Let \(F\) be a finite extension of degree \(n\) of an imaginary quadratic field \(K\). Deligne has conjectured the algebraicity of special values of \(L\)-functions attached to certain Hecke characters of type \(A_ 0\) of \(F\). These conjectures were proved by Harder. The purpose of this paper is to provide a more elementary, though still very technical, proof of these conjectures under some special, but still fairly general, assumptions.
The main theorem: Let \(\psi\) be a Hecke character of \(K\) whose infinity type is \(\bar z,\) and \(\vartheta\) a Dirichlet character of \(F\). Set \(\phi_ k=\vartheta (\psi^ k\circ N_{F/K})\). Let \(t\) be an integer. Then the value of the \(L\)-function of \(\phi_ k\) at \(t\) divided by \(\omega^{nk}\pi^{n(t-k)}\) is an algebraic number, where \(\omega\) is the real period of an elliptic curve with complex multiplication by \(k\) defined over \(\overline {\mathbb Q}\), when
(i) \(n=2\), \(1\leq t\leq k\), \(F\) arbitrary, or
(ii) \(n\geq 3\), \(t=k\), and the image of the action of \(\mathrm{Gal}(\overline{K}/K)\) on the embeddings of \(F\) in \(\mathbb C\) (lying over a fixed embedding of \(K\) in \(\mathbb C\) contains the alternating group of order \(n\).
An explicit formula for the special values is given, in terms of derivatives of generalized Kronecker-Eisenstein series.
The author’s tools include a refined version of Shintani’s method to find the sum of certain series indexed by lattices of units, and a distribution-theoretic study of the resulting generalized Kronecker-Eisenstein series.