Granville, Andrew; Stark, H. M. \(ABC\) implies no “Siegel zeros” for \(L\)-functions of characters with negative discriminant. (English) Zbl 0967.11033 Invent. Math. 139, No. 3, 509-523 (2000). 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. Reviewer: Roger Heath-Brown (Oxford) Cited in 1 ReviewCited in 20 Documents MSC: 11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\) 11D25 Cubic and quartic Diophantine equations 11R42 Zeta functions and \(L\)-functions of number fields 11F11 Holomorphic modular forms of integral weight Keywords:class field; Dirichlet \(L\)-function; real zero; exceptional zero; imaginary quadratic field; \(abc\)-conjecture Citations:Zbl 0609.14011 PDFBibTeX XMLCite \textit{A. Granville} and \textit{H. M. Stark}, Invent. Math. 139, No. 3, 509--523 (2000; Zbl 0967.11033) Full Text: DOI