an isogeny $\phi:C\to C'$ there is an injection $$C'(K)/\phi(C(K)) \hookrightarrow H^1(\hbox{Gal}(\overline K/K),\hbox{Ker}(\phi)).$$ If $\hbox{Ker}(\phi)$ is defined over $K$ and is (say) cyclic of order $n$, or more generally if the kernel is isomorphic as a Galois module to $\mathbf{\mu}_n$, then $H^1(\hbox{Gal}(\overline K/K),\hbox{Ker}(\phi))\cong K^*/(K^*)^n$ by Hilbert Theorem 90. So one gets $$\alpha:C'(K)/\phi(C(K)) \hookrightarrow K^*/(K^*)^n.$$ Doing the same thing with the dual isogeny gives $$\alpha':C(K)/\hat\phi(C'(K)) \hookrightarrow K^*/(K^*)^n.$$ Combining these and a little information about rational $n$-torsion, one can get a formula relating $2^{\text{rank} C(K)}$ to $\#\text{Image}(\alpha)\cdot\#\text{Image}(\alpha')$.

Of course, the hard part is computing those images, especially if the Tate-Shafarevich group has non-trivial $n$-torsion.

BTW, this is all explained in Chapter 10 of my book The Arithmetic of Elliptic Curves.