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](https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACR0014.0001.001) 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](https://math.stackexchange.com/questions/2726917/holomorphic-1-form-on-projective-curve?rq=1) 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! 



The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve
I'm getting back to the question of describing holomorphic $1$-forms on a plane curve. 


**Affine curves:** Let $X$ be a smooth curve in $\mathbb{A}^2$, given by the equation $F(x,y)$. Then $F\_x dx + F\_y dy=0$. Since $F=0$ is smooth, the functions $F\_x$ and $F\_y$ have no common zero on $X$ and we can define a $1$-form on $X$ by $\omega = \tfrac{dx}{F\_y} = - \tfrac{dy}{F\_x}$. Moreover, $\omega$ is nowhere vanishing, so every $1$-form on $X$ is of the form $h \omega$ for some holomorphic function $h$ on $X$. For example, if $X$ is a hyperelliptic curve $y^2 = x^{2g+1} + a\_{2g} x^{2g} + \cdots + a\_1 x$, then $\omega = \tfrac{dx}{2y}$ and integrating $\omega$ along paths in $X$ corresponds concretely to computing integrals like $\int \tfrac{dx}{\sqrt{x^{2g+1} + a\_{2g} x^{2g} + \cdots + a\_1 x+a\_0}}$.


Incidently, there is a more conceptual way to describe $\omega$: It is the residue of the $2$-form $\tfrac{dx \wedge dy}{F(x,y)}$ along $X$. Since every automorphism of $\mathbb{A}^2$ multiplies $dx \wedge dy$ by a scalar, and every automorphism taking $X$ to itself (setwise) multiplies $F$ by a scalar, this shows that $\omega$ is well defined up to scalar multiple independent of the choice of coordinates on $\mathbb{A}^2$.


**Projective curves** Let $\overline{X}$ be the smooth projective completion of $X$. Then we can ask whether or not $h \omega$ extends to a global holomorphic $1$-form on $\overline{X}$. The vector space of holomorphic $1$-forms on $\overline{X}$ always has dimension $g$. Describing which $h \omega$ extend to $\overline{X}$ in general involves describing how to compute the smooth projective completion of $X$, which is a little complicated, so I'll just give the most important special cases.


**Hyperelliptic curves** Let $X$ be given by $y^2 = x^{2g+1} + a\_{2g} x^{2g} + \cdots + a\_1 x+a\_0$ for some square free polynomial of degree $2g+1$. Then a basis for the $1$-forms on $\overline{X}$ is $x^j \omega$ for $0 \leq j < g$. So the periods of this curve are concretely integrals of the form $\int \tfrac{x^j dx}{\sqrt{x^{2g+1} + a\_{2g} x^{2g} + \cdots + a\_1 x+a\_0}}$ for $j$ in this range.


**Smooth projective planar curves** Let $f(x,y,z)$ be a smooth degree $d$ hypersurface, so $F(x,y) = f(x,y,1)$. Then a basis of $1$-forms on $\overline{X}$ is $x^i y^j \omega$ for $i+j \leq d-3$. 


**Curves transverse to a toric compatification** The following includes both of the previous cases. Let $F(x,y) = \sum\_{(i,j) \in A} F\_{ij} x^i y^j$ for some finite set $A$ of exponents. Let $\Delta$ be the convex hull of $A$, this is a convex polytope. I will impose a condition for each edge of $\Delta$. Let $(p,q)$, $(p,q)+(u,v)$, $(p,q)+2 (u,v)$, ..., $(p,q) + k (u,v)$ be the lattice points on an edge. Suppose that the single variable polynomial $\sum F\_{(p+ku, q+kv)} z^k$ has no nonzero multiple roots. Then the closure of $X$ in the toric variety associated to $\Delta$ is smooth. In that case, a basis for the $1$-forms is $x^{i-1} y^{j-1} \omega$, where $(i,j)$ runs over the lattice points in the interior of $\Delta$. 


