Arithmeticity of discrete subgroups and automorphic forms (Q1266185)

The authors have obtained a very interesting result which establishes a connection between 1) the arithmeticity of a noncompact lattice \(\Gamma\) in the group \(PSL_2(\mathbb{R})\) and 2) the existence of a cuspidal automorphic form \(f\) in \(L^2 [\Gamma\setminus G]\), whose Dirichlet series \(L(s,f)\) has a local factor at a finite prime \(p\). To be precise, let \(G\) be a connected linear semisimple Lie group. Define a noncompact lattice \(\Gamma\) of \(G\) (i.e., a discrete subgroup of \(G\) of finite covolume such that \(\Gamma\setminus G\) is not compact) to be arithmetic if there exists a semisimple algebraic group \({\mathbf G}\) over \(Q\) such that \({\mathbf G}(\mathbb{R})= G\) up to connected components and \(\Gamma\) is commensurate with \({\mathbf G}(\mathbb{Z})\). [In the case of cocompact \(\Gamma\), the equality \(G(\mathbb{R})= G\) holds only up to compact factors.] Now specialise to \(G= SL_2 (\mathbb{R})\) and let \(\Gamma\subset G\) be any noncompact lattice. One knows that \(\Gamma\) has unipotent elements. After a conjugation in \(G\), one may assume that \(\Gamma\) contains elements of the form \[ \left\{ \begin{pmatrix} 1&m\lambda\\ 0&1\end{pmatrix},\;m\in\mathbb{Z}\right\} \quad\text{and}\quad \left\{ \begin{pmatrix} 1&0\\ n\lambda&1 \end{pmatrix},\;n\in\mathbb{Z} \right\} \] for some \(\lambda\in \mathbb{R}\). For a given prime \(p\), consider the Hecke operators \(T_p\) at \(p\), which act on \(K\)-finite elements \(f\) of a cuspidal representation \(\pi\subset L^2[\subset \Gamma\setminus G]\), as follows: \[ (T_pf)(g)= \frac 1p \left\{ \sum_{i=0}^{p-1} f\left[ \begin{pmatrix} 1&i\lambda\\ 0&1\end{pmatrix} g\right]+ f\begin{pmatrix} p&0\\ 0&1\end{pmatrix} g\right\}. \] Using the Whittaker model of \(\pi\), the authors define an \(L\)-function of \(f\) (this is an extension of the classical \(L\)-function of a cusp form on a congruence subgroup of \(SL_2(\mathbb{Z})\)). The authors prove (1) If \(f\) is a \(K\)-eigenvector and is an eigenform for \(T_p\) for some prime \(p\) with \((T_pf)=\lambda_pf\), then the \(L\)-function \(L(s,f)\) has a local \(p\)-factor of the form \((1- \frac{\lambda_p}{p^s}+ \frac{p}{p^{2s}})^{-1}\). (2) \(\Gamma\) is arithmetic if and only if there exists a cusp form \(f\) on \(L^2[\Gamma\setminus G]\) which is an eigenform for \(T_p\) for some prime \(p\).