Differentials of the second kind
================================


Gross and Rohrlich in the paper *On the periods of abelian integrals and a formula of Chowla and Selberg* state the claim below without citation (pg. 198), giving an explicit determination of the cohomology classes of the Fermat curve $X\_d := \{x^d + y^d - z^d = 0\} \subset \mathbb{P}^2$ using differentials of the second kind.


Working on the affine chart $\{z=1\} \subset \mathbb{P}^2$ they define the following meramorphic 1-forms on $X\_d$:
$$
 \eta\_{r,s,t} = x^{r-1}y^{s-d} \mathrm{d} x.
$$
These forms can be integrated over homology classes, giving the cohomology classes that we need. 



> 
> **Claim:** The middle cohomology of $X\_d$ is generated by $\eta\_{r,s,t}$ where $0 < r,s,t < d$ and $r+s+t \equiv 0 \,(\text{mod } d)$. 
> 
> 
> 


*Example:* Let $d=3$. Then, $X\_3$ is elliptic and we have $\mathrm{H}^1(X\_3,\mathbb{C}) = \langle \frac{\mathrm{d}x}{y^2} , \frac{x\mathrm{d}x}{y}\rangle$. 



> 
> **Main problem:** How does one prove the claim above?
> 
> 
> 


Perhaps using Griffiths residues
================================


We could try to prove this statement using the Griffiths' residue map:
$$
 \operatorname{res}: \bigoplus\_{\ell \ge 1} \mathrm{H}^0(\mathbb{P}^2,\Omega\_{\mathbb{P}^2}(\ell X\_d)) \to \mathrm{H}^1(X\_d,\mathbb{C}),
$$
which implies that the residues of the following meramorphic forms generate the image:
$$
 \omega\_{r,s,t} = \frac{x^{r-1}y^{s-1}z^{t-1}}{(x^d+y^d-z^d)^\ell }\cdot \Omega,
$$
where $0 < r,s,t < d$, $r+s+t = \ell d$ and $\Omega = x \mathrm{d}y\mathrm{d}z - y \mathrm{d}x\mathrm{d}z + z\mathrm{d}x\mathrm{d}y$.


Therefore it remains to integrate the forms $\omega\_{r,s,t}$ over a suitable tubular neighbourhood of $X\_d$. Representation theoretic arguments imply that $\operatorname{res} \omega\_{r,s,t} = c\_{r,s,t} \eta\_{r,s,t}$ for a scalar $c\_{r,s,t} \in \mathbb{C}^\*$, provided Gross and Rohrlich are correct.


However, an explicit computation reveals that the residue of $\omega\_{r,s,t}$ is in fact a scalar multiple of $x^{r-1}y^{s-\ell d} \mathrm{d}x$ instead of $x^{r-1}y^{s-d} \mathrm{d}x$. Therefore we can answer Question 1 if we can answer the following.



> 
> **Equivalent problem:** How can we lower the pole order of $x^{r-1}y^{s-\ell d} \mathrm{d}x$ modulo cohomological equivalence to get $x^{r-1}y^{s-d}\mathrm{d}x$?
> 
> 
> 



How to compute cohomology using differentials of the second kind on a Fermat curve?
If I recall correctly, you can find a proof of the claim in Lang's book ["Introduction to Algebraic and Abelian Functions"](https://books.google.co.cr/books?id=p4aA6DKzoeIC). He has a chapter on the Fermat curve and in fact (after looking at the google preview of the book) I think that the claim is essentially Theorem 2.2 in Chapter II of Lang's book.


