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Patrick Meisner: Statistics of point counts for certain families of curves over finite fields

Time: Wed 2018-10-17 13.15

Location: Room 306, House 6, Kräftriket, Department of Mathematics, Stockholm University

Participating: Patrick Meisner (KTH)

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The generalized Riemann hypothesis (GRH) for function fields tells us that if C is a curve over \(\mathbb{F}_q\) with L-function LC(u), then we can find a unitary symplectic matrix \(\Theta_C\), called the Frobenius of C, such that \(L_C(u) = \det(1-uq^{1/2}\Theta_C)\). Moreover, if Nn(C) denotes the number of \(\mathbb{F}_{q^n}\)-rational points on C, then we have the equation

\(N_n(C) = q^n+1-q^{n/2} \operatorname{Tr}(\Theta_C^n)\)

In this talk we will discuss results related to statistics of \(\operatorname{Tr}(\Theta_C^n)\). In particular, the probability that \(\operatorname{Tr}(\Theta_C)\) takes a certain value and the expected value of \(\operatorname{Tr}(\Theta_C)\) as C runs over certain families of curves.

Belongs to: Stockholm Mathematics Centre
Last changed: Oct 13, 2018