On the density of abelian l-extensions

2015/09/04 by Chuang, Chih-Yun, Kuan, Yen-Liang
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1509.01345

Abstract

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let ℓ be a rational prime and K a rational function field \Bbb F_q(t) with ℓ \nmid q. Let \textupDisc_f(F/K) denote the finite discriminant of F over K. Denote the number of abelian ℓ-extensions F/K with \textupdeg(\textupDisc_f(F/K)) = (ℓ-1)αn by a_ℓ(n), where α=α(q, ℓ) is the order of q in the multiplicative group (\Bbb Z/ℓ \Bbb Z)^×. We give a explicit asymptotic formula for a_ℓ(n). In the case of cubic extensions with q≡ 2 \pmod 3, our formula gives an exact analogue of Cohn's classical formula.

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