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Harder's conjecture II

This paper proves Harder's conjecture by establishing a congruence modulo a prime ideal between the Hecke eigenvalues of a specific Siegel modular form constructed via Klingen-Eisenstein and Saito-Kurokawa lifts and those of a Hecke eigenform, under conditions where the prime divides the algebraic part of a specific L-value.

Original authors: Hiraku Atobe, Masataka Chida, Tomoyoshi Ibukiyama, Hidenori Katsurada, Takuya Yamauchi

Published 2026-06-17

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CC BY 4.0

Original authors: Hiraku Atobe, Masataka Chida, Tomoyoshi Ibukiyama, Hidenori Katsurada, Takuya Yamauchi

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

The Big Picture: A Cosmic Puzzle of Numbers

Imagine the world of mathematics as a vast, intricate library filled with books of "music." In this library, the most famous composers are Modular Forms. These aren't songs you hear with your ears, but complex mathematical patterns that describe how numbers behave, much like the rhythm of a drum or the harmony of a chord.

Some of these musical patterns are "primitive" (the original, solo compositions), while others are "lifts" (orchestral arrangements created by taking a solo piece and expanding it into a larger symphony).

The Problem:
For a long time, mathematicians have been trying to solve a specific puzzle known as Harder's Conjecture. The puzzle asks: If you take a specific solo piece (a primitive form) and create a special "lifted" version of it, can you find a completely different, independent symphony (a Hecke eigenform) that sounds almost exactly the same, but only when you listen to them through a very specific, high-pitched filter (a prime number)?

In math terms, this is called a congruence. It means that if you look at the "notes" (coefficients) of both symphonies, they differ only by a tiny amount that disappears when you divide by a specific large number.

The Cast of Characters

To understand this paper, let's meet the main players using an analogy of a Music Factory:

The Soloist ( $f$ ): A primitive modular form. Think of this as a famous, solo violinist. They play a specific tune defined by their "weight" (how complex the music is).
The Saito-Kurokawa Lift ( $I_2(f)$ ): This is a machine that takes the solo violinist and creates a duet. It's a specific way of turning a solo piece into a piece for two instruments (degree 2).
The Klingen-Eisenstein Lift ( $[I_2(f)]_k$ ): This is a second machine. It takes the duet and expands it into a massive, 4-instrument orchestra piece. This is the "standard arrangement" everyone expects.
The Mystery Symphony ( $F$ ): The paper claims there exists a different composer who wrote a 4-instrument piece that sounds almost identical to the standard arrangement, but it wasn't made by the standard machine. It's a "hidden" symphony.
The Filter ( $p$ ): A giant prime number. The paper proves that if you listen to the standard arrangement and the mystery symphony through this filter, they sound indistinguishable.

What This Paper Actually Does

The authors (Atobe, Chida, Ibukiyama, Katsurada, and Yamauchi) are the detectives who finally solved the case of the Mystery Symphony.

1. The Previous Mystery (The Sequel):
In a previous paper, the team proposed a theory: "If the soloist's tune has a special property (related to a specific value called an L-value), then this Mystery Symphony must exist." They proved it for a few specific, easy cases, like checking a few specific songs in the library.

2. The New Breakthrough (The Extended Version):
This new paper is the "Extended Version." The authors say, "We aren't just going to check a few songs. We are going to prove this works for almost any song you can throw at us, provided you check a few simple conditions first."

They created a checklist (Theorems 5.5 and 5.7) that acts like a quality control test. If a specific song passes these tests (checking things like whether the "notes" are divisible by the filter number), then the Mystery Symphony is guaranteed to exist.

3. The Detective Work (Galois Representations):
How did they prove it? They didn't just listen to the music; they looked at the "DNA" of the symphonies.

In math, every symphony has a hidden "genetic code" called a Galois representation.
The authors showed that if the Mystery Symphony didn't exist, the DNA of the standard arrangement would have to look like a "Frankenstein's monster" (a mix of different parts that shouldn't fit together).
By using advanced tools (like Selmer groups, which act like a security system for these genetic codes), they proved that the "Frankenstein" DNA is impossible. Therefore, the only logical conclusion is that the Mystery Symphony must exist to make the DNA match.

The "Harder" Part

The paper is titled "Harder's Conjecture." This refers to a mathematician named Günter Harder. The conjecture is essentially a prediction about how these different musical arrangements relate to each other.

The authors prove that:

The Standard Arrangement (the Klingen-Eisenstein lift) and The Mystery Symphony (a lift of a degree-2 form) are congruent modulo $p$ .
This confirms Harder's prediction: The "notes" (eigenvalues) of the Mystery Symphony are exactly what you would get if you added the soloist's notes to some predictable background noise ( $\ell^{k-2} + \ell^{j+k-1}$ ).

The "Proof" in the Paper

The paper is very technical, but the core logic is:

Assume the opposite: Suppose the Mystery Symphony doesn't exist.
Find a contradiction: If it doesn't exist, the mathematical "DNA" (Galois representation) of the standard arrangement would have to be a weird, broken mix of parts.
Use the Checklist: The authors show that under their specific conditions (the "easy to check" conditions in the abstract), this broken DNA is impossible.
Conclusion: Therefore, the Mystery Symphony must exist.

Real-World Examples (Section 10)

The paper doesn't just stay in theory. In the final section, they actually go into the library and find specific examples.

They pick specific "soloists" (like $\phi_+$ and $\phi_-$ ).
They calculate the "filter" numbers (primes like 4289, 67021, etc.).
They verify that the conditions are met.
Result: They confirm that for these specific cases, the Mystery Symphony definitely exists and matches the prediction.

Summary

Think of this paper as the final piece of a massive jigsaw puzzle.

The Puzzle: Connecting different types of mathematical music (modular forms) across different dimensions.
The Missing Piece: Proving that a specific "hidden" symphony exists and matches a known one under a specific filter.
The Solution: The authors built a robust framework (using Galois representations and Selmer groups) to prove this connection holds true for a vast range of cases, not just a few lucky examples. They provided the full "blueprint" (proofs) for how to find these hidden symphonies whenever the conditions are right.

They didn't invent new music; they proved that a specific, hidden harmony must exist in the mathematical universe, and they gave us the tools to find it.

Technical Summary of "Harder's Conjecture II: Extended Version with Full Proofs"

Problem Statement
This paper addresses Harder's conjecture, which posits a congruence relation between the Hecke eigenvalues of a Siegel modular form of degree 2 and specific quantities derived from the Hecke eigenvalues of an elliptic modular form. Specifically, let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(\mathbb{Z})$ , and let $p$ be a prime ideal dividing the algebraic part of the $L$ -value $L(k+j, f)$ . The conjecture asserts the existence of a Hecke eigenform $F$ in the space of cusp forms of weight $(k+j, k)$ for $Sp_2(\mathbb{Z})$ such that the eigenvalues of $F$ satisfy a specific congruence modulo $p$ involving the eigenvalues of $f$ and powers of primes.

In a previous work (arXiv:2306.07582v2), the authors proposed a stronger conjecture (Conjecture 1.1) concerning the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda (DII) lift of $f$ , denoted $[I_2(f)]_k$ , and a specific lift of type $A(I)$ , denoted $A^{(I)}_4(F)$ , of the form $F$ . They proved that Conjecture 1.1 implies Harder's conjecture. However, the previous proof relied on case-by-case verification of non-congruence conditions for specific weights $(k, j)$ .

Methodology
The authors establish Conjecture 1.1 in a general setting (for even $k$ and $j \equiv 0 \pmod 4$ ) under a set of checkable conditions. The methodology relies on two primary pillars:

Congruence of Klingen-Eisenstein Lifts: Building on previous results, the authors utilize the fact that if $p$ divides the algebraic part of $L(k+j, f)$ , there exists a Hecke eigenform $G$ (not a constant multiple of $[I_2(f)]_k$ ) such that $G \equiv_{ev} [I_2(f)]_k \pmod p$ . To prove the conjecture, it suffices to show that any such $G$ must be the lift of type $A(I)$ .
Non-Congruence via Galois Representations: The core technical innovation is the use of Galois representations to distinguish between different types of modular forms. The authors analyze the semi-simplification of the mod $p$ Galois representation $\bar{\rho}_{G, St}$ attached to a candidate form $G$ .
- For the Klingen-Eisenstein lift $[I_2(f)]_k$ , the semi-simplified representation is known to decompose into a direct sum involving twists of the representation $\bar{\rho}_f$ attached to $f$ and powers of the cyclotomic character $\chi_p$ .
- The authors prove that if $G$ is not the lift of type $A(I)$ , its associated Galois representation cannot exhibit this specific decomposition structure under the given conditions.
- This is achieved by analyzing the extension classes in the Bloch-Kato Selmer groups and utilizing properties of short crystalline representations. Specifically, they demonstrate that if the decomposition were to occur for a non- $A(I)$ form, it would imply the existence of non-trivial elements in specific Selmer groups or class groups (e.g., $C_p^{\omega^{-j}}$ ), which are ruled out by the imposed conditions (such as Vandiver's conjecture or regularity of $p$ ).

Key Contributions

Generalization of Previous Results: The paper extends the proof of Conjecture 1.1 (and consequently Harder's conjecture) from specific cases $(k, j) = (10, 4), (14, 4), (4, 24)$ to a general setting for even $k$ and $j \equiv 0 \pmod 4$ .
Full Proofs of Auxiliary Lemmas: The authors provide complete proofs for Lemmas 7.2, 7.3 and Corollaries 7.4, 7.5, which were previously stated without proof in the earlier version. These lemmas concern the structure of lattices in Galois representations and the reducibility of representations based on extension groups.
Algorithmic Verification: The paper details an algorithm (Proposition 6.2) to compute Fourier coefficients of Klingen-Eisenstein series and critical $L$ -values. This allows for the effective checking of the "easily checkable" conditions required for the theorem to hold in concrete examples.
Refined Conditions: The authors formulate the main theorem (Theorem 5.5) with a list of conditions (C.1–C.8) involving the non-divisibility of Fourier coefficients, $L$ -values, and class numbers. They further refine these in Theorem 5.7 to conditions (C.2') and (C.8') that are independent of the choice of periods $c^\pm(f)$ .

Results
The main result is Theorem 5.5 (and its variant Theorem 5.7), which states that under the specified conditions (including $p > 2k+j-2$ , absolute irreducibility of $\bar{\rho}_f$ , and non-divisibility of specific $L$ -values and class numbers), there exists a Hecke eigenform $F \in S_{(k+j, k)}(\Gamma(2))$ such that:
$A^{(I)}_4(F) \equiv_{ev} [I_2(f)]_k \pmod p$
As a direct consequence, Theorem 5.9 confirms Harder's conjecture for these parameters, establishing the congruence of Hecke eigenvalues:
$\lambda_F(T(\ell)) \equiv a(\ell, f) + \ell^{k-2} + \ell^{j+k-1} \pmod{p'}$
Furthermore, Theorem 5.10 connects these congruences to arithmetic geometry, proving that the Bloch-Kato Selmer group $Sel(\mathbb{Q}, T(k+j))$ associated with the representation $T$ has a $p$ -torsion element.

In Section 10, the authors apply these results to confirm Harder's conjecture for new concrete examples, specifically $(k, j) = (12, 8), (10, 12),$ and $(8, 16)$ , using specific primitive forms derived from Eisenstein series and the Ramanujan delta function.

Significance
The paper claims significance by providing a rigorous, general proof of Harder's conjecture in a broad class of cases, moving beyond the limitations of case-by-case verification. By linking the congruence of modular forms to the structure of Galois representations and Selmer groups, the work deepens the understanding of the arithmetic properties of Siegel modular forms and their relationship to elliptic modular forms. The explicit conditions provided allow for the systematic verification of the conjecture for any given pair of weights $(k, j)$ , facilitating further computational exploration in the field.