The large sieve, monodromy and zeta functions of algebraic curves, II: independence of the zeros

Details The large sieve, monodromy and zeta functions of algebraic curves, II: independence of the zeros The large sieve, monodromy and zeta functions of algebraic curves, II: independence of the zeros

2008-07-14

arxiv.org/abs/0807.2118v1

Mathematics

Number Theory

39 pages

Scientific paper

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.

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