Biased statistics for traces of cyclic 𝑝-fold covers over finite fields

Abstract

In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model.

The fluctuations in the number of points on a family of curves over a finite field

“…Recently, extending the results of Kurlberg and Rudnick [6], Bucur, David, Feigon and Lalín in a series of two beautiful papers [3,4] successfully obtain interesting results on the distribution of the trace of the Frobenius endomorphism Frob C over moduli spaces of cyclic l-fold covers of genus g when g → ∞. Interested reader may refer to their papers [3,4] for more details and for other results related with the subject. The proofs of [6,3,4] are similar and are based on an ingenious counting argument.…”

“…The idea of the proof is similar to that of [6,3,4], though it is presented in a slightly different way via character sums.…”

“…Related to this work [6], problems of similar flavor with various arithmetic and geometric applications have been considered before by Larsen [11], Knizhnerman and Sokolinskii [9,10] and Bergström [1]. Recently, extending the results of Kurlberg and Rudnick [6], Bucur, David, Feigon and Lalín in a series of two beautiful papers [3,4] successfully obtain interesting results on the distribution of the trace of the Frobenius endomorphism Frob C over moduli spaces of cyclic l-fold covers of genus g when g → ∞. Interested reader may refer to their papers [3,4] for more details and for other results related with the subject.…”

“…Interested reader may refer to their papers [3,4] for more details and for other results related with the subject. The proofs of [6,3,4] are similar and are based on an ingenious counting argument. The main purpose of this paper is to give a slightly different treatment of the proof.…”

“…The results of [6,3,4] then can be derived directly. Our proofs follow the ideas of [6,3,4], however, the properties of character sums will be used in an essential way.…”