There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations 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 $g\in S_{2k}$ is bounded by $k^{1-\frac{1}{210}+ε}$. This improves the previously known bound $k$ for this quantity.
We prove a result on the distribution of Hecke eigenvalues, $μ_F(p^r)$ (for $r=1,2$ or $3$) of a non Saito--Kurokawa lift $F$ of degree $2$. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an $F$, which is the best possible in terms of orders of magnitude.
We prove an explicit central value formula for a family of complex $L$-series of degree $6$ for $\mathrm{GL}_2 \times \mathrm{GL}_3$ 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.
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 pull…
A new- and old-form theory for Bessel periods of Saito-Kurokawa representations is given. We introduce arithmetic subgroups so that a local Bessel vector fixed by the subgroup indexed by the conductor of the representation is unique up to scalars. The global Langlands L-function of a holomorphic Saito-Kurokawa representation coincides with a canonically settled Piatetski-Shapiro zeta integral of the newform.
We build a one-variable $p$-adic $L$-function attached to two Hida families of ordinary $p$-stabilised newforms $\mathbf{f}$, $\mathbf{g}$, interpolating the algebraic part of the central values of the complex $L$-series $L(f \otimes \textrm{Ad}(g), s)$ when $f$ and $g$ range over the classical specialisations of $\mathbf{f}$, $\mathbf{g}$ 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 previo…
We consider a special theta lift $θ(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $θ(f)$ 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 $θ(f)$ to those of $f$, and in particular prove that $θ(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $θ(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily…
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 $\mathrm{S}_{2k-2}(Γ_0(N))$ ordinary at a prime $p\nmid N$ with sign $ε_f=-1$ and write $α$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $π_α$ 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,…
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 representat…
We prove (under certain assumptions) the irreducibility of the limit $σ_2$ of a sequence of irreducible essentially self-dual Galois representations $σ_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$ approaches 2 in a $p$-adic sense) which mod $p$ reduce (after semi-simplifying) to $1 \oplus ρ\oplus χ$ with $ρ$ irreducible, two-dimensional of determinant $χ$, where $χ$ is the mod $p$ cyclotomic character. More precisely, we assume that $σ_k$ are crystalline (with a particular choice of weights) and Siegel-ordinary at $p$. Such representations arise in the study of $p$-adi…
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 asym…
We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\barσ$ with $\barσ^{\rm ss}=χ^{k-2} \oplus ρ\oplus χ^{k-1}$, where $χ$ is the mod $p$ cyclotomic character and $ρ$ is an absolutely irreducible reduction of the Galois representation $ρ_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f(\mathbf{Q}, ρ_f(1-k)\otimes \mathbf{Q}_p/\mathbf{Z}_p)$ and $H^1_f(\mathbf{Q}, ρ(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\barσ$ t…
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 $Γ_0(N)$. We finally obtain for all prime numbers $p$ local Maass relations. This generalizes known results for level $N=1$.
We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce new CAP representations of metaplectic, symplectic and orthogonal groups. These constructions partially generalize the theories of Waldspurger on the Shimura correspondence and of Piatetski-Shapiro on the Saito-Kurokawa lifting to higher dimensions. Applications include a relation between Fourier coefficients of Hilbert cusp forms of weight k+1/2 and a weighted sum of the representation numbers of a quadratic form of rank 2k by a quadratic form of rank 4k.
We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito-Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelof hypothesis for the corresponding Koecher-Maass series. The ingredients include a new relative trace formula for pairs of Heegner periods.
The twisted elliptic genera of a $K3$ surface associated with the conjugacy classes of the Mathieu group $M_{24}$ are known to be weak Jacobi forms of weight $0$. In 2010, Cheng constructed formal infinite products from the twisted elliptic genera and conjectured that they define Siegel modular forms of degree two. In this paper we prove that for each conjugacy class of level $N_g$ the associated product is a meromorphic Borcherds product on the lattice $U(N_g)\oplus U \oplus A_1$ in a strict sense. We also compute the divisors of these products and determine for which conjugacy classes the pr…
Kyoto University（京都大学）；Tokyo Denki University（东京电机大学）；Osaka University（大阪大学）；Hokkaido University（北海道大学）；Muroran Institute of Technology（室兰工业大学）；Tohoku University（东北大学）
