Simple zeros of degree 2 \(L\)-functions (Q277531)

The purpose of this paper is to prove that the complete \(L\)-function of a normalized holomorphic cuspidal new Hecke eigenform on the upper half-plane, of arbitrary weight, level and Nebentypus, has infinitely many simple zeros.NEWLINENEWLINEThis was previously known only for the Ramanujan \(\Delta\) cusp form, by the work of \textit{J. B. Conrey} and \textit{A. Ghosh} [Invent. Math. 94, No. 2, 403--419 (1988; Zbl 0653.10038)]. Their result for \(\Delta\) is a bit stronger, as it provides an asymptotic lower bound in \(T\) for the number of simple zeroes with the imaginary part in \([0,T]\). A similar asymptotic lower bounds for the case considered in this paper is also obtained in the concurrent work [\textit{M. B. Milinovich} and \textit{N. Ng}, Proc. Lond. Math. Soc. (3) 109, No. 6, 1465--1506 (2014; Zbl 1318.11115)], but assuming the grand Riemann hypothesis. In a recent paper [\textit{P. J. Cho}, Int. J. Number Theory 9, No. 1, 167--178 (2013; Zbl 1275.11120)], it was proved, generalizing [\textit{J. B. Conrey} and \textit{A. Ghosh}, Invent. Math. 94, No. 2, 403--419 (1988; Zbl 0653.10038)], that the \(L\)-functions of the first few Maaß cusp forms of level one have infinitely many simple zeros.NEWLINENEWLINEThe idea of the proof is to consider certain (twisted) Dirichlet series, which has certain analytic properties under the assumption that the \(L\)-function of the considered holomorphic cusp form has finitely many simple zeros. These analytic properties lead to a contradiction, unless the local \(L\)-function on every unramified prime is a square, which cannot happen for holomorphic cusp forms.