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Pramath Anamby

Publications and source records attributed to Pramath Anamby.

10 recordsLinked to original sources
arXiv2026

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

In the first half of the paper, we lay down a classical approach to the study of Saito-Kurokawa (SK) lifts of (Hecke congruence) square-free level, including the allied new-oldform theory. Our treatment of this relies on a novel idea of computing ranks of certain matrices whose entries are $L^2$-norms of eigenforms. For computing the $L^2$ norms we work with the Hecke algebra of $\mathrm{GSp}(2)$. In the second half, we formulate precise conjectures on the $L^\infty$ size of the space of SK lifts of square-free level, measured by the supremum of its Bergman kernel, and prove bounds towards them using the results from the first half. Here we rely on counting points on lattices, and on the geometric side of the Bergman kernels of spaces of Jacobi forms underlying the SK lifts. Along the way, we prove a non-trivial bound for the sup-norm of a Jacobi newform of square-free level and also discuss about their size on average.

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arXiv2026

Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels

We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $Γ_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$ has a non-zero Fourier coefficient $A(F,T_0)$ such that $(T_0(n,n),N)=1$. Then there exist infinitely many odd and square-free (and thus fundamental) integers $m$ such that $m=\mathrm{discriminant}(T)$ and $A(F,T)\neq 0$. In the case of odd degrees, we prove a stronger result by replacing odd and square-free with odd and prime. We also prove quantitative results in this direction. As a consequence, we can show in particular that the statement of the main result in arXiv:2408.03442 about the algebraicity of certain critical values (and the expected functional equation) of the spinor $L$-functions of holomorphic newforms (in the ambit of Deligne's conjectures) on congruence subgroups of $\mathrm{GSp}(3)$ is unconditional.

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arXiv2025

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

We obtain the full spectral decomposition of the pullback of a Saito-Kurokawa (SK) newform $F$ of odd, square-free level; and show that the projections onto the elements $\mathbf g \otimes \mathbf g$ of an arithmetically orthogonalized old-basis are either zero or whose squares are given by the certain $\mathrm{GL}(3)\times \mathrm{GL}(2)$ central $L$-values $L(f\otimes \mathrm{sym}^2 g, \frac{1}{2})$, where $F$ is the lift of the $\mathrm{GL}(2)$ newform $f$ and $g$ is the newform underlying $\mathbf g$. Based on this, we work out a conjectural formula for the $L^2$-mass of the pullback of $F$ via the CFKRS heuristics, which becomes a weighted average (over $g$) of the central $L$-values. We show that on average over $f$, the main term predicted by the above heuristics matches with the actual main term. We also provide several results and sufficient conditions that ensure the non-vanishing of the pullbacks.

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arXiv2024

Determination of a pair of newforms from the product of their twisted central values

We show that a pair of newforms $(f,g)$ can be uniquely determined by the product of the central $L$-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted $L$-functions- $L(1/2, f \times χ)L(1/2, g \times χψ)$, where $(f,g)$ is a pair of newforms. The average is taken over the primitive Dirichlet characters $χ$ and $ψ$ of distinct prime moduli.

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arXiv2023

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 $L(1/2, f \otimes χ_D)$ with $f$ varying over newforms of level a prime $p$ and even weight $k$ as $k,p \to \infty$ and $D$ being (explicitly) polynomially bounded by $k,p$. Here $χ_D$ 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 $k^3$.

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arXiv2022

Non-vanishing of theta components of Jacobi forms with level and an application

We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_μ$ with either $(μ,2m_1m_2)=1$ or $(μ,2m_1m_2)\nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.

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arXiv2020

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

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.

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arXiv2020

Hecke-Siegel type threshold for square-free Fourier coefficients: an improvement

We prove that if $f$ is a non zero cusp form of weight $k$ on $Γ_0(N)$ with character $χ$ such that $N/(\text{conductor }χ)$ square-free, then there exists a square-free $n\ll_ε k^{3+ε}N^{7/2+ε}$ such that $a(f,n)\neq 0$. This significantly improves the already known existential and quantitative result from previous works.

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arXiv2020

Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients

We prove that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which are also treated in this paper.

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