Harder's conjecture and Hermitian automorphic forms

Technical Summary: Harder's Conjecture and Hermitian Automorphic Forms

Problem Statement
This paper addresses Harder's conjecture, which predicts congruences between elliptic modular forms and Siegel modular forms of degree two. Specifically, let $k \ge 4$ and $j \ge 2$ be integers with $j$ even, and let $f$ be a primitive elliptic cusp form of weight $2k + j - 2$ for $SL_2(\mathbb{Z})$. The conjecture posits the existence of a Siegel cusp eigenform $F \in S_{\det^k \Sym^j}(Sp_2)$ such that, for a prime ideal $\mathfrak{p}$ dividing the algebraic critical value $L^{\text{alg}}(k+j, f)$, the spinor $L$-polynomial of $F$ satisfies a specific congruence modulo a prime ideal $\mathfrak{P}$ above $\mathfrak{p}$:
$$L_\ell(X, F, \text{Spin}) \equiv L_\ell(X, f)(1 - \ell^{k-2}X)(1 - \ell^{k+j-1}X) \pmod{\mathfrak{P}}$$
for every rational prime $\ell$. While previous works have established cases of this conjecture (e.g., via Shimura-type correspondences, even unimodular lattices, or Miyawaki lifts), this paper aims to derive the conjecture conditionally from congruences involving Hermitian automorphic forms on the quasi-split unitary group $U_{2,2}$.

Methodology
The authors employ a two-pronged strategy combining automorphic representation theory and Galois representation theory:

1. Automorphic Side (Construction of Lifts):

   • The authors utilize Mok's endoscopic classification for quasi-split unitary groups to construct "Hermitian spin lifts." For a Siegel cusp eigenform $F$, they construct a Hermitian cusp eigenform $\text{Lift}_E(F)$ on $U_{2,2}$ (where $E$ is an auxiliary imaginary quadratic field).
   • They establish that the standard $L$-polynomial of this lift factors into the product of the spinor $L$-polynomial of $F$ and its $\chi_E$-twist.
   • By comparing this factorization with the explicit standard $L$-polynomial of a Hermitian Klingen–Eisenstein lift associated with $f$, they prove that a congruence between the Hermitian cusp form $G$ and the Klingen–Eisenstein lift implies the desired spinor $L$-polynomial congruence, provided $G$ is indeed the lift of some $F$.

2. Galois Side (Descent and Irreducibility):

   • The core difficulty lies in proving that the Hermitian cusp form $G$ obtained from the Klingen–Eisenstein congruence is actually a spin lift.
   • The authors use Galois representations attached to unitary groups (following Skinner) to analyze the residual extension classes arising from the congruence $G \equiv [f]_{j+4}^2 \pmod{\mathfrak{P}}$.
   • They employ Selmer-group vanishing arguments to exclude "unwanted" residual extension classes. Specifically, they prove that under explicit arithmetic hypotheses, the associated Galois representation $R^\natural_G$ is absolutely irreducible and conjugate-invariant.
   • Conjugate invariance allows the global Arthur parameter of $G$ to descend from $GL_4(\mathbb{A}_E)$ to $GL_4(\mathbb{A}_{\mathbb{Q}})$. This descent identifies $G$ as the transfer of a Siegel cusp eigenform, thereby establishing $G = \text{Lift}_E(F)$.

Key Contributions and Results

• Main Theorem (Theorem 5.8): The paper proves a conditional version of Harder's conjecture. It establishes that if specific arithmetic conditions are met regarding the prime $\mathfrak{p}$, the imaginary quadratic field $E$, and the non-congruence of $f$, then the spinor $L$-polynomial congruence holds.
• Technical Hypotheses: The theorem requires ten explicit conditions, including:
  • $p > 2k + j - 2$ and $p$ unramified in $E$.
  • The class number of $E$ is one, with a unique ramified prime $q \neq p$.
  • Non-divisibility of specific $L$-values (e.g., $L(0, \tilde{\omega}^{j+1})$, $L^{\text{alg}}(k+j, f \otimes \chi_E)$) by $\mathfrak{P}$.
  • A "Tame" condition on the ramified prime $q$ ($q^{j+2} \not\equiv 1 \pmod p$).
  • The image of the Galois representation attached to $f$ containing $SL_2(\mathcal{O}_P)$.
• Numerical Verification: The authors provide a table of numerical examples (Table 1) where these hypotheses are verified. This includes recovering known cases (e.g., $(k,j,p) = (10,4,41)$) and establishing new cases for both even and odd $k$ (e.g., $(7, 12, 73)$, $(5, 20, 193)$).

Significance
The paper claims significance in providing a unified framework for Harder-type congruences that treats even and odd $k$ uniformly (for $k \ge 4$). By shifting the problem to the setting of Hermitian automorphic forms and utilizing the endoscopic classification of Mok alongside Galois-theoretic descent, the authors offer a conditional proof that bypasses some limitations of previous methods (such as the parity restrictions on $k$ in earlier works). The work demonstrates that the existence of the Siegel form $F$ can be deduced from the existence of a Hermitian congruence, provided the relevant Selmer groups vanish and the Galois representations satisfy specific irreducibility and invariance properties. The paper explicitly notes that the case $k=3$ requires separate discussion due to boundary issues in Mok's classification but suggests it is accessible via minor modifications.