Mass equidistribution for Hecke eigenforms. (English) Zbl 1211.11050

Holowinsky, Roman; Soundararajan, Kannan

Ann. Math. (2) 172, No. 2, 1517-1528 (2010).

Let \(f\) be a Hecke eigencuspform of weight \(k\) for \(SL_2({\mathbb Z})\) and \(F_k(z):=y^{\frac{k}{2}}f(z),z=x+iy\). Let \(\phi\) be a Maass cusp form which is also an eigenfunction of all Hecke operators. The authors prove that \[ |\langle\phi F_k,F_k\rangle|\ll_{\phi,\varepsilon}(\log k)^{\frac{-1}{30}+\varepsilon}, \] where \(\langle\cdot,\cdot\rangle\) is the Petersson scalar product. Let \(\psi\) be a fixed smooth function compactly supported in \((0,\infty)\). Then \[ |\langle E(\cdot\mid \psi)F_k,F_k\rangle-\frac{3}{\pi}\langle E(\cdot\mid \psi),1\rangle|\ll_{\phi,\varepsilon}(\log k)^{\frac{-2}{15}+\varepsilon}, \] where \(E(z\mid \psi)\) is the incomplete Eisenstein series associated to \(\psi\).

Reviewer: Florin Nicolae (Berlin)

MSC:

11F11	Holomorphic modular forms of integral weight
11F25	Hecke-Petersson operators, differential operators (one variable)

Keywords:

Hecke eigenform; Maass cusp form; equidistribution