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

Pramath Anamby, Soumya Das

TL;DR

This work develops a classical, lattice-based approach to the Saito–Kurokawa lifting problem for square-free levels, integrating old/newform theory in degree 2 with a detailed analysis of inner products via a 4×4 matrix M_p. It provides a concrete realisation of SK lifts as Eichler–Zagier–Ibukiyama (EZI) lifts and uses Maaß relations to distinguish EZI lifts, while also treating oldforms that are not EZI lifts through an explicit orthonormal basis and W_p-invariant structures. The paper then pivots to the level-aspect sup-norm problem, establishing nontrivial bounds for Jacobi forms of index 1 and analyzing the Bergman kernel for SK lifts by decomposing into Type 0, Type 1, and Type 2 regions, with Fourier, Fourier–Jacobi, and counting-matrix techniques. It culminates in conjectures for the size of SK_k(N) and the sup-norms, supported by lower bounds via Waldspurger-type formulas and average L-values, and provides comprehensive ancillary results including an appendix covering the case n=1. The results advance understanding of SUP-norm phenomena for higher-rank automorphic forms and illuminate the interplay between Jacobi forms, half-integral weight forms, and Siegel modular forms in the level aspect.

Abstract

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.

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

TL;DR

This work develops a classical, lattice-based approach to the Saito–Kurokawa lifting problem for square-free levels, integrating old/newform theory in degree 2 with a detailed analysis of inner products via a 4×4 matrix M_p. It provides a concrete realisation of SK lifts as Eichler–Zagier–Ibukiyama (EZI) lifts and uses Maaß relations to distinguish EZI lifts, while also treating oldforms that are not EZI lifts through an explicit orthonormal basis and W_p-invariant structures. The paper then pivots to the level-aspect sup-norm problem, establishing nontrivial bounds for Jacobi forms of index 1 and analyzing the Bergman kernel for SK lifts by decomposing into Type 0, Type 1, and Type 2 regions, with Fourier, Fourier–Jacobi, and counting-matrix techniques. It culminates in conjectures for the size of SK_k(N) and the sup-norms, supported by lower bounds via Waldspurger-type formulas and average L-values, and provides comprehensive ancillary results including an appendix covering the case n=1. The results advance understanding of SUP-norm phenomena for higher-rank automorphic forms and illuminate the interplay between Jacobi forms, half-integral weight forms, and Siegel modular forms in the level aspect.

Abstract

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 -norms of eigenforms. For computing the norms we work with the Hecke algebra of . In the second half, we formulate precise conjectures on the 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.
Paper Structure (63 sections, 43 theorems, 344 equations, 2 figures)

This paper contains 63 sections, 43 theorems, 344 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Old and new SK lifts vs. old and new EZI lifts
  3. Maaß relations characterize EZI lifts
  4. The matrix Mp that controls SK lifts
  5. Computations in the Hecke algebra
  6. Some background on the sup-norm problem in the level aspect
  7. Conjectures and results on sup-norms
  8. Size of Jacobi forms of index 1
  9. Non-trivial bound for a Jacobi newform
  10. Size of spaces of SK lifts
  11. Discussion about the proofs
  12. The three regions: Type 0, 1, 2
  13. Type 0 -- Fourier expansion
  14. Type 1 -- Fourier-Jacobi expansion
  15. Type 2 -- counting matrices
  16. ...and 48 more sections

Key Result

Theorem 1.2

Let $N \ge 1$ be square-free and $F \in S^{(2)}_k(N)$ be a eigenform with eigenvalues $\lambda_F(n)$ for $(n,N)=1$. Then the following are equivalent.

Figures (2)

  • Figure 1: The SK and EZI maps on the newspace with $f\in S_{2k-2}^{new}(N)$.
  • Figure 2: The SK and EZI maps on the oldspace with $f \in S_{2k-2}(N/p)$.

Theorems & Definitions (85)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Theorem 1.10
  • ...and 75 more