On The Ono Invariants Of Imaginary Quadratic Number Fields

The finite extension K of Q is called algebraic number field. All algebraic integers of K form a ring, denoted Z_k. Let I and J be two ideals of Z_k, if there are non-zero elements α,β ∈ Z_k such that (α)I = (β)J, we say that / and J are equivalent. This is an equivalence relation between ideals of Z_k. The number of equivalence classes is called the class number of K. Finding the class number of K is one of the main topics in computational number theory. However, this problem is so difficult that there are no efficient algorithms to get class numbers of general number fields nowadays. Even for quadratic number fields, finding class numbers is still quite hard. Therefore, the research on the property of quadratic number fields is helpful to find efficient algorithms to get class numbers.Let E_d(x) denote the "Euler polynomial " x² + x + (1 - d)/4 if d ≡ 1 mod 4 and x² — d if d ≡ 2,3 mod 4. Set Ω(n) = the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariant Ono_d of K = Q(d^1/2) is defined to be m&x{Ω{E_d{b)) : b = 0,1, ... , |D|/4 - 1} except when d = -1,-3 in which case Ono_d is defined to be 1, where D denote the discriminant of K. Finally, let h_d denote the class number of K. In 2002 J. Cohen and J. Sonn gave the following conjecture. Conjecture(Cohen-Sonn). h_d = 3 iff Ono_d = 3 and — d = p ≡ 3 mod 4 is a prime.Cohen and Sonn verified that the conjecture is true for p < 1.5 × 10⁷. Moreove, they used a result of Bach in algebraic number fields to prove that the conjecture holds for p > 10¹⁷ assuming the Extended Riemann Hypothesis.In this paper, we first use searching method on computers to show that the Cohen-Sonn conjecture is true for p ∈ [l,10⁹]. Then we prove that for p > 10⁹, if p ≡ 7 mod 8,p ≡ 11 mod 12, p ≡ 11,19 mod 20, p ≡ 3,19, 27 mod 28, p ≡ 7,19,35,39,43 mod 44 or p ≡ 3,23,27,35,43,51 mod 52, then Ono_d > 3. Moreover we prove that if there is an odd prime q≤ (p/4-1)^1/3 , such that (-p/q) = 1, then Ono_d > 3. Thus we get a constructive method to show that the Cohen-Sonn conjecture is true for p ∈ [10⁹,10¹³] by the aid of computer.