Discrepancy results for modular forms

Abstract

Let F be a totally real number field, r = [F : Q], and N be an integral ideal. Let Ak(N, ω) be the space of holomorphic Hilbert cusp forms with respect to K1(N), weight k = (k1, ..., kr) with kj > 2, kj even for all j and central Hecke character ω. For a fixed level N, we study the behavior of the Petersson trace formula for the Hecke operators acting on Ak(N, ω) as k0 → ∞ where k0 = min(k1, ..., kr) subjected to a given condition. We give an asymptotic formula for the Petersson formula under certain conditions. As an application, we generalize a discrepancy result for classical cusp forms with squarefree levels by Jung and Sardari to Hilbert cusp forms for F with the ring of integers O having class number 1, odd narrow class number, and the ideals being generated by numbers belonging to Z.

In the second part, we restrict ourselves to classical cusp forms i.e. when the field is Q. We obtain a generalization for the discrepancy result in the context of levels (of the form 2a × b with b odd, a = 0, 1, 2) and the space of old forms. Then we get a similar kind of lower bound for λp2 (f) for an eigenform f.