The Large Sieve, Monodromy, and Zeta Functions of Algebraic Curves, 2: Independence of the Zeros

Abstract

Using the sieve for Frobenius developed earlier by the author, we show that in a certain sense, the roots of the L-functions of most algebraic curves over finite fields do not satisfy any nontrivial (linear or multiplicative) dependency relations. This can be seen as an analogue of conjectures of Q-linear independence among ordinates of zeros of L-functions over number fields. As a corollary of independent interest, we find for “most” pairs of distinct algebraic curves over a finite field the form of the distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The method of proof also emphasizes the relevance of random matrix models for this type of arithmetic questions. We also describe an alternate approach, suggested by Katz, which relies on Serre's theory of Frobenius tori.