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 \(\Theta _C\). Following the work of Rudnick (Acta Arith. 143(1), 81–99, 2010), we compute the expected value of \({{\mathrm{tr}}}(\Theta _C^n)\) over the moduli space of hyperelliptic curves of genus g, over a fixed finite field \({\mathbb F}_q\), in the limit of large genus. As an application, we compute the expected value of the number of points on C in \({\mathbb F}_{q^n}\) as the genus tends to infinity. We also look at biases in both expected values for small values of n.
Highlights
Let C be a smooth projective curve of genus g ≥ 1 defined over a fixed finite field Fq of odd cardinality q
If we let #C(Fqn) denote the number of points on C in finite extensions Fqn of degree n of Fq, the Zeta function associated to the curve C is defined by ZC(u) := exp
We study the traces of high powers of the Frobenius class of CQ over Hg over a fixed finite field Fq of odd cardinality q as g → ∞
Summary
Let C be a smooth projective curve of genus g ≥ 1 defined over a fixed finite field Fq of odd cardinality q. Our results for odd n are exact and coincide with the RMT results for all values of g At first glance, this may seem counterintuitive as one expects to have an error term, as in the even case and as in [1]. We continue to get deviations from the RMT results for even n ≥ 4, our results hold for n = 2 and our deviations are different from those obtained in [1] Another approach to computing #CQ(Fqn) Hg is the work of Alzahrani [5] who uses the distribution of points on Hg over Fq in Fqn. Another approach to computing #CQ(Fqn) Hg is the work of Alzahrani [5] who uses the distribution of points on Hg over Fq in Fqn Using these methods, the results of Alzahrani agree with the Corollary above (albeit with a larger error term). Milione in their study of statistics for biquadratic curves; their work is collected in [9]
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