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Consider the plane algebraic curve $$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$ Its compactification results in a Riemann surface $C_1$ of genus $1$. Hence, it can be transformed into an ellptic curve $$ w^2 + z^3 - \frac{640}{3}z + \frac{5888}{27} = 0\tag{2} $$ through the birational transformation $$x = \frac{1}{256}z^2 - \frac{5}{48}z - \frac{29}{36} - \frac{1}{32}w, \ \ y = -\frac{1}{16}z - \frac{1}{6}.\label{3}\tag{3}$$

I want to perform integration on $C_1$, especially the integration of Abelian differentials along the canonical cycles, that is, periods of Abelian differentials. However, I don't know how to proceed. Is it possible to calculate elliptic integrals and then obtain the integral of the corresponding differential on $C_1$ through transformation \eqref{3}?

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  • $\begingroup$ Do you mean an exact or numerical computation? $\endgroup$
    – François Brunault
    Commented Sep 8, 2023 at 20:01
  • $\begingroup$ @FrançoisBrunault Numerical computation is sufficient. $\endgroup$
    – mxjia
    Commented Sep 9, 2023 at 1:39
  • $\begingroup$ I think you need to specify which cycles in $C_1$ you wish to integrate over. $\endgroup$
    – François Brunault
    Commented Sep 9, 2023 at 9:38
  • $\begingroup$ @FrançoisBrunault Yes, you are right. I'm currently reading a reference book (A. I. Bobenko, C. Klein (eds.), Computational Approach to Riemann Surfaces, 2011) on this subject. Thank you. $\endgroup$
    – mxjia
    Commented Sep 9, 2023 at 23:30
  • $\begingroup$ It’s hard to understand what exactly you are trying to compute. The differentials of both curves are related to the differentials of the respective planes via adjunction, and the differentials of the two planes are related by the exact sequence of differentials of blow ups (that would be your binational transformation). This is all rather explicit, but you didn’t say what you plan to integrate these differentials on. $\endgroup$
    – David Lehavi
    Commented Dec 29, 2023 at 20:33

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