Summation methods and distribution of eigenvalues of Hecke operators
Let \(S_k(N)\) be the space of cusp forms of level \(N\) and even weight \(k\), and let \(d_k(N)\) be its dimension. For primes \(p\) and \(1\leq i\leq d_k(N)\), consider the eigenvalues \(\alpha_{p,i}\) of the normalized \(p\)th Hecke operator on \(S_k(N)\). Then \(-2\leq\alpha_{p,i}\leq 2\), according to Deligne's theorem. The Sato-Tate conjecture predicts that, if \(N\), \(k\) are fixed and \(p\) varies, the values \(\alpha_{p,i}= 2\cos\theta_{p,i}\) are equidistributed in \([-2,2]\) with respect to the measure \({1\over\pi} \sqrt{1-{x^2\over 3}}\,dx\). \textit{J.-P. Serre} [J. Am. Math. Soc. 10, No. 1, 75--102 (1997; Zbl 0871.11032)] proved that, when \(p\) is fixed and \(N\), \(k\) varies over infinite families such that \(N+ k\to\infty\), the values \(\alpha_{p,i}\) are equidistributed with respect to a measure which depends on \(p\). In the paper under review the authors construct an infinite matrix \(A= (a_{N,n})\) with the property that for fixed \(p\) the eigenvalues \(\alpha_{p,i}\) are \(A\)-equidistributed with respect to the measure \(\mu={1\over 2\pi}(1-{x^2\over 4})^{-1/2}dx\) which is in dependent from \(p\). Here a sequence of numbers \(x_n\) in \([-2, 2]\) is said to be \(A\)-equidistributed with respect to \(\mu\) if \[ \lim_{N\to\infty} \sum^\infty_{n=1} a_{N,n}f(x_n)= \int^2_{-2} f(x)\,d\mu \] for all real-valued continuous functions \(f\) on \([-2, 2]\). As a main tool, the authors use a Petersson trace formula which gives an asymptotic expression for \(\sum_i \alpha_{p,i}\) as \(N\to\infty\).
scientific article; zbMATH DE number 5635547