# Period Homomorphisms and closed 1-forms

This is from Otto Forster's "lectures on Riemann Surfaces", on integration of forms.

Let $\Gamma = \alpha_1 \mathbb{Z} + \alpha_2 \mathbb{Z}$ be a lattice in $\mathbb{C}$ (i.e. $\alpha_i \in \mathbb{C}$ and are lin. independent over $\mathbb{R}$), and $X := \mathbb{C} / \Gamma$ a torus. Given any homomorphism (of groups) \begin{equation*} a: \pi_1(X) \to \mathbb{C} \end{equation*} show that there exists a closed 1-form $\omega \in \mathcal{E}^1(X)$ (smooth 1-forms on $X$) whose period homomorphism is equal to $a$.

My Attempt (so far):

So first off, recall that the "period homomorphism" of a closed 1-form $\omega \in \mathcal{E}^1(X)$ is the map \begin{equation*} \gamma \mapsto \int_\gamma \omega \quad \text{ $\gamma \in \pi_1(X)$} \end{equation*} (which is nice and well-defined, since the integral of a closed 1-form is invariant under homotopic deformations of $\gamma$). Since $\pi_1(X) \cong \mathbb{Z} \times \mathbb{Z}$, the given map $a$ is uniquely determined by the images of the two fundamental cycles, which are the images of the two curves $\gamma_i: [0,1] \to \mathbb{C}$ given by $t \mapsto t\alpha_i$, under the canonical projection $p: \mathbb{C} \to \mathbb{C}/\Gamma$.

What I want to do is find some nice function $F : \mathbb{C} \to \mathbb{C}$ that is additively automorphic (w/ constant summands of automorphy) under $\text{Deck}(\mathbb{C}/ X)$ so that there exists a closed 1-form $\omega \in \mathcal{E}^1(X)$ such that $dF = p^*\omega$. For then we have \begin{equation*} \int_{p \circ \gamma_i} \omega = \int_{\gamma_i} dF = F(\alpha_i)-F(0) \end{equation*} and by (maybe) scaling $F$ in different ways, I can force the integral to be equal to $a(p \circ \gamma_i)$.

Or am I way off? Anybody?

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You are right. Note that you can choose F to be real-linear by prescribing its values at $\alpha _{1}, \alpha _{2}$. Also, such functions have the needed behaviour under the group of deck transformations (which acts by translations).