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Transcendental values of class group \(L\)-functions. (English) Zbl 1281.11071
In this article, the authors consider the question of linear independence of class group \(L\)-functions. Let \(K\) be an algebraic number field and \(f\) a complex-valued function of the ideal class group \({\mathfrak H}_K\) of \(K\). Here the authors consider the Dirichlet series \[ L(s,f) := \sum_{\mathfrak a} \frac{f({\mathfrak a})}{{\mathbf N}(\mathfrak a )^s}, \] where the summation is over all integral ideals \(\mathfrak a\) of the ring of integers \(\mathcal{O_K}\) of \(K\). If \(f\) is identically 1, then \(L(s,f)\) is the Dedekind zeta function of \(K\). If \(f\) is a character \(\chi\) of the ideal class group \({\mathfrak H}_K\) of \(K\), then, \(L(s,\chi)\) is a Hecke \(L\)-function. The goal of the authors is to investigate special values of \(L(s,f)\) at \(s=1\) when \(K\) is an imaginary quadratic field and \(f\) takes algebraic values.

In particular, they investigate the transcendental nature of \(L(1,\chi)\) when \(\chi\) is an ideal class character and \(K\) is imaginary quadratic and prove the following beautiful theorem:

Let \(K\) be an imaginary quadratic field and \({\mathfrak H}_K\) its ideal class group. The values \(L(1,\chi)\), as \(\chi\) ranges over the non-trivial characters of the class group (modulo the action of complex conjugation), and \(\pi\) are linearly independent over \(\overline{\mathbb Q}\).

The basic tools are Kronecker’s limit formula and Baker’s theory of linear forms in logarithms.

In the second part of the paper, they derive an upper bound of the number of non-trivial characters \(\chi \!\!\pmod{q}\) for which \(L'(1,\chi)=0\). Conjecturally, \(L'(1,\chi)\) is never equal to zero which is a deep arithmetic question.

11J81 Transcendence (general theory)
11J86 Linear forms in logarithms; Baker’s method
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11R42 Zeta functions and \(L\)-functions of number fields
11R47 Other analytic theory
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