On the Siegel zeros of quadratic fields and their applications
In this presentation, we will discuss the locations of the possible exceptional real zeros of Dirichlet \(L\)-functions. These are known as the Siegel zeros or Landau–Siegel zeros. In 1975, Goldfeld and Schinzel provided absolute effective constants for bounds of such zeros. We obtain explicit upper bounds of Siegel zeros for quadratic fields. This is achieved by studying the structure of class groups together with the uses of the latest known computational results. We also present two natural applications of the explicit bounds, namely an up-to-date version of the zero-free region of Dirichlet L-functions and a result on the estimate of the Euler function of imaginary quadratic fields. In addition, we revisit a result of Clark and Pollack on the size of the torsion subgroup of elliptic curves with complex multiplication over a degree \(d\) number field. In 2015, they showed that this quantity is bounded by \( cd \log \log d\) (\(c\) is an absolute constant and is not specified). We provide the first explicit value of \(c\).