Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$.

I know that to construct the Jacobian variety associated to $C$, one integrates a basis of global holomorphic differential forms over the contours of the curve's homology group. I'm looking for information that is oriented toward actually computing things for given concrete examples; everything I've seen so far, however, has been uselessly abstract or non-specific. Note: I'm new to this—I'm an analyst who knows next to nothing about algebra and even less about differential geometry or topology.

In my quest for a sensible answer, I turned to a H.F. Baker's wonderful (though densely written) text from the start of the 20th century. Just reading through the first few pages makes it *abundantly* clear that there is a general procedure for constructing a basis of holomorphic differential forms for a given curve. Ted Shifrin's comment on this math-stack-exchange problem only makes me more certain than ever that the answers I seek are out there, somewhere.

Broadly speaking, my goals are as follows. In all of these, my aim is to be able to use the answers to these questions to compute various specific examples, either by hand, or with the assistance of a computer algebra system. So, I'm looking for formulae, explanations and/or step-by-step procedures/algorithms, and/or pertinent reference/reading material.

(1) In the case where $F$ is a polynomial, what is/are the procedure(s) for determining a basis of holomorphic differential 1-forms over $F$? If the procedure varies depending on certain properties of $F$ (say, if $F$ is an affine curve, or a projective curve, or of a certain form, or some detail like that), what are those variations?

(2) In the case where $F$ is a polynomial of $x$-degree $d_{x}$, $y$-degree $d_{y}$, and $C$ is a curve of genus $g$, I know that the basis of holomorphic differential 1-forms for $C$ will be of dimension $g$. In the case, say, where $C$ is an elliptic curve, with:

$$F\left(x,y\right)=4x^{3}-g_{2}x-g_{3}-y^{2}$$

the classical *Jacobi Inversion Problem* arises from considering a function $\wp\left(z\right)$ which parameterizes $C$, in the sense that $F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)$ is identically zero. Using the equation: $$F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)=0$$ we can write: $$\wp^{-1}\left(z\right)=\int_{z_{0}}^{z}\frac{ds}{4s^{3}-g_{2}s-g_{3}}$$ and know that the multivaluedness of the integral then reflects the structure of the Jacobian variety associated to $C$.

That being said, in the case where $C$ is of genus $g\geq2$, and where we can write $F\left(x,y\right)=0$
as: $$y=\textrm{algebraic function of }x$$ nothing stops us from performing the exact same computation as for the case with an elliptic curve. Of course, this computation must be wrong; my question is: *where and how does it go wrong*? How would the parameterizing function thus obtained relate to the "true" parameterizing function—the multivariable Abelian function associated to $C$? Moreover, how—if at all—can this computation be modified to produce the correct parameterizing function (the Abelian function)?

(3) My hope is that by understanding both (1) and (2), I'll be in a position to see what happens when these classical techniques are applied to *non-algebraic* plane curves defined but with $F$ now being an analytic function (incorporating exponentials, and other transcendental functions, in addition to polynomials). Of particular interest to me are the transcendental curves associated to exponential diophantine equations such as: $$a^{x}-b^{y}=c$$
$$y^{n}=b^{x}-a$$

That being said, I wonder: has this already been done? If so, links and references would be much appreciated.

Even if it has, though, I would still like to know the answers to my previous questions, even if it's merely for my personal edification alone.

Thanks in advance!

processby which such varieties were (classically) constructed: the basis of differential forms, the methods of dealing with the Jacobi inversion problem, etc. $\endgroup$ – MCS Mar 11 '19 at 1:14