\(ABC\) implies no ``Siegel zeros'' for \(L\)-functions of characters with negative discriminant
\textit{P. Vojta} [Diophantine approximations and value distribution theory. Lecture Notes in Mathematics 1239, Springer-Verlag (1987; Zbl 0609.14011)] has given a version of the \(abc\)-conjecture for number fields, with full uniformity with respect to the field. The present paper shows how this conjecture implies the existence of a positive constant \(c\) such that \(L(s, \chi)\neq 0\) for \(s> 1- c/\log d\), when \(\chi\) is the character corresponding to the imaginary quadratic field \(\mathbb{Q}(\sqrt{-d})\). The proof goes via an estimate for the class number, which enters through a bound for the height of the \(j\)-invariant, evaluated at either \(\tau= (-1+ \sqrt{-d})/2\) or \(\sqrt{-d}/2\) as appropriate. The \(j\)-function appears in the Diophantine equation \[ j(\tau)= \gamma_2(\tau)^3= \gamma_3(\tau)^3+ 1728 \] where \(\gamma_2(\tau)\) and \(\gamma_3(\tau)\) lie in a certain ray class field. The paper also investigates other Diophantine equations over class fields, which arise from other modular relations. This produces new (conjecturally) best possible examples in the uniform \(abc\)-conjecture.
scientific article; zbMATH DE number 1443585