An algebraic relation between any two elliptic functions with the same periods 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.
 A: There are three fundamental facts in play here. Probably one could also use some kind of algorithm from commutative algebra to constructively obtain an explicit algebraic relation between $f$ and $g$ given $R_1,R_2,R_3,R_4$ but we don't need to actually compute the relationship, so I'll ignore that.
Lemma I. The differential equation $\dot{\wp}^2=4\wp^3-g_2\wp-g_3$ is satisfied.
Since $\wp$ has a pole of order $2$ at zero, $\dot\wp$ has a pole of order $3$, and so we should be able to say $\dot\wp^2\sim C\wp^3$ for some constant $C$. We may calculate the first few terms of $\wp$'s Laurent expansion (and hence that of $\dot\wp$'s as well) by expanding the individual terms in $\wp$'s defining sum via geometric series formula, then interchanging order of summation. With the Laurent expansion we can determine that $C=4$, and then look at the expansion of $\dot\wp^2-4\wp^3$: we see this difference is equal to $A\wp +B+O(z)$ around $z=0$ for an appropriate choice of constants $A=-60G_4$ and $B=-140G_6$. Thus the difference between the two sides of the differential equation is a bounded entire function, which must be constant, which must be $0$ since near $z=0$ it is $O(z)$.
This is repeated in Theorem 2 here.
Lemma II. The function field of doubly periodic functions is $\Bbb C(\wp,\dot\wp)$.
Every function $f$ can be split as $f=f_{\rm even}+f_{\rm odd}$ into even and odd parts. If $f$ is an odd elliptic function with given period lattice, then $f(z)/\dot\wp(z)$ is an even elliptic function with the same period lattice since we know $\dot\wp$ is odd. Thus $f(z)=A(z)+B(z)\dot\wp$ for even elliptic functions $A$ and $B$. We want to see that even elliptic functions are rational functions in $\wp$, which means ratios of linear factors of the form $\wp(z)-w$. As $\wp$ is surjective, we can think of these factors as $\wp(z)-\wp(u)$ for poles or roots $u$ of the function $f$. This is the subject of this MSE question.
Lemma III. Transcendence degrees of field extensions are well-defined.
Let $L/K$ be any extension of fields. By Zorn's lemma there exists a subset $S\subset L$ which is algebraically independent over $K$ and for which $L/K(S)$ is algebraic, and any algebraically independent subset is contained in a maximal such one $S$. The cardinality $|S|$ is an invariant of the extension $L/K$; it does not depend on which $S$ was chosen. It is known as the transcendence degree. It is analogous to the invariant of a dimension for vector spaces (the size of a basis is independent of which basis is chosen).
See Theorem 115 in Pete L Clark's notes on Field Theory.
Assume $\wp$ is transcendental (satisfies no algebraic relation) over $\Bbb C$. We know $\Bbb C(\wp,\dot\wp)$ is an algebraic extension of $\Bbb C(\wp)$ since it's made by adjoining a square root of $4\wp^3-g_2\wp-g_3$, and so $\{\wp\}$ is a transcendence basis for $\Bbb C(\wp,\dot\wp)/\Bbb C$. This implies the extension has transcendence degree equal to one. If $f,g\in\Bbb C(\wp,\dot\wp)$ are elliptic functions, $\{f,g\}$ cannot be algebraically independent as that would imply transcendence degree $\ge2$, a contradiction.
If $\wp$ were algebraic over $\Bbb C$ then $\Bbb C(\wp,\dot\wp)/\Bbb C$ would be algebraic and so no two elements would be algebraically independent. I assume $\wp$ is transcendental for any choice of period lattice and never algebraic, but now that I think about it I've never seen that explicitly stated anywhere.
A: 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 $F(f,g)=0$.
