On the change of root numbers under twisting and applications (Q2839346)

Let \(f\) be a cuspidal modular form of level \(N\), weight \(k\) and Nebentypus \(\chi\), which is an eigenform for Hecke operators. Passing to the adèlic setting, any such \(f\) gives rise to an irreducible cuspidal automorphic representation \(\pi(f)\) of \(\mathrm{GL}_2(\mathbb{A})\), where \(\mathbb{A}\) is the ring of adèles of \(\mathbb{Q}\). At the real place the local component of \(\pi(f)\) is a discrete series representation of \(\mathrm{GL}_2(\mathbb{R})\) determined by the weight of \(f\). At the primes \(p\) not dividing the level \(N\), the local component of \(\pi(f)\) is an unramified representation of \(\mathrm{GL}_2(\mathbb{Q}_p)\) with Satake parameters determined by the weight, Nebentypus and \(p\)th Fourier coefficient of \(f\). The paper under review is concerned with determining the type of local components of \(\pi(f)\) at the primes dividing \(N\), that is, whether it is principal series, Steinberg or supercuspidal representation. Additional assumption made throughout the paper is that the Nebentypus \(\chi\) of \(f\) is trivial.NEWLINENEWLINENEWLINEThe method considers the root number attached to \(f\), that is, the factor appearing in the functional equation for the standard \(L\)-function attached to \(f\). More precisely, the authors study how the root number changes under twisting by appropriate quadratic Hecke characters. The computations are made in terms of the corresponding two-dimensional representations of the Weil-Deligne group of \(\mathbb{Q}_p\), as in [\textit{H. Carayol}, Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 409--468 (1986; Zbl 0616.10025)]. It turns out that for odd primes \(p\), assuming that \(f\) is of trivial Nebentypus, the change of the root number under twisting determines the type of the representation attached to \(f\). However, for \(p=2\) the twisting is not enough, and some additional information is required.NEWLINENEWLINEThe same problem for \(f\) of arbitrary Nebentypus was recently also considered by \textit{D. Loeffler} and \textit{J. Weinstein} [Math. Comput. 81, No. 278, 1179--1200 (2012; Zbl 1332.11056)], using a different approach, which provides explicit local data for \(\pi(f)\) at \(p\).