# Higher Genus Surfaces That “Look Like” Genus-1 Surfaces

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?

• 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. – Donu Arapura Aug 10 '17 at 16:54