A family of pairs of imaginary cyclic fields of degree (p-1)/2 with both class numbers divisible by p - Aoki, Miho; Kishi, Yasuhiro (The Ramanujan journal, 2020)
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 ) .