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

This paper establishes a classical framework for studying Saito-Kurokawa lifts of square-free level using $L^2$ -norms and Hecke algebras to derive new-oldform theory, which is then applied to prove bounds on the sup-norms of these lifts and their underlying Jacobi forms while formulating precise conjectures on their $L^\infty$ size.

Original authors: Pramath Anamby, Soumya Das

Published 2026-03-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pramath Anamby, Soumya Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the size and shape of a massive, invisible cathedral built from pure mathematics. This cathedral is made of Saito-Kurokawa (SK) lifts.

In the world of advanced number theory, these "lifts" are special structures built by taking simpler, well-understood buildings (called elliptic modular forms) and elevating them into a more complex, higher-dimensional space (called Siegel modular forms).

The authors of this paper, Pramath Anamby and Soumya Das, are trying to answer two main questions about this cathedral:

How do we tell the new parts from the old parts? (The "New-Oldform" theory).
How big is the cathedral, and how tall can a single pillar get? (The "Sup-norm" problem).

Here is a breakdown of their work using everyday analogies.

1. The "New" vs. "Old" Confusion

Imagine you have a library of books. Some books are brand new originals. Others are copies, or copies of copies, or copies that have been slightly altered (like adding a new chapter or changing the font).

In math, when you increase the "level" (think of this as the complexity or the number of rules the building must follow), you get Old Forms (copies of lower-level structures) and New Forms (genuinely new structures that couldn't exist at lower levels).

The Problem:
For a long time, mathematicians had a perfect way to sort these books for simple structures (elliptic forms). But for these complex SK lifts, the sorting method was messy. Sometimes, a book looked like a copy, but it was actually a unique original. Sometimes, a "copy" didn't look like a copy at all.

The Solution (The Matrix Detective):
The authors developed a new detective tool. Instead of looking at the text of the books (which is hard to read), they measured the "weight" of the books using a special scale called the $L^2$ norm (think of this as the total energy or volume of the book).

They created a 4x4 grid (a matrix) where they compared the weights of four different versions of the same book.

If the grid had a specific pattern (Rank 3), the book was a Saito-Kurokawa lift.
If the grid was full of unique information (Rank 4), it was not a lift.

This allowed them to perfectly separate the "New" lifts from the "Old" ones, even in complex situations where previous methods failed. They also discovered that some "Old" forms are like hybrids—they are made of three parts, but one part is a "ghost" that doesn't fit the standard copying rules.

2. The "Sup-Norm" Problem: How Tall is the Building?

Now that they know how to sort the books, they want to know how "loud" or "tall" these mathematical structures can get.

In math, the Sup-norm is like asking: "What is the highest point a wave can reach in this ocean?"

The "Space" Size: If you add up all the waves in the ocean, how big is the total splash?
The "Single Form" Size: How high can one single wave get?

The Analogy of the "Catch":
The authors found a tricky "catch" when measuring these waves.

Imagine you are measuring the height of a wave. Usually, as you go further out (higher level), the waves get smaller and smaller.
However, for these specific SK lifts, there is a "ghost wave" (related to something called Theta series) that gets bigger as you go out.
It's like trying to measure the height of a crowd of people. Usually, the crowd gets smaller as you move away. But here, there's a group of people holding giant balloons that get bigger the further you go, messing up your measurement.

Their Discovery:
They managed to prove that even with these giant balloons, the waves don't get infinitely tall.

They proved a non-trivial bound: The waves are actually shrinking, just slower than we thought.
They formulated a Conjecture: They believe they know the exact formula for how tall the waves get based on the level $N$ . It turns out the "size" of the whole space drops off very predictably (like $N^{-2}$ ).

3. The Three Regions of the Cathedral

To measure the height of the building, they couldn't just look at it from one angle. They had to split the cathedral into three zones, like exploring a cave:

Region 0 (The Safe Zone): Here, the building is very stable. They could use standard math (Fourier expansion) to measure it easily. It was like measuring a flat floor.
Region 1 (The Middle Zone): This was trickier. They had to use a "Fourier-Jacobi expansion," which is like looking at the building through a prism that breaks the light into colors. They realized the size of the building here depends heavily on the size of the "ghost waves" (Jacobi forms) they mentioned earlier.
Region 2 (The Dangerous Zone): This is the hardest part. The building is unstable here. Standard measuring tools (Fourier expansion) break down.
- The Solution: They used a Counting Argument. Instead of measuring the wave directly, they counted how many "tiles" (matrices) fit into a specific space. It's like estimating the size of a forest not by measuring every tree, but by counting how many trees fit in a square mile. This allowed them to get a precise bound even in the most chaotic part of the building.

4. Why Does This Matter?

You might ask, "Why do we care about the height of these invisible mathematical waves?"

The Riemann Hypothesis Connection: These waves are deeply connected to the distribution of prime numbers (the atoms of math). Understanding their size helps us understand the fundamental laws of numbers.
New Tools: The "Matrix Detective" method they invented (using $L^2$ norms to sort forms) is a new tool that can be used to solve other difficult problems in higher-dimensional math, not just for SK lifts.
Correcting the Record: They found that some previous assumptions about how these forms behave were slightly wrong (specifically regarding "self-adjoint" operators, which is a fancy way of saying "symmetry"). They fixed these errors, ensuring future mathematicians build on solid ground.

Summary

In simple terms, Anamby and Das built a new sorting machine to separate the genuine mathematical structures from the copies. Then, they went on a surveying expedition to measure the maximum height of these structures. They found that while the structures are complex and have some "ghostly" interference, they are actually very well-behaved and follow a predictable pattern of shrinking as they get more complex. They did this by combining algebraic tricks, geometric counting, and a deep understanding of how these mathematical waves interact.

1. Problem Statement and Context

The paper addresses the sup-norm problem for holomorphic Siegel cusp forms of degree 2, specifically focusing on the subspace of Saito-Kurokawa (SK) lifts of square-free level $N$ .

The Sup-Norm Problem: This involves estimating the $L^\infty$ size (supremum norm) of automorphic forms. For a space of forms, this is measured by the supremum of its Bergman Kernel (BK). For a single eigenform, it is the maximum value of the invariant function $v^{k/2}|f(z)|$ .
The Challenge: While significant progress has been made for elliptic modular forms (degree 1) and in the weight aspect for higher rank groups, the level aspect for degree 2 (symplectic group $GSp(4)$) remains nascent.
Specific Obstacles: SK lifts are functorial lifts from elliptic forms (or Jacobi forms) but lack global Whittaker models, making standard analytic tools (like the geometric side of the BK) difficult to apply directly. Furthermore, the classical theory of "new" and "old" forms for SK lifts at higher levels was not fully developed in the literature, particularly regarding the distinction between SK lifts and the broader class of Siegel forms.

2. Methodology

The authors employ a hybrid approach combining classical modular form theory with spectral analysis and geometric counting arguments.

A. Classical New-Oldform Theory via L2 Norms

Instead of relying on Fourier expansions at various cusps (which is technically difficult for SK lifts), the authors develop a novel classical approach based on $L^2$ -norms:

The Inner Product Matrix ( $M_p$ ): For a form $F$ of level $N$ and a prime $p \nmid N$ , they define a $4 \times 4$ matrix $M_p$ whose entries are inner products of $F$ and its images under four operators: Identity, $U_S(p)$ , $W_p$ , and $U_S(p)W_p$ .
Rank Characterization: They prove that the rank of $M_p$ characterizes whether $F$ is an SK lift. Specifically, if $F$ is an SK lift, the rank is 3; otherwise, it is 4. This allows them to construct explicit orthonormal bases for the oldspace of SK lifts.
Hecke Algebra Computations: They perform extensive calculations within the Hecke algebra of $GSp(2)$ to derive exact relations between the inner products of these operators, avoiding the need for explicit Fourier expansions at cusps.

B. Characterization of EZI Lifts

The authors distinguish between SK lifts (defined via L-functions) and Eichler-Zagier-Ibukiyama (EZI) lifts (defined via classical lifting from Jacobi forms).

They show that while all EZI lifts are SK lifts, not all SK lifts are EZI lifts (specifically in the oldspace).
They introduce Maass-Ibukiyama relations as the unique characterization for EZI lifts within the space of SK lifts.

C. Sup-Norm Bounds via Reduction to Jacobi Forms

To bound the sup-norm of SK lifts, they reduce the problem to the study of Jacobi forms of index 1 and half-integral weight forms:

Theta Decomposition: They utilize the isomorphism between Jacobi forms and half-integral weight forms (via the Eichler-Zagier map).
Fundamental Domain Decomposition: They decompose the fundamental domain of $\Gamma_0^{(2)}(p)$ $Γ_{0}^{(2)} (p)$ into three regions (Type 0, 1, and 2) based on the coset representatives of the symplectic group.
- Type 0 (Large $Y$ ): Bounded via Fourier expansion.
- Type 1 (Intermediate): Bounded via Fourier-Jacobi expansion, reducing to the size of Jacobi forms.
- Type 2 (Small $Y$ ): The most difficult region. They use a counting argument on discrete matrices (related to the geometric side of the Bergman kernel) to bound the sum over forms.

3. Key Contributions and Results

A. Structural Theory of SK Lifts

New-Oldform Basis: They provide an explicit orthonormal basis for the oldspace of SK lifts of square-free level, which is invariant under the Atkin-Lehner involution $W_p$ .
Rank Theorem (Theorem 1.2): They establish that a Hecke eigenform $F$ is an SK lift if and only if the inner product matrix $M_p$ has rank 3 (for $p \ge 17$ ).
Maass-Ibukiyama Relations: They prove that satisfying the Maass-Ibukiyama relations is a necessary and sufficient condition for an SK lift to be an EZI lift (Theorem 4.12).

B. Sup-Norm Bounds and Conjectures

The paper formulates precise conjectures and proves bounds towards them for the size of the space of SK lifts, denoted $\sup(SK_k(N))$ .

Conjecture 1.9 (Global Size): For even $k \ge 4$ and square-free $N$ , the size of the space is:
$\sup(SK_k(N)) \asymp_k N^{-2}$
(Note: This is significantly smaller than the expected size for general Siegel forms, which is conjectured to be $O(1)$ ).
Theorem 1.8 (Prime Level Bounds): For a prime level $p$ , they prove:
$p^{-2} \ll_k \sup(SK_k(p)) \ll_{k, \epsilon} p^{-1+\epsilon}$
The lower bound $p^{-2}$ is derived using Waldspurger's formula and average central L-values. The upper bound is obtained by analyzing the three regions of the fundamental domain.
Jacobi Form Results:
- Theorem 1.4: The size of the space of Jacobi forms of index 1 is bounded by $O(N^{1/2})$ .
- Theorem 1.6: A non-trivial bound for a single normalized Jacobi newform: $\|\phi\|_\infty \ll N^{-1/36+\epsilon}$ .
- Conjecture 1.7: The expected bound for a single Jacobi newform is $N^{-1/2+\epsilon}$ .

C. Technical Appendices

Appendix 1: Proves that for elliptic cusp forms of square-free level, $\sup(S_k(N)) \asymp 1$ for all $k > 2$ (including half-integral weights), using classical Poincaré series arguments.
Appendix 2: Proves that the Hecke operator $U_J(p)$ on Jacobi forms is not self-adjoint on the full space (only on the new subspace), correcting a potential misconception in previous literature.

4. Significance

Foundational Classical Theory: The paper fills a critical gap by providing the first comprehensive classical treatment of new and old SK lifts for higher levels, a topic previously only understood via representation theory (CAP representations).
Methodological Innovation: The use of $L^2$ -norm matrices to characterize new/oldforms and distinguish SK lifts from general Siegel forms is a novel technique that avoids the complexities of Fourier expansions at cusps.
Progress on Sup-Norms: It provides the first non-trivial bounds for the sup-norm of SK lifts in the level aspect. The results suggest that SK lifts are "smaller" (in terms of sup-norm growth) than general Siegel forms, likely due to their functorial origin from lower-rank groups.
Future Directions: The results lay the groundwork for studying the sup-norm problem for general Siegel cusp forms by isolating the SK subspace. The techniques (counting arguments in Type 2 regions, norm relations) are expected to be adaptable to other spaces, such as Hermitian modular forms.

In summary, Anamby and Das successfully bridge the gap between the representation-theoretic definition of SK lifts and classical analytic number theory, providing rigorous bounds on their size and clarifying the structure of their oldforms.