On the distribution of the rational points on cyclic covers in the absence of roots of unity

2017/11/13 by Bary-Soroker, Lior, Meisner, Patrick · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1711.04684

Abstract

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: Let ℓ be a prime, q a prime power and consider the ensemble H_g,ℓ of ℓ-cyclic covers of ℙ¹_\mathbbF_q of genus g. We assume that q\not≡ 0,1\mod ℓ. If 2g+2ℓ-2\not≡0\mod (ℓ-1)\rm ord_ℓ(q), then H_g,ℓ is empty. Otherwise, the number of rational points on a random curve in H_g,ℓ distributes as ∑_i=1^q+1 X_i as g→ ∞, where X₁,…, X_q+1 are i.i.d. random variables taking the values 0 and ℓ with probabilities (ℓ-1)/(ℓ) and (1)/(ℓ), respectively. The novelty of our result is that it works in the absence of a primitive ℓ-th-root of unity, the presence of which was crucial in previous studies.

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