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- W2077681785 abstract "Let $cC$ be a smooth absolutely irreducible curve of genus $g ge 1$ defined over $F_q$, the finite field of $q$ elements. Let $# cC(F_{q^n})$ be the number of $F_{q^n}$-rational points on $cC$. Under a certain multiplicative independence condition on the roots of the zeta-function of $cC$, we derive an asymptotic formula for the number of $n =1, ldots, N$ such that $(# cC(F_{q^n}) - q^n -1)/2gq^{n/2}$ belongs to a given interval $cI subseteq [-1,1]$. This can be considered as an analogue of the Sato--Tate distribution which covers the case when the curve $E$ is defined over $Q$ and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E.~Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$." @default.
- W2077681785 created "2016-06-24" @default.
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- W2077681785 date "2010-01-01" @default.
- W2077681785 modified "2023-10-14" @default.
- W2077681785 title "On the distribution of the number of points on algebraic curves in extensions of finite fields" @default.
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- W2077681785 doi "https://doi.org/10.4310/mrl.2010.v17.n4.a9" @default.
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