Vlad Matei

TL;DR: In this article, the authors studied the geometry associated to the distribution of arithmetic functions, including the von Mangoldt function and the Mobius function, in short intervals of polynomials over a finite field, using the Grothendieck-Lefschetz trace formula.

TL;DR: In this article, the authors studied the geometry associated to the distribution of arithmetic functions, including the von Mangoldt function and the Mobius function, in short intervals of polynomials over a finite field, using the Grothendieck-Lefschetz trace formula.

TL;DR: The methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

TL;DR: In this paper, the problem of counting the number of plane cubic curves defined over a finite field that do not share a common component and intersect in exactly rational points is investigated, and the main inputs to the proof include counting pairs of cubic curves that do share common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.

TL;DR: In this article, the problem of counting the number of plane cubic curves defined over a finite field that do not share a common component and intersect in exactly the same rational points was investigated.