2009/03/23 by Florent Jouve, F. Jouve, Jouve, F.
Mathematics · Social Sciences · #11C08 (Secondary) #11E08 #11G25 (Primary) #11N36 #14D10 #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical and Political Studies #Number Theory (math.NT) #math.NT #msc:11C08 #msc:11E08 #msc:11G25 #msc:11N36 #msc:14D10
paper · pdf · doi:10.48550/arxiv.0903.3898
20 pages
arxiv created 2009/03/23 · openalex publication_date 2009/03/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a quantitative version of a result due to N. Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's Theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes "as irreducible as possible" when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result which gives an estimate for the proportion of elliptic curves studied whose L-functions have "maximal" Galois group . To do so we make use of E. Kowalski's idea to apply large sieve methods in algebro-geometric contexts. Besides large sieve techniques, we use results of C. Hall on finite orthogonal monodromy and previous work of the author on orthogonal groups over finite fields.