Question 1: Suppose you have an algebraic curve Y with an action by a finite group G. Then G acts on the space of global holomorphic differentials (regular 1-forms) on Y, which is a finite dimensional vector space. This representation of G has a character (the traces of the matrices corresponding to elements of G), whose values lie in a finite extension of Q. What are the conditions (on G, or on G and Y) under which the character values are all in Q itself? Question 2: Choose a positive integer n, and another positive integer h relatively prime to n. Notice that some power of h (call it h^k) will be 1 mod n. Now for each j from 1 to n, add up j (mod n) + jh (mod n) + jh2 (mod n) + ... j h^{k-1} (mod n). Do you get the same sum, no matter which j you started with? Is it always a multiple of n? Can you find values of h for which you *don´t* always get the same sum? In this talk I will explain why these questions are related, and report on recent results with Ted Chinburg. We have a complete answer to Question 2, which it turns out answers Question 1 in the tame case.