A function on the upper half space which is analogous to the imaginary part of \(\log\eta(z)\). (English) Zbl 0601.10021

The upper half space \(\mathbb H=\mathbb C\times ]0,\infty [\) consists of all Hamiltonian quaternions \(u=z+jv\) where \(z\in\mathbb C\), \(v>0\). The group \(\mathrm{SL}(2, \mathbb C)\) operates on \(\mathbb H\) as \(Mu=(au+b)(cu+d)^{-1}\) for \(M=\left( \begin{matrix} a\\ c\end{matrix} \begin{matrix} b\\ d\end{matrix} \right)\). The analogue of \(\text{Im}(\log\eta(z))\) which is announced in the title is a function \(h\) on \(\mathbb H\). For its definition a lattice \(L\) in \(\mathbb C\) has to be fixed. The role of the modular group is then taken by \(\mathrm{SL}(2,\mathcal O)\) where \(\mathcal O=\{m\in\mathbb C\mid mL\subset L\}\). A main result of the paper is the transformation formula \[ h(Au)=h(u)+\Phi (A) \] for \(u\in\mathbb H\), \(A\in \mathrm{SL}(2,\mathcal O)\). Here, \(\Phi\) is the homomorphism of \(\mathrm{SL}(2,\mathcal O)\) into \(\mathbb C\) which was introduced by R. Sczech [Invent. Math. 76, 523–551 (1984; Zbl 0521.10021)] and whose definition involves an analogue of Dedekind sums.
Another main result is a relation between \(\Phi(A)\) and special values of certain \(L\)-series. In fact, the author proves a more general transformation formula for a function \(h(u;p,q)\), involving parameters \(p,q\in\mathbb C\) and satisfying \(h(u)=h(u;0,0)\). This function is an integral of a differential form on \(\mathbb H\) which is defined by means of certain Eisenstein series.


11F20 Dedekind eta function, Dedekind sums
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols


Zbl 0521.10021
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