JOUVE, Florent

doi:10.1093/imrn/rnp066

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JOUVE, Florent
Institut de Mathématiques de Bordeaux [IMB]

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International Mathematics Research Notices. 2009-05-21p. ?

Oxford University Press (OUP)

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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.< Réduire

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Maximal Galois group of L-functions of elliptic curves

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