On the change of root numbers under twisting and applications

Abstract

Abstract. The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime p, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at p by analyzing the change of sign under a suitable twist. We also explain the case p = 2, where twisting is not enough in general.

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

“…We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti [16]. Our method however is completely different from that of Pacetti and we use representation theory crucially.…”

“…Applying the above lemma when K|Q p is ramified quadratic, we see that if (K, χ) is a minimal admissible pair, then m is odd and so a(χ) = l(χ) + 1 is even [15, Theorem 3.3]. We can find the valuation of the level of the modular form with arbitrary nebentypus from the following proposition (see also [16,Corollary 3.1] for Γ 0 (N )). The local factor in the ramified case can be computed from that.…”

“…Pacetti [16] studied the variance of local root numbers in the context of twisting by a quadratic character for modular forms with trivial nebentypus and determined the type of local automorphic representations at p as an application. In this article, we explore the same and investigate what properties of modular forms with arbitrary nebentypus are encoded therein.…”

Heegner Points on Cartan Non-split Curves

“…(1) We compute the local type at each prime dividing N. This can be done either by looking at the reduced curve and the field where it gets semi-stable reduction or by considering twists, as in [Pac13]. Using the local type information, we compute the newforms of smaller level that appear in Theorem 2.2.…”

On Heegner points for primes of additive reduction ramifying in the base field

“…If we impose the extra condition gcd(cond(χ), N cond(η)) = 1, then at primes dividing the conductor of E/K the equation becomes ε p (E/K) = η p (−1), where ε p (E/K) is the local root number at p of the base change of E to K (it is equal to ε p (E)ε p (E ⊗ η)). This root number is easy to compute if p = 2, 3 (see [Pac13]):…”