A family of pairs of imaginary cyclic fields of degree (p-1)/2 with both class numbers divisible by p

• Published in: The Ramanujan journal Vol. 52; no. 1; pp. 133 - 161
• Main Authors: Aoki, Miho, Kishi, Yasuhiro
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2020; Springer Nature B.V
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Integers; Mathematics; Mathematics and Statistics; Number Theory; Polynomials; 11R29; Abelian number fields; Gauss sums; Class numbers; Fundamental units; 11R11; Linear recurrence sequences; 11R16; Jacobi sums
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-018-0085-9

Let p be a prime number with p ≡ 5 ( mod 8 ) . We construct a new infinite family of pairs of imaginary cyclic fields of degree ( p - 1 ) / 2 with both class numbers divisible by p . Let k 0 be the unique subfield of Q ( ζ p ) of degree ( p - 1 ) / 4 and u p = ( t + b p ) / 2 ( > 1 ) be the fundamental unit of k : = Q ( p ) . We put D m , n : = L m ( 2 F m - F n L m ) b for integers m and n , where { F n } and { L n } are linear recurrence sequences of degree two associated to the characteristic polynomial P ( X ) = X 2 - t X - 1 . We assume that there exists a pair ( m 0 , n 0 ) of integers satisfying certain congruence relations. Then we show that there exists a positive integer N q which satisfies the both class numbers of k 0 ( D m , n ) and k 0 ( p D m , n ) are divisible by p for any pairs ( m ,  n ) with m ≡ m 0 ( mod N q ) , n ≡ n 0 ( mod N q ) and n > 3 . Furthermore, we show that if we assume that ERH holds, then there exists the pair ( m 0 , n 0 ) .