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

The Saito-Kurokawa lifting and Darmon points

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…

Newforms of Saito-Kurokawa lifts

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.

On $p$-adic $L$-functions for $\operatorname{GL}(2)\times\operatorname{GL}(3)$ 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 $\mathbf{f}$, $\mathbf{g}$, 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…

A quaternionic Saito-Kurokawa lift and cusp forms on $G_2$

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…

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 $\mathrm{S}_{2k-2}(Γ_0(N))$ 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,…

Saito-Kurokawa提升的质量等分布

Mass equidistribution for Saito-Kurokawa lifts

本文在广义黎曼假设（GRH）下，证明了Saito–Kurokawa型全纯尖点Hecke特征型的质量随权趋于无穷在Siegel模簇上等分布，并推出其零除子也满足等分布性质。

Maass relations for Saito-Kurokawa 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 representat…

Irreducibility of limits of Galois representations of Saito-Kurokawa type

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…

新的和旧的Saito-Kurokawa提升通过L^2范数和其上范数的界：水平方面

New and old Saito-Kurokawa lifts classically via $L^2$ norms and bounds on their supnorms: level aspect

本文通过L^2范数和上范数界研究Saito-Kurokawa提升，结合新旧形式理论，提出关于平方自由水平空间的L^∞大小猜想并给出界证。

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 asym…

Deformations of Saito-Kurokawa type and the Paramodular Conjecture (with an appendix by Cris Poor, Jerry Shurman, and David S. Yuen)

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…

无平方因子水平的 Saito-Kurokawa 提升的回拉、其非零性与 $L^2$-质量

Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass

本文研究奇数无平方因子水平 SK 新形式回拉的谱分解，将投影平方联系于 GL(3)×GL(2) 中心 L-值，并利用 CFKRS 启发法给出 L2-质量公式及非零性条件。

Local and global Maass relations (expanded version)

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…

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 $Γ_0(N)$. We finally obtain for all prime numbers $p$ local Maass relations. This generalizes known results for level $N=1$.

Richelot-Brandt图的结构迹恒等式与认证谱

The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra

本文证明Richelot-Brandt图相关的$R(\pi)$迹的结构公式，提出主亏格重数猜想的符号精细化版本，经精确算术认证验证了11至149之间所有素数的该预测。

2次西格尔尖点形式的傅里叶系数界与全局上确界范数

Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2

假设广义黎曼假设，针对非斋藤-黑川提升的权$k$、$L^2$标准化Hecke特征2次西格尔尖点形式，证明其傅里叶系数和全局上确界范数的界，前者依赖于精细全局Gan-Gross-Prasad猜想相关局部积分的新界。

超几何4流形上提升的质数等分布

Mass equidistribution for lifts on hyperbolic $4$-manifolds

本文证明了超几何4流形上Pitale提升序列的量子唯一混浊性猜想，通过构造具有有利几何性质的放大器，首次成功应用于逃离非 tempered 子群的情形。