Saito-Kurokawa Lifts Research Articles

Mass Equidistribution for Saito-Kurokawa Lifts

Let F be a holomorphic cuspidal Hecke eigenform for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{Sp}_{4}({\\mathbb{Z}})$\\end{document} of weight k that is a Saito–Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of F equidistributes on the Siegel modular variety as k⟶∞. As a corollary, we show under GRH that the zero divisors of Saito–Kurokawa lifts equidistribute as their weights tend to infinity.

On newforms and Saito-Kurokawa lifts

On newforms and Saito-Kurokawa lifts

On p-adic L-functions for GL(2)×GL(3) via pullbacks of Saito–Kurokawa lifts

We build a one-variable p-adic L-function attached to two Hida families of ordinary p-stabilised newforms f, g, interpolating the algebraic part of the central values of the complex L-series L(f⊗Ad0(g),s) when f and g range over the classical specialisations of f, g on a suitable line of the weight space. The construction rests on two major results: an explicit formula for the relevant complex central L-values, and the existence of non-trivial Λ-adic Shintani liftings and Saito–Kurokawa liftings studied in a previous work by the authors. We also illustrate that, under an appropriate sign assumption, this p-adic L-function arises as a factor of a triple product p-adic L-function attached to f, g, and g.

Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average

We formulate a precise conjecture about the size of the $$L^\infty $$ -mass of the space of Jacobi forms on $$\mathbb {H}_n \times \mathbb C^{g \times n}$$ of matrix index S of size g. This $$L^\infty $$ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when $$n=1$$ , $$g \ge 1$$ . When $$n=1$$ and $$g=1$$ , we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger’s formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values $$L(1/2, f \otimes \chi _D)$$ with f varying over newforms of level a prime p and even weight k as $$k,p \rightarrow \infty $$ and D being (explicitly) polynomially bounded by k, p. Here $$\chi _D$$ is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is $$k^{5/2}$$ by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded $$\mathbb {H}\times \mathbb {H}$$ is $$k^2$$ . In an appendix, the same question is answered for the pullbacks of the whole space $$S^2_k$$ , the size here being $$k^3$$ .

Comparing Hecke eigenvalues of Siegel eigenforms

Abstract This article deals with various kinds of quantitative results about the comparison between the normalized Hecke eigenvalues of two distinct Siegel cuspidal Hecke eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts. We also prove some simultaneous sign change results for their eigenvalues.

On Siegel eigenvarieties at Saito–Kurokawa points

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(\Gamma_0(N))$ ordinary at a prime $p\nmid N$ with sign $\epsilon_f=-1$ and write $\alpha$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $\pi_\alpha$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))$ attached to $f$ is at most one and some mild assumptions on the automorphic representation attached to $f$, we show that $\mathcal{E}$ is smooth at the point corresponding to $\pi_\alpha$, and that the irreducible component of $\mathcal{E}$ specializing to $\pi_\alpha$ is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of $\mathcal{E}$ at $\pi_\alpha$ yields that $\dim H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))\geq 2$.

Some Remarks on the Saito Kurokawa Lift for Orthogonal Forms

Duke and Imamoglu showed that the Saito–Kurokawa lift for Siegel modular forms can in an elegant manner be obtained from a converse theorem by Imai using spectral analysis of the hyperbolic Laplacian. In an earlier paper we gave a simplified approach without any appeal to spectral analysis. Here we want to show, that this generalizes to some orthogonal groups of signature (2,m+2) with m>2.

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

AbstractFollowing Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinorL-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

A note on arithmetic behavior of Hecke eigenvalues of Siegel cusp forms of degree two

Let [Formula: see text] and [Formula: see text] be Siegel cusp forms for the group [Formula: see text] with weights [Formula: see text], [Formula: see text], respectively. Suppose that neither [Formula: see text] nor [Formula: see text] is a Saito–Kurokawa lift. Further suppose that [Formula: see text] and [Formula: see text] are Hecke eigenforms lying in distinct eigenspaces. In this paper, we investigate simultaneous arithmetic behavior and related problems of Hecke eigenvalues of these Hecke eigenforms, some of which improve upon results of Gun et al.

A quaternionic Saito–Kurokawa lift and cusp forms on G2

We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\theta(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $\theta(f)$ to those of $f$, and in particular prove that $\theta(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $\theta(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily large weight with all algebraic Fourier coefficients. In the case of level one, we obtain precise formulas for the Fourier coefficients of $\theta(f)$ in terms of those of $f$. In particular, we construct nonzero cuspidal modular forms on $G_2$ of level one with all integer Fourier coefficients.

CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

AbstractIt has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

Maass relations for Saito\u2013Kurokawa lifts of higher levels

It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms of level 1. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for mathrm {GSp}_4 is predicted to fail only in case of CAP representations. It is not known though whether the associated Siegel modular forms (of various levels) still satisfy a version of Maass relations. We show that this is indeed the case for the ones related to P-CAP representations. Our method generalises an approach of Pitale, Saha and Schmidt who employed representation–theoretic techniques to (re)prove this statement in case of level 1. In particular, we compute and express certain values of a global Bessel period in terms of Fourier coefficients of the associated Siegel modular form. Moreover, we derive a local–global relation satisfied by Bessel periods, which allows us to combine those computations with a characterization of local components of CAP representations.

Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

Let F and G be Siegel cusp forms for \({\mathrm{Sp}}_4({{\mathbb {Z}}})\) and weights \(k_1, k_2\), respectively. Also let F and G be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither F nor G is a Saito–Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.

Pullback formulae for nearly holomorphic Saito–Kurokawa lifts

We give explicit pullback formulae for nearly holomorphic Saito-Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulae provide non-trivial examples for the refined Gan-Gross-Prasad conjecture for $({\rm SO}_5,{\rm SO}_4)$ in the non-tempered cases. As an application, we obtain Deligne's conjecture for critical values of certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also expect to apply our pullback formulae to construct two-variables $p$-adic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$ in the future.

Pullbacks of Saito-Kurokawa Lifts and a Central Value Formula for Degree 6 $L$-Series

We prove an explicit central value formula for a family of complex L -series of degree 6 for \text{GL}_2 \times \text{GL}_3 which arise as factors of certain Garret-Rankin triple product L -series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito-Kurokawa lifts, and as an application we prove Deligne's conjecture stating the algebraicity of the central values of the considered L -series up to the relevant periods.

Bounds for the Petersson norms of the pullbacks of Saito–Kurokawa lifts

Using the amplification technique, we prove that ‘mass’ of the pullback of the Saito–Kurokawa lift of a Hecke eigen form g∈S2k is bounded by k1−1210+ε. This improves the previously known bound k for this quantity.

Maass Spezialschar of level N

In this paper the image of the Saito–Kurokawa lift of level N with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level N by symmetries involving Hecke operators related to $$\Gamma _0(N)$$ . We finally obtain for all prime numbers p local Maass relations. This generalizes known results for level $$N=1$$ .

The Saito–Kurokawa lift and Siegel’s theta series

The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.

Local and global Maass relations

We characterize the irreducible, admissible, spherical representations of mathrm{GSp}_4(F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito–Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito–Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacobi forms. As an additional application of our methods, we give a new characterization of Saito–Kurokawa lifts involving a certain average of Fourier coefficients.

Fourier–Jacobi periods of classical Saito–Kurokawa lifts

Kohnen–Skoruppa (Invent Math 95(3): 541–558, 1989) proved a formula for the ratio of the Petersson inner products of the half integral weight modular form and its Saito–Kurokawa lifting. We give an interpretation of this formula in the framework of the refined Gan–Gross–Prasad conjecture. This provides us with an example of the refined Gan–Gross–Prasad conjecture for the nontempered representations.