Values of zeta functions and class number 1 criterion for the simplest cubic fields (Q2709971)

Let \(K\) be the simplest cubic field defined by the irreducible polynomial \(f(x)=x^3+mx^2-(m+3)x+1\), where \(m\) is a positive integer such that \(D=m^2+3m+9\) is square-free. Let \(h_m\) and \(\zeta_K(s)\) be the class number and the Dedekind zeta function of \(K\), respectively, and let \(\zeta_K(s,C)\) be the class zeta function of \(K\) belonging to the principal ideal class of \(K\). NEWLINENEWLINENEWLINEThe authors use results of Siegel and Zagier to compute \(\zeta_K(-1)\), which easily gives \(\zeta_K(2)\) by means of the functional equation. They exploit Byeon's formula for \(\zeta_K(2,C)\) and obtain a necessary and sufficient condition for the equality \(\zeta_K(2)=\zeta_K(2,C)\) to hold, which is equivalent to \(h_m=1\). NEWLINENEWLINENEWLINELet \(\rho\) be a root of \(f(x)\), let \({\mathfrak d}_K=(f'(\rho))\) be the different of \(K\). Let \(S=\{\nu\in{\mathfrak d}_K^{-1}\mid \nu\gg 0, \text{tr}(\nu)=1\}\) be the set of all totally positive elements of \({\mathfrak d}_K^{-1}\) of trace one. Each \(\nu\in S\) is of the form \(\nu=(a+b\rho+c\rho^2)/D\) with \(a,b,c\in\mathbb{Z}\) and mapping \(\eta(\nu)=(c,(a+2c-3)/m)\) gives a one-to-one correspondence \(\eta\) between \(S\) and the set \(\mathfrak S\) of all lattice points inside an explicitly described triangle in the \((c,t)\)-plane. NEWLINENEWLINENEWLINEFor each \(\nu\in S\) with \(\eta(\nu)=(c,t)\) the authors obtain a formula for the norm \(N((\nu){\mathfrak d}_K)=|f_m(c,t)|\) where \(f_m(c,t)\) is at most quadratic polynomial in \(m\). The main result of the paper is the following criterion: \(h_m=1\) if and only if \(|f_m(c,t)|\) is a prime for each \((c,t)\in{\mathfrak S}\setminus \{(1,0),(m+1,1),(-m-2,m+2)\}\). An easy corollary is the following statement: if \(h_m=1\) then \(2m+3\) is a prime, which has been proved already by Byeon.