Modular \(L\)-values of cubic level
Let \(\chi\) be a fixed primitive Dirichlet character modulo \(D\) and \(N\) a squarefree integer coprime to \(D\). The paper studies certain averages of \(\chi\)-twisted \(L\)-values over (i) even Maass newforms on \(\Gamma_0(N^3)\) of weight \(k=0\) with trivial central character, or (ii) holomorphic newforms on \(\Gamma_0(N^3)\) of weight \(k > 2\) whose central character satisfies \(\omega'(-1)=(-1)^k\), as \(N \to \infty\). These results correspond to Theorem 5.4 and Theorem 4.1, respectively. In the Maass case there is even an exact formula. There are two ingredients in their approach. The global one is Jacquet's relative trace formula for \(\mathrm{GL}(2)\) with respect to its spherical subgroups \(N\) and \(M\), against suitable characters. Here \(N\) is the upper triangular unipotent subgroup and \(M\) is the diagonal torus \(\mathrm{diag}(\cdot, 1)\). Integration over \(N\) produces Fourier coefficients \(a_r(u)\), whereas a classical result by Hecke asserts that integration over \(M\) produces \(L\)-values. In order to exploit the relative trace formula, one chooses suitable test functions to project to some given spectrum \(A_k(\hat{\sigma}, \omega)\), then one computes the relevant terms in the trace formula. The local ingredient concerns the choice of local data (supercuspidal representations) at \(p\)-adic places. Such representations have explicit matrix coefficients (in terms of Kloosterman sums!) at least for \(\mathrm{GL}(2)\). The paper contains many details for the relevant explicit computations.
scientific article; zbMATH DE number 6136442