## Discriminant of Hecke fields and twisted adjoint $$L$$-values for $$GL(2)$$.(English)Zbl 0924.11035

Let $$f$$ be an elliptic cusp form and let $$\chi$$ be a characteristic given by using Legendre symbol. This paper is considering numerical relations between the special value at $$s=1$$ of the twisted adjoint $$L$$-function $$L(s, \text{Ad}(f)\otimes \chi)$$ and the discriminant of Hecke fields, and gives some generalization. The authors give some conjectures and numerical data supporting them with historical remarks. The conjectures and their consequence are too complicated to summarize here, but they are part of the outcomes of their studies on this theme over 20 years.
{K. Goto [J. Number Theory 73, 34-46 (1998; Zbl 0924.11036 below)] calculated the numerical examples}.
Reviewer: M.Muro (Yanagido)

### MSC:

 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols

Zbl 0924.11036
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