Determination of Hilbert modular forms using squarefree coefficients

Technical Summary: Determination of Hilbert Modular Forms Using Squarefree Coefficients

Problem Statement
The paper addresses the problem of uniquely determining half-integral weight Hilbert modular forms over a totally real number field $F$ (of narrow class number 1) by a subset of their Fourier coefficients. While the full collection of Fourier coefficients uniquely determines a modular form, the authors investigate whether a "minimal" set suffices. Specifically, they focus on coefficients supported on squarefree algebraic integers. This inquiry extends previous work by Kohnen (restricted to the "plus space" over $\mathbb{Q}$) and Luo-Ramakrishnan (concerning integral weight forms and central $L$-values) to the more general setting of Hilbert modular forms with varying weights and levels.

Methodology
The authors employ a combination of analytic number theory techniques, specifically leveraging the Shimura correspondence, Waldspurger's formula, and Rankin-Selberg theory.

1. Shimura Correspondence and Functional Equations: The core strategy involves relating the half-integral weight forms $f$ and $g$ to their integral weight Shimura lifts, denoted $\tilde{f}$ and $\tilde{g}$. The authors establish the functional equations for the Dirichlet series associated with these forms. A critical step involves the Fricke involution and the $W(n)$ operator, which allow the authors to relate the $L$-function of a form to that of its twist.
2. Comparison of Dirichlet Series: Assuming the normalized Fourier coefficients of $f$ and $g$ agree on all squarefree integers (up to a constant $\kappa$), the authors construct a ratio of their completed $L$-functions, $C(s) = \Lambda(s, f) / \Lambda(s, g)$.
3. Analytic Continuation and Contradiction: By analyzing the functional equations of the $L$-functions and the properties of the Gamma factors involved, the authors demonstrate that $C(s)$ must be a constant. This involves a proof by contradiction regarding the weights and levels of the forms, utilizing the Legendre duplication formula and the behavior of the $L$-functions at specific points to rule out mismatched parameters.
4. Uniqueness via Twists: Once the equality of the $L$-functions is established, the authors utilize a lemma (Lemma 2) concerning the vanishing of twisted Dirichlet series. This lemma, derived from the orthogonality of characters and properties of ray class groups, allows them to conclude that if the $L$-functions of the twists are equal, the forms themselves must be equal.
5. Rankin-Selberg Theory: For the non-vanishing result, the authors utilize the Rankin-Selberg $L$-function $L(s, f \times f)$ and its pole at $s=1$. By estimating the sum of squared coefficients and applying the Ramanujan bound, they derive a lower bound for the magnitude of coefficients supported on squarefree integers.

Key Contributions and Results

• Theorem 3 (Main Determination Result): The authors prove that if $f$ and $g$ are half-integral weight cuspidal Hilbert Hecke eigenforms of varying weights and levels, and their normalized Fourier coefficients satisfy $\lambda_f(\tau) = \kappa \lambda_g(\tau)$ for every squarefree $\tau \in \mathcal{O}_F$ (with $\kappa \neq 0$), then $f = \kappa g$. This generalizes Kohnen's result by removing the restriction to the "plus space" and allowing for varying weights and levels, though it retains the requirement that the equality holds for every squarefree integer rather than "almost all."
• Theorem 4 (Non-Vanishing Result): The paper provides a "soft proof" that for a non-zero Hecke eigenform $f$, the limit superior of the absolute values of its Fourier coefficients supported on squarefree integers is non-zero. Specifically, $\limsup_{T \to \infty, N(\tau)<T, \tau \text{ sq.free}} |\lambda_f(\tau)| \gg_{f,F} 1$. This confirms that infinitely many such coefficients are non-vanishing.
• Corollary 1 (Plus Space Specialization): The authors recover a result analogous to Kohnen's Theorem 1 for the Kohnen plus space over $F$, showing that equality of coefficients on fundamental discriminants implies the forms are identical.

Significance and Claims
The paper claims to provide a rigorous formulation of the philosophical expectation that squarefree coefficients carry all necessary information to uniquely determine a half-integral weight Hilbert modular form. The authors position Theorem 3 as an intermediate result between Kohnen's specific case (fixed weight/level, plus space) and the Luo-Ramakrishnan result (integral weight, central $L$-values).

The authors are modest regarding the scope of their generalization. They explicitly state that their current methods do not appear capable of removing the requirement that the coefficient equality holds for every squarefree integer (as opposed to "almost all") or of relaxing the equality of coefficients to an equality of their magnitudes. Furthermore, while Theorem 4 establishes non-vanishing, the authors note that proving the stronger result that the limit superior is infinite (i.e., $\limsup |\lambda_f(\tau)| = \infty$) remains an open problem, though they cite recent work by GKS24 as a step in that direction.

The assumption that the narrow class number of $F$ is 1 is noted as a convenience and a partial necessity for the current proof structure, though the authors suggest that parts of the argument could be adapted for fields where the discriminant is a principal ideal. The work relies heavily on the robustness of the Shimura correspondence and the functional equations of the associated $L$-functions to bridge the gap between the half-integral and integral weight realms.