Journal Article

Maximal Galois Group of <i>L</i>-Functions of Elliptic Curves

Florent Jouve

in International Mathematics Research Notices

Volume 2009, issue 19, pages 3557-3594
Published in print January 2009 | ISSN: 1073-7928
Published online May 2009 | e-ISSN: 1687-0247 | DOI: http://dx.doi.org/10.1093/imrn/rnp066
Maximal Galois Group of L-Functions of Elliptic Curves

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We give a quantitative version of a result due to 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 “big” Galois group. To do so, we make use of Kowalski's idea to apply large sieve methods in algebro-geometric contexts. Besides large sieve techniques, we use results of Hall on finite orthogonal monodromy and previous work of the author on orthogonal groups over finite fields.

Journal Article.  11188 words.  Illustrated.

Subjects: Mathematics

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