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 $p$-adic families of Siegel modular forms and properties of their limits as $k\to 2$ appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by $p$-adic $L$-values of an elliptic modular form (giving rise to $ρ$) which we assume are non-zero.
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 study the analytic behavior of the restriction of a Siegel modular form to $\mathbb{H} \times \mathbb{H}$ in the case that the Siegel form is a Saito-Kurokawa lift. A formula of Ichino links this behavior to a family of $GL_3 \times GL_2$ $L$-functions.
Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer gro…
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…
Let $E_{/_\Q}$ be an elliptic curve of conductor $Np$ with $p\nmid N$ and let $f$ be its associated newform of weight 2. Denote by $f_\infty$ the $p$-adic Hida family passing though $f$, and by $F_\infty$ its $Λ$-adic Saito-Kurokawa lift. The $p$-adic family $F_\infty$ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde A_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$. We relate explicitly certain global points on $E$ (coming from the theory of Stark-Heegne…
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 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…
We characterize the irreducible, admissible, spherical representations of 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 term…
In this paper we prove a general theorem about congruences between automorphic forms on a reductive group G which is compact at infinity modulo the center. If the rank is one, this essentially reduces to Ribet's level-raising theorem. We then specialize to the higher rank case where G is an inner form of GSp(4). Here we get congruences with automorphic forms having a generic local component. In particular, a Saito-Kurokawa form is congruent to a form which is not of Saito-Kurokawa type. We get similar results for U(3) at split primes.
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 $N=1$.
A work of Sorensen is rewritten here to include nontrivial types at the infinite places. This extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{\times}, where D is a definite quaternion algebra over a totally real field F. This is done for any automorphic representations πof an arbitrary reductive group G over F which is compact at infinity. It is not assumed that π_\infty is trivial. If λis a finite place of \bar{\Q}, and w is a place where π_w is unramified and π_w is congruent to the trivial representation mod λ, then under some mild additional ass…
The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…
