Traces of High Powers of the Frobenius Class in the Moduli Space of Hyperelliptic Curves

2015/10/21 by Chinis, Iakovos Jake
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1510.06350

Abstract

The Zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ_C. Following the work of Rudnick, we compute the expected value of tr(Θ_Cⁿ) over the moduli space of hyperelliptic curves of genus g, over a fixed finite field \mathbbF_q, in the limit of large genus. As an application, we compute the expected value of the number of points on C in \mathbbF_qⁿ as the genus tends to infinity. We also look at biases in both expected values for small values of n.

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