Doubly periodic entire function with periods that are linearly independent over $\mathbb{Q}$ is constant.

Solution 1:

First assume that $\lambda_1,\lambda_2$ are linearly independent not only over $\mathbb Q$ but over $\mathbb R.$

The convex hull of $\{\,0,\,\lambda_1,\,\lambda_2,\,\lambda_1+\lambda_2\,\}$ is a parallelogram and thus a compact set. Entire functions are continuous. A continuous function on a compact set is bounded. By periodicity conjoined with the fact that $\lambda_1,\lambda_2$ are linearly independent over $\mathbb R$ and $\mathbb C$ is $2$-dimensional over $\mathbb R,$ we conclude that this entire function is bounded. Liouville's theorem says bounded entire functions are constant.

Now weaken the linear-independence hypothesis to being over $\mathbb Q.$ If $\lambda_1,\lambda_2$ are not linearly independent over $\mathbb R,$ then there is some irrational number $a\ne0$ such that $\lambda_1=a\lambda_2.$ Since $a$ can be approximated as closely as desired by rational numbers, one can show that $0$ can be approximated as closely as desired by integer linear combinations of $\lambda_1$ and $\lambda_2.$ This yields arbitrarily small periods of the periodic function, and for continuous functions, this means the function must be constant on the line $\{ c\lambda_1 : c\in\mathbb R\}.$ This implies its derivative is $0$ on that line. A holomorphic function whose derivative is $0$ on a set of points that has a limit point is constant.