Saito–Kurokawa lifts of square-free level

Abstract

Let f ∈ S2κ−2(Γ0(M)) be a Hecke eigenform with κ ≥ 2 even and M ≥ 1 and odd and square-free. In this paper we survey the construction of the Saito-Kurokawa lifting from the classical and representation theoretic point of view. We also provide some arithmetic results on the Fourier coefficients of Saito-Kurokawa liftings. We then calculate the norm of the Saito-Kurokawa lift.

CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINORL-VALUES

“…So far we have only considered forms of level 1. In Section 4, we generalise the results of Sections 2 and 3 to the case of (2) 0 (M)-level, for odd, square-free M. Agarwal and Brown have proved the Hecke eigenvalue congruences analogous to those of Katsurada and Brown in Theorem 1.2, namely Theorem 4.2. While the uniqueness condition (leading to a congruence of Fourier coefficients) may be less practical here, we are motivated by the work of Dickson, Pitale, Saha and Schmidt, showing that in this case the direct analogue of Conjecture 1.1 (i.e.…”

“…Let f = a f (n)q n ∈ S 2k−2 ( 0 (M)) be a normalised newform (of genus 1), with k ≥ 2 even and M odd and square-free. As explained in [2], there is a Saito-Kurokawa lift f ∈ S k ( (2) 0 (M)). It is a Hecke eigenform for T, with μ f (T(p)) = a f (p) + p k−2 + p k−1 for all primes p M. THEOREM 1.13 of [16] is that the following is a consequence of Yifeng Liu's refined Gan-Gross-Prasad conjecture.…”

“…To prove that this factor actually occurs, in the denominator of an algebraic number obtained by dividing L(k − 1, F, spin) by a suitable twisted value L(k − 1, F, spin, χ d ), we need something on central twisted spinor L-values for F. Our main results are conditional on conjectures of Böcherer type, relating these to linear combinations of Fourier coefficients of F. In the remainder of this introduction, we go into considerably more detail, at least in the case of level 1, then briefly summarise the contents of the paper, which cover also odd, square-free (2) 0 (M) level and paramodular level. Let F(Z) = S a(F, S)e 2πitr(SZ) be a Siegel cusp form of genus 2 and weight k for Let F, F be the Petersson norm of F (normalised as in [16]).…”

Hermitian Maass lift for general level

“…Also, we adopt the notation e [x] := e 2πix . Explicitly, one has, for example 2 (N) such that there exists a function α F : Z ≥0 → C so that the Fourier coefficients of F can be described using α F in the following way…”

“…The Saito-Kurokawa lift, established by the work of many authors (classically by Maass [19,20,21], Andrianov [3], Eichler-Zagier [9] for the full level and reinterpreted in representation theoretic language by Piatetski-Shapiro [26]), has been of interest and importance in number theory, for instance, in proving part of the Bloch-Kato conjecture by Skinner-Urban [30], providing evidence for the Bloch-Kato conjecture by Brown [7] and Agarwal-Brown [1]. For these applications, one needs a generalization of the Saito-Kurokawa lift to higher level, which was established by the work of Manickam-Ramakrishnan-Vasudevan [22,23,24], Ibukiyama [11], Agarwal-Brown [2] and Schmidt [28] for square-free level. (Saito-Kurokawa lift is known to exist for all level.…”