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

2008/07/14 by Emmanuel Kowalski, Kowalski, Emmanuel
Mathematics · #11G25 #11M99 #11N35 #14D10 #14G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G25 #msc:11M99 #msc:11N35 #msc:14D10 #msc:14G15

paper · pdf · doi:10.48550/arxiv.0807.2118

39 pages

arxiv created 2008/07/14 · openalex publication_date 2008/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• The Tate conjecture over finite fields (AIM talk)
• Independence of ℓ of monodromy groups
• The conjugate dimension of algebraic numbers

Related

• The large sieve, monodromy and zeta functions of curves
• Independence of the zeros of elliptic curve L-functions over function fields
• Counting Number Fields in Fibers
• Applications of the square sieve to a conjecture of Lang and Trotter for\n a pair of elliptic curves over the rationals
• Maximal Galois group of L-functions of elliptic curves