  Using the sieve for Frobenius, we show that, in a certain sense, the roots of
the L-functions of "most" algebraic curves over finite fields do not satisfy
any non-trivial (linear or multiplicative) rational dependency relations. This
can be seen as an analogue of conjectures of linear independence among
ordinates of zeros of L-functions over number fields. As a corollary, we find,
for "most" pairs of distinct algebraic curves over a finite field, the limiting
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 alternative approach, which relies on Serre's
theory of Frobenius tori, and we give a number of examples.
