Hoffmann–Rosenbaum–Yoshida Model

• Hoffmann–Rosenbaum–Yoshida Model is a framework for constructing vector-valued Yoshida lifts using theta correspondence and automorphic representation theory.
• The model provides explicit inner product formulas linking Petersson norms to special values of Asai and Rankin–Selberg L-functions, crucial for theoretical and computational insights.
• It facilitates the computation of Bessel periods and non-vanishing results that are essential for establishing congruences, constructing Selmer groups, and analyzing Galois representations.

The Hoffmann–Rosenbaum–Yoshida (HRY) Model encompasses the explicit construction, analysis, and arithmetic applications of vector-valued Yoshida lifts—Siegel modular forms of degree two—within the framework of theta correspondence and automorphic representation theory. Central topics include the derivation of inner product formulas for such lifts, the link with special values of Asai L-functions, explicit computation of Bessel periods, and consequences for arithmetic non-vanishing and Galois representations.

1. Foundations of Yoshida Lifts and the HRY Model

Yoshida lifts are a specific class of Siegel modular forms of genus 2 created via the theta correspondence from automorphic forms on orthogonal groups associated to quadratic spaces of signature (4, 0) or (3, 1). For $F = \mathbb{Q} \oplus \mathbb{Q}$, the source data consists of a pair of elliptic modular newforms $(f_1, f_2)$, while for $F$ a real quadratic field, it is a Hilbert modular newform. The theta lift, parameterized by explicitly chosen vector-valued pluri-harmonic polynomials and Bruhat–Schwartz functions, produces a modular form on $\mathrm{Sp}(4)$ of weight $$\mathrm{Sym}^{2k_2}(\mathbb{C}^2)\otimes\det^{k_1-k_2+2}$$.

The term "HRY Model" refers to the overarching framework for constructing and analyzing such Yoshida lifts in a representation-theoretic context, emphasizing their role as explicit, non-endoscopic, generic supercuspidal automorphic representations and their connections to the arithmetic of L-values and Galois representations.

2. Explicit Construction and Fourier Expansion

The construction proceeds via global theta lifting. Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ of discriminant $N^-$, and $(N_1^+, N_2^+)$ positive integers prime to $N^-$. For newforms $f_i$ of weights $(2k_i+2)$ and levels $N_i = N^-N_i^+$, Jacquet–Langlands correspondence associates to $f_i$ a modular form on $D^\times \backslash D^\times_\mathbf{A}$. The Yoshida lift $\theta^*_\mathbf{f}$ emerges as a theta integral, employing a test function $\varphi$ whose finite part is supported on $$V(\widehat{\mathbf{Z}})\oplus V(\widehat{\mathbf{Z}})$$ and whose infinite part involves a pluri-harmonic polynomial $P_{\ul{k}}$.

On the Siegel upper half-space $\mathfrak{H}_2$, the resulting modular form admits a Fourier expansion: $$\theta^*_{\mathbf{f}}(Z) = \sum_S \mathbf{a}(S) q^S,$$ with coefficients given explicitly by: $\mathbf{a}(S) = \sum_{h_f \in [\mathcal{E}_{\mathbf{z}}]} w_{\mathbf{z},h_f} \cdot P_{\ul{k}}(\mathbf{z})(\mathbf{f}(h_f)),$ where $\mathbf{z} \in \mathbf{X}$ with $S = S_\mathbf{z}$, and $[\mathcal{E}_{\mathbf{z}}]$ is a finite double coset. This explicit formula is critical for both theoretical and computational aspects relating to the arithmetic of these forms.

3. Inner Product Formula and Relation to L-functions

A central achievement within the HRY model is the derivation of explicit formulas for the Petersson norm (self-inner product) of Yoshida lifts. If $f$ is a modular form over $F$ and $\theta^*_f$ its Yoshida lift, then: $$\frac{ \|\theta_f^*\|_{\mathfrak{H}_2}^2 }{ \|f\|_R^2 } = L(\mathrm{As}^+(f),k_1+k_2+2)\cdot (4\pi)^{-(2k_1+3)} \Gamma(k_1+k_2+2)\Gamma(k_1-k_2+1) \cdot \mathcal{E},$$ where $\mathcal{E}$ is an explicit product of local quantities depending on the level $N$, number of ramified primes in $F$, Atkin–Lehner eigenvalues $\varepsilon_p$, and discriminant factors.

For $F = \mathbb{Q}\oplus\mathbb{Q}$, $L(\mathrm{As}^+(f),s)$ becomes the Rankin–Selberg convolution $L(f_1\times f_2, s)$. These explicit inner product formulas are obtained by realizing the Yoshida lift as a theta lift and applying the Rallis inner product formula, with explicit computation of the requisite local zeta integrals at both finite and infinite places.

Summary Table: Petersson Norm Formula

Quantity	Interpretation / Definition
$\|\theta^*_f\|_{\mathfrak{H}_2}$	Petersson norm on Siegel space
$L(\mathrm{As}^+(f), s)$	(Plus) Asai L-function of source modular form
$\mathcal{E}$	Explicit local factors, including eigenvalues and level terms

The formula generalizes prior scalar-valued results to the vector-valued setting and provides precise links between automorphic periods and special L-values.

4. Bessel Periods and Their Arithmetic Role

For a vector-valued Yoshida lift, Bessel periods are integrals against certain automorphic forms or Hecke characters, detailed as follows: $$\mathbf{B}_{\theta, S, \phi}^* = e(\mathbf{f}, \phi)\cdot \Theta_C(\mathbf{f}, \phi \circ N_{E/K}),$$ with $e(\mathbf{f}, \phi)$ an explicit local constant and $\Theta_C$ a toric period sum over a finite set. In the split case ($F = \mathbb{Q}\oplus\mathbb{Q}$, $E = K \oplus K$), the Bessel period factors as a product of toric periods associated to each constituent modular form: $$P(\mathbf{f}, \phi \circ N_{E/K}, \varsigma^{(C)}) = P(\mathbf{f}_1, \phi, \varsigma^{(C)}) P(\mathbf{f}_2, \phi^{-1}, \varsigma^{(C)}).$$ This explicit linkage provides the basis for arithmetic applications, notably to congruences and Selmer groups, by connecting Fourier coefficients mod primes to the non-vanishing of Bessel periods.

5. Non-vanishing Results and Arithmetic Applications

A key arithmetic consequence derived using the explicit norm and period formulas is the non-vanishing of Yoshida lifts, both over $\mathbb{C}$ and modulo primes. Specifically, if the Atkin–Lehner eigenvalues $\varepsilon_p=1$ for all $p|N$, then the Yoshida lift $\theta_f^*$ is non-zero. This generalizes earlier non-vanishing results and extends them to vector-valued and real quadratic settings.

Non-vanishing Modulo Primes:

Under additional conditions on $f_1, f_2$—including conditions on the prime $\ell$, residual irreducibility, and Atkin–Lehner sign parity—the Yoshida lift $\theta^*_{\mathbf{f}_1\otimes\mathbf{f}_2}$ has $\lambda$-integral Fourier expansion, and infinitely many Fourier coefficients are nonzero modulo $\lambda$. The argument exploits the correspondence between certain Bessel periods and Fourier coefficients, together with prior results on toric periods, to deduce the desired non-vanishing modulo $\ell$.

Summary Table: Non-vanishing Conditions

Context	Key Non-vanishing Condition	Consequence
Complex coefficients	$\varepsilon_p = 1$ for all $p|N$	$\theta^*_f \neq 0$
Modulo $\lambda$	Atkin–Lehner sign parity; residual irreducibility	Infinite Fourier coefficients nonzero mod $\lambda$

These arithmetic facts are crucial for the construction of congruences between automorphic forms, as in the Yoshida congruence method, and for generating explicit Selmer group elements used in the Bloch–Kato conjecture.

6. Broader Connections and Impact

Within the HRY model, Yoshida lifts function as concrete examples of non-endoscopic, generic supercuspidal automorphic representations for $\mathrm{GSp}_4$. Their explicit construction enables the analysis of period ratios, algebraicity of special L-values, and the existence of congruences between automorphic forms. Applications include:

• Parametrization in Local Langlands Correspondence: Yoshida lifts contribute to the explicit realization of the local Langlands correspondence for symplectic groups in non-endoscopic settings.
• Algebraicity of L-values and Period Ratios: The Petersson norm formula facilitates the paper of algebraic and $p$-adic properties of critical L-values.
• Modularity Lifting and Selmer Groups: Non-vanishing modulo primes is pivotal in the arithmetic of congruences and the construction of non-trivial Bloch–Kato Selmer groups.
• Congruences and Galois Representations: Explicit period and inner product computations underpin congruence relations yielding Galois representations with targeted properties.

A plausible implication is that similar theta lifting and period formula techniques may be generalized to higher genus Siegel modular forms and other reductive groups, extending the reach of the HRY model to broader classes of automorphic forms.

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Summary Table: HRY Model Features

Feature	Description
Object of Study	Vector-valued Yoshida Siegel modular forms (genus 2)
Main Formula	Petersson norm in terms of $L(\mathrm{As}^+(f), s)$ with explicit factors
Arithmetic Impact	Explicit non-vanishing results, mod $\ell$ and over $\mathbb{C}$
Key Tools	Rallis formula, explicit zeta integrals, Jacquet–Langlands transfer
Primary Applications	Congruences, special values, Galois and Selmer group constructions

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The HRY model, through its explicit constructions, formulas, and arithmetic consequences, consolidates the theory and arithmetic of vector-valued Yoshida lifts, positioning them as central objects in the paper of non-endoscopic automorphic forms, special value formulas, and their deep connections with the Galois representations predicted by the Langlands program (Hsieh et al., 2016, Hsieh et al., 2015).