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ON THE DISTRIBUTION OF THE RATIONAL POINTS ON CYCLIC COVERS IN THE ABSENCE OF ROOTS OF UNITY
Author(s) -
BarySoroker Lior,
Meisner Patrick
Publication year - 2019
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579319000111
Subject(s) - mathematics , projective line , prime power , prime (order theory) , root of unity , distribution (mathematics) , combinatorics , independent and identically distributed random variables , pure mathematics , random variable , discrete mathematics , projective test , mathematical analysis , statistics , projective space , physics , quantum mechanics , quantum
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 , ℓof ℓ ‐cyclic covers of P F q 1 of genus g . We assume that q ≢ 0 , 1 mod ℓ . If 2 g + 2 ℓ − 2 ≢ 0 mod( ℓ − 1 ) ord ℓ ( q ) , then H g , ℓis empty. Otherwise, the number of rational points on a random curve in H g , ℓdistributes as∑ i = 1 q + 1X ias g → ∞ , whereX 1 , … , X q + 1are independent and identically distributed 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.