Selberg trace formula for odd weight. I, II (Q1119961)

There are five paragraphs in the note, the first four are contained in Part I, the last one in Part II. Let \(\Gamma\) be a Fuchsian group of the first kind which does not contain -I. The first aim of the note is to rewrite the Selberg trace formula for odd weight and the group \(\Gamma\) in a form clarifying the difference between the contributions of regular and irregular cusps. A new Eisenstein series attached to each \(\Gamma\)-equivalent class of cusps is introduced. The formula is too long to reproduce here. The second aim is to give a dimension formula for weight one, using the residue of the Selberg type zeta function which has a functional equation. This improves the result obtained by the author, T. Hiramatsu and Y. Tanigama [see \textit{T. Hiramatsu}, J. Reine Angew. Math. 402, 166- 180 (1989; see the following review)], where the zeta function used there has no functional equation.