Lower bounds for the number of rational points on curves over finite fields
Christophe Ritzenthaler
IRMAR, Université de Rennes 1 & CIMPA, Nice
https://perso.univ-rennes1.fr/christophe.ritzenthaler/
Date(s) : 29/09/2022 iCal
14h30 - 15h30
The number of rational points on a curve of genus g over Fq is upper bounded by 1+q+2g √q. But how good is this bound in general?
If the situation for fixed q and g going to infinity has been studied for a while, much less was known for g fixed and q going to infinity. As a consequence of Katz-Sarnak theory, we’ll first get for any given g > 0, any ε > 0 and all q large enough, the existence of a curve of genus g over Fq with at least 1 + q + (2g − ε)√q rational points. Then using a distinct method, we get weaker bounds of the form 1 + q + 4 √q − 32 but which are valid for any q > q0, with q0 explicit and g>1.
If the situation for fixed q and g going to infinity has been studied for a while, much less was known for g fixed and q going to infinity. As a consequence of Katz-Sarnak theory, we’ll first get for any given g > 0, any ε > 0 and all q large enough, the existence of a curve of genus g over Fq with at least 1 + q + (2g − ε)√q rational points. Then using a distinct method, we get weaker bounds of the form 1 + q + 4 √q − 32 but which are valid for any q > q0, with q0 explicit and g>1.
This is a joint work with Jonas Bergström, Everett Howe and Elisa Lorenzo García.
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