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Upper bounds on \(| L(1,\chi)|\) and applications. (English) Zbl 0912.11046

Let \(K/k\) be an abelian extension of number fields with conductor \(f\), and \(\chi\) a non-trivial character of its Galois group. In a recent paper, the author derived the upper bound \[ | L(1,\chi)| \leq R_k ((\log f)/2 + 2\mu_k), \] where \(R_k\) is the residue of Dedekind’s zeta function of \(k\) at \(s=1\) and \(\mu_k\) some constant depending on \(k\). This paper presents a new proof for this result, and it generalizes the previously known bound \(R_k\mu_k \leq \frac 18 \log^2 d_k\) for real quadratic fields with discriminant \(d_k\) to totally real number fields \(k\) by proving that \[ R_k\mu_k \leq (\log^n d_k)/(2^n n!) \] for large enough discriminants \(d_k\). Moreover, techniques for computing the product \(R_k\mu_k\) efficiently in concrete cases are explained. In order to demonstrate the improvement on earlier results, several applications to class number one problems are given. Some of these results were used recently to show that there is only one CM-field with Galois group \(\text{SL}_2(\mathbb F_3)\) and class number \(1\), namely the second Hilbert class field in the strict sense of the cyclic cubic field with conductor \(163\).

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11R29 Class numbers, class groups, discriminants
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