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It is my understanding that a genus-$g$ Riemann surface has $2g$ independent cycles that satisfy the usual intersection rules:

$$a_i \cap a_j = 0$$ $$b_i \cap b_j = 0$$ $$a_i \cap b_j = \delta_{ij}$$

If one would like to define a period matrix, one considers $g$ linearly independent holomorphic differentials $\omega_j$ that are normalized in such a way that:

$$\oint_{a_j} \omega_i = \delta_{ij}$$

Then, one can define a period matrix

$$\Omega_{ij} = \oint_{b_j} \omega_i$$

which will in general be a $g \times g$ matrix.

At this point, my question becomes a bit speculative: I am interested in Riemann surfaces that look or behave like genus-$1$ curves. My first question is:

  1. Let us imagine a surface where all the period integrals are proportional to each other by some constant of proportionality, so in effect there is only one independent period, and all the information about the surface captured by the other cycles is obtained by simply multiplying by the appropriate constant. Is there a way to write down an "effective genus-$1$ curve" for this higher genus surface?

  2. If someone hands me a period matrix, is there a way to reconstruct the curve it came from?

Any related comments/helpful links will be much appreciated.

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    $\begingroup$ The answer to both questions is "no". (For the first question, such Riemann surfaces do not exist). See any book on Riemann surfaces, for example Siegel, Topics in complex function theory. $\endgroup$ – Alexandre Eremenko Aug 10 '17 at 11:48
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Eremenko already commented about Q1 (but see comments below). To clarify further about Q2: Any period matrix can be shown to be symmetric with positive definite imaginary part, although in general, not all matrices of this type come from Riemann surfaces. However, if you happen know the matrix comes from a Riemann surface, then the surface can be reconstructed from it, in theory, by Torelli's theorem. You would need to look at the proofs, of which there are several, to do this explicitly.

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    $\begingroup$ For Q1, why do no such curves exist? $\endgroup$ – Will Sawin Aug 10 '17 at 16:29
  • $\begingroup$ Actually, thinking about it some more, they probably do exist e.g $\begin{pmatrix}i&\epsilon \\ \epsilon & i\end{pmatrix}$ ought to come from a Riemann surface. I don't understand the rest of the question, however. $\endgroup$ – Donu Arapura Aug 10 '17 at 16:54
  • $\begingroup$ @Donu Arapura Thanks for your reply! As a follow-up: do you know of any kinds of surface that exhibit such properties, i.e. that they have many periods, but the periods themselves only differ at most by some constants? A hyperelliptic curve, perhaps? $\endgroup$ – Madhusudhan Raman Aug 10 '17 at 17:04

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