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).



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


