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Abstract: Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ definedover $\F q$, the finite field of $q$ elements. Let $# \cC\F {q^n}$ be thenumber of $\F {q^n}$-rational points on $\cC$. Under a certain multiplicativeindependence condition on the roots of the zeta-function of $\cC$, we derive anasymptotic formula for the number of $n =1,

., N$ such that $# \cC\F {q^n}- q^n -1-2gq^{n-2}$ belongs to a given interval $\cI \subseteq -1,1$. Thiscan be considered as an analogue of the Sato-Tate distribution which covers thecase when the curve $\E$ is defined over $\Q$ and considered modulo consecutiveprimes $p$, although in our scenario the distribution function is different.The above multiplicative independence condition has, recently, been consideredby E. Kowalski in statistical settings. It is trivially satisfied for ordinaryelliptic curves and we also establish it for a natural family of curves ofgenus $g=2$.



Author: Omran Ahmadi, Igor E. Shparlinski

Source: https://arxiv.org/







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