Some exact formulae for the numbers of representations of integers by ternary quadratic forms
From the author's introduction: ``In this paper, we consider some ternary quadratic forms with class number two for their genus, and give exact formulae for the numbers of representations of an integer by these forms. The main idea is as follows. For a positive definite ternary form \(f\), let \(f\) and \(g\) be the representatives of classes in the genus of \(f\). On the one hand, some linear combination of the numbers of representations of an integer by \(f\) and \(g\) can be related to the class number of a certain quadratic field; on the other hand, in certain cases, we can find another linear combination of these numbers which is related to the \(L\)-function of an elliptic curve. By these two linear combinations, in terms of the class number of a quadratic field and the special value of the \(L\)-function of an elliptic curve, we can get exact formulae for the numbers of representations of an integer by \(f\) and \(g\), respectively. This also shows the difficulty of the classical problem mentioned above because of the mysterious properties of the special values of \(L\)-functions and class numbers.'' The results are too lengthy to state here.
scientific article; zbMATH DE number 2119254
scientific article