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ON THE RELATIVE ZETA FUNCTION IN FUNCTION FIELDS
Shiomi, Daisuke Korean Mathematical Society 2012 대한수학회논문집 Vol.27 No.3
In the previous paper [8], the author gave a determinant formula of relative zeta function for cyclotomic function fields. Our purpose of this paper is to extend this result for more general function fields. As an application of our formula, we will give determinant formulas of class numbers for constant field extensions.
THE p-PART OF DIVISOR CLASS NUMBERS FOR CYCLOTOMIC FUNCTION FIELDS
Daisuke Shiomi Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.3
In this paper, we construct explicitly an infinite family of primes P with h<sup>±</sup><sub>P</sub> ≡ 0 (mod q<sup>deg</sup> <sup>P</sup>), where h<sup>±</sup><sub>P</sub> are the plus and minus parts of the divisor class number of the P-th cyclotomic function field over 𝔽<sub>q</sub>(T). By using this result and Dirichlet's theorem, we give a condition of A, M ∈ 𝔽<sub>q</sub>[T] such that there are infinitely many primes P satisfying with h<sup>±</sup><sub>P</sub> ≡ 0 (mod p<sup>e</sup>) and P ≡ A (mod M).