TITLE: An algebraic relation between any two elliptic functions with the same periods
QUESTION [6 upvotes]: I'm sorry if my question is trivial. In his book "Elements of the Theory of Elliptic Functions", page 44, Akhiezer proves, that any two elliptic functions with the same periods are connected by an algebraic relation. For such functions $f$ and $g$ he takes their representation $$f(u)=R_1(\wp)+R_2(\wp)\wp^\prime,\ \ \ \ \ (1)$$
$$g(u)=R_3(\wp)+R_4(\wp)\wp^\prime,\ \ \ \ \ (2)$$ where $R_1,R_2,R_3,R_4$ are rational functions of their argument and the relation $$[\wp^\prime]^2=4\wp^3-g_2\wp-g_3.\ \ \ \ \ \ \ (3)$$ Then he writes that we can eliminate $\wp$, $\wp^\prime$ from (1), (2), (3) and we get a relation $F(f,g)$, where $F$ is a polynomial.
I don't understand, how can we eliminate $\wp$ from the relations above. It's easy to eliminate $\wp^\prime$ from, for example, (1): $$\wp^\prime=\frac{f(u)-R_1(\wp)}{R_2(\wp)}.$$ But in the general case, $\wp$ can't be expressed as a rational function of $f$, $g$ and $\wp^\prime$, so I don't understand, how we get polynomial $F$. What is meant by the word "eliminate"? Thanks.

REPLY [3 votes]: Perhaps this answer is too late to be of help to the original question poser, but I think they were seeking a more concrete answer than the ones provided so far, which give a justification for why the relation $F(f,g)=0$ exists but do not give a practical way to compute $F$, in the spirit of Akhiezer's description.
After clearing denominators, (1),(2) & (3) can be regarded as a polynomial system in $\wp$ and $\wp'$ with $f,g$ and other quantities appearing as coefficients. So indeed one can first eliminate $\wp'$ from (1) & (3) to get a relation between $f$ and $\wp$, (R1) say, and also eliminate $\wp'$ from (2) & (3) to get a relation between $g$ and $\wp$, (R2) say; these can both be written as polynomial relations, and then one can eliminate $\wp$ from these two relations (R1) & (R2) by computing the resultant w.r.t. $\wp$, which will be the desired polynomial relation $F(f,g)=0$.