The period polynomial r f (z) for an even weight k ≥ 4 newform f ∈ S k (Γ 0 (N)) is the generating function for the critical values of L(f , s). It has a functional equation relating r f (z) to r f (− 1 Nz ). We prove the Riemann hypothesis for these polynomials: that the zeros of r f (z) lie on the circle jzj = 1= ffiffiffiffi N p . We prove that these zeros are equidistributed when either k or N is large. Λðf , sÞ = N s=2
Suppose that O L is the ring of integers of a number field L, and suppose that(note: q := e 2πiz ) is a normalized Hecke eigenform for SL 2 (Z). We say that f is non-ordinary at a prime p if there is a prime ideal p ⊂ O L above p for which a f (p) ≡ 0 (mod p).For any finite set of primes S, we prove that there are normalized Hecke eigenforms which are non-ordinary for each p ∈ S. The proof is elementary and follows from a generalization of work of Choie, Kohnen and the third author [1].
Asai and Friedberg studied the imaginary Doi-Naganuma lifting which sends elliptic modular forms to automorphic forms over an imaginary quadratic field. In this paper we extend this lifting to weak Maass forms by using regularized integral. We construct an automorphic object with singularities on the quadratic upper half-plane H 1 by the regularized theta lifting of a weak Maass form. We also give the convergence region and describe its singularity type. Finally we compute the Fourier coefficients of the lifted form explicitly and present the case of Poincaré series as an example.
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