The $p$-part of divisor class numbers for cyclotomic function fields
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In this paper, we construct explicitly an infinite family of primes $P$ with $h_P^{\pm} \equiv 0 \pmod {q^{\deg P}}$, where $h_P^{\pm}$ are the plus and minus parts of the divisor class number of the $P$-th cyclotomic function field over $\mathbb{F}_q(T)$. By using this result and Dirichlet's theorem, we give a condition of $A, M \in \mathbb{F}_q[T]$ such that there are infinitely many primes $P$ satisfying with $h_P^{\pm} \equiv 0 \pmod {p^e}$ and $P \equiv A \pmod M$.
Abstract
In this paper, we construct explicitly an infinite family of primes $P$ with $h_P^{\pm} \equiv 0 \pmod {q^{\deg P}}$, where $h_P^{\pm}$ are the plus and minus parts of the divisor class number of the $P$-th cyclotomic function field over $\mathbb{F}_q(T)$. By using this result and Dirichlet's theorem, we give a condition of $A, M \in \mathbb{F}_q[T]$ such that there are infinitely many primes $P$ satisfying with $h_P^{\pm} \equiv 0 \pmod {p^e}$ and $P \equiv A \pmod M$.
참고문헌 (11)
- K. Q. Feng, "A note on irregular prime polynomials in cyclotomic function field theory", J. Number Theory, 22(2), pp.240~245 (1986)[DOI]
- E. -U. Gekeler, "On power sums of polynomials over finite fields", J. Number Theory, 30(1), pp.11~26 (1988)[DOI]
- D. M. Goss, "Kummer and Herbrand criterion in the theory of function fields", Duke Math. J, 49(2), pp.377~384 (1982)[DOI]
- D. M. Goss, "Class-groups of function fields", Duke Math. J, 52(2), pp.507~516 (1985)[DOI]
- H. L. Lee, "Power sums of polynomials in a Galois field", Duke Math. J, 10, pp.277~292 (1943)[DOI]
- J. Lee, "Infinite families of irregular primes in cyclotomic function fields", J. Number Theory, 207, pp.1~21 (2020)[DOI]
- R. Lidl, "Finite Fields, Encyclopedia of Mathematics and its Applications, 20" (1997), Cambridge Univ. Press
- M. I. Rosen, "Number Theory in Function Fields, Graduate Texts in Mathematics, 210" (2002), Springer
- H. Stichtenoth, "Algebraic Function Fields and Codes, Graduate Texts in Mathematics, 254" (2009), Springer
- D. S. Thakur, "Function Field Arithmetic" (2004), World Sci. Publishing, Inc
- E. Le Yaouanc, "On real quadratic function fields", J. Number Theory, 123(1), pp.27~34 (2007)[DOI]