On two of Liouville's quaternary forms
The quaternary forms referred to in the title are \(x^ 2+2y^ 2+3z^ 2+6t^ 2\) and \(x^ 2+y^ 2+6z^ 2+6t^ 2.\) The generating theta series of these forms are decomposed into Eisenstein series and the cusp eigenform \(H(z)=\eta (z)\eta (2z)\eta (3z)\eta (6z),\) where \(\eta\) denotes Dedekind's eta function. This yields Liouville's and Kloosterman's formulas for the numbers L(n) and K(n) of representations of n by these forms and, in particular, \(K(p)=L(p)=p+1+\lambda (p)\) for all primes \(p>3\), where \(\lambda\) (p) is the p-th Fourier coefficient of H(z). An analysis of L(p) and K(p) modulo small powers of 2 and the theory of ternary quadratic forms yield results on class numbers \(h(-n)\) of imaginary quadratic fields \({\mathbb{Q}}(\sqrt{-n}).\) One of the results is \[ h(-3p)\equiv h(-6p)\equiv (p+1+\lambda (p))(mod 16)\quad for\quad primes\quad p\equiv -1(mod 24).\quad This\quad implies\quad a\quad wea\ker \quad assertion\quad on\quad the\quad sum\quad \] \(h(-3p)+h(-6p)\) which is contained as a special case in \textit{A. Pizer}'s work [J. Number Theory 8, 184-192 (1976; Zbl 0329.12003)].
scientific article; zbMATH DE number 4097476
scientific article