An Artin-Schreier curve X (associated with an equation of the form y^p- y = f(x)) must satisfy the familiar Weil bound
 
||X(F_{p^n} )| - (p^n + 1)| < (degf - 1)(p - 1)p^{n/2}
 
but in many cases stronger bounds hold. In particular, Rojas-Leon and Wan proved such a curve must satisfy a bound of the form

||X(F_{p^n}) - (p^n +1)|<C_{d,n} p^{(n+1)/2}
 
where C{d,n} is a constant that depends on d := degf and n but not p. I will talk about how to use the representation theory of the symmetric group to prove both similar bounds and related statements about the zeros of the zeta function of X. Specifically, I will define a class of auxiliary varieties Y_n, each with an action of S_n, and explain how the S_n representation H^{n-1}(Y_n) contains useful arithmetic information about X. To provide an example, I will use these techniques to show that if d is “small” relative to p, then (a/b)^p = 1 for “most” pairs of X zeta zeroes a and b.