Shortest nonzero lattice points in a totally real multi-quadratic number field and applications

Technical Summary of "Shortest Nonzero Lattice Points in a Totally Real Multi-Quadratic Number Field and Applications"

Problem Statement
The paper addresses two interconnected problems in algebraic number theory and the theory of automorphic forms. First, it seeks an explicit characterization of the set of shortest nonzero lattice points, denoted $B_M = \{s \in M : \|\sigma(s)\| = \min(M)\}$, for a fractional ideal $M$ within a totally real multi-quadratic number field $F = \mathbb{Q}(\sqrt{d_1}, \dots, \sqrt{d_n})$. Here, $\|\sigma(s)\| = \sqrt{\sum_{i=1}^r (\sigma_i(s))^2}$ represents the Euclidean norm of the vector of embeddings of $s$. Second, the paper aims to apply this characterization to refine the asymptotic expansion of the Petersson trace formula for Hilbert cusp forms over such fields. The motivation stems from the need to understand the discrepancy between probability measures involving eigenvalues of cusp forms, where precise knowledge of the minimal norm $\min(M)$ and the structure of $B_M$ is required to sharpen existing asymptotic estimates.

Methodology
The author employs a combination of algebraic identities, Diophantine analysis, and analytic number theory:

1. Algebraic Identities: The paper establishes a general algebraic identity (Theorem 4) regarding the sum of squares of linear combinations involving signs determined by binary tuples. This identity is applied to the embeddings of elements in a multi-quadratic field to derive an explicit formula for $\|\sigma(s)\|^2$ (Theorem 5).
2. Diophantine Classification: By equating the derived norm formula to the square of the minimal norm, the author classifies the shortest nonzero lattice points as rational solutions to a specific Diophantine equation (Theorem 2). The paper distinguishes between "trivial solutions" (where at most one variable is non-zero) and non-trivial solutions.
3. Trace Formula Refinement: The author revisits the Petersson trace formula for Hilbert cusp forms (Theorem 6, based on Knightly and Li). The asymptotic behavior as the weight $k_0 \to \infty$ is analyzed. The key step involves analyzing the set $A_i$ of elements in the ideal class representatives that satisfy specific norm bounds. The author demonstrates that for multi-quadratic fields, if the Diophantine equation admits a trivial solution, the set $A_i$ reduces to a singleton containing only the element corresponding to that trivial solution. If no trivial solution exists, the set $A_i$ is empty.

Key Contributions and Results

• Explicit Norm Formula (Theorem 1 & 5): The paper provides an explicit formula for $\|\sigma(s)\|$ for an element $s \in \mathbb{Q}(\sqrt{d_1}, \dots, \sqrt{d_n})$. If $s$ is expressed in the basis of square roots, the squared norm is shown to be $2^n$ times the sum of the squares of the coefficients weighted by the square-free integers $d_i$.
• Trace and Cross-Term Formulas (Corollary 1.1): As a consequence, explicit formulas for the trace $\text{Tr}(s)$ and the sum of products of distinct embeddings $\sum_{i<j} \sigma_i(s)\sigma_j(s)$ are derived.
• Characterization of Shortest Vectors (Theorem 2): The set of shortest nonzero lattice points for a fractional ideal $M$ is precisely identified as the set of elements in $M$ whose coefficients form a rational solution to the Diophantine equation:
  $$2^n \left( X_0^2 + \sum_{i=1}^n d_i X_i^2 + \sum_{i'=2}^n \binom{n}{i'} \sum_{j'=1}^{\binom{n}{i'}} Y_{i',j'}^2 \prod_{\alpha=1}^{i'} d_{s_\alpha} \right) = (\min(M))^2$$
• Refined Asymptotic for Petersson Trace Formula (Theorem 3): The paper presents a refined asymptotic formula for the Petersson trace formula.
  • If the associated Diophantine equation has a trivial solution for a given index $i$, the triple sum in the original asymptotic (Theorem 7) collapses into a double sum, and the error term is explicitly bounded.
  • If no trivial solution exists for any index $i$, the entire sum over the set $A_i$ vanishes, resulting in a simpler asymptotic with a smaller error term.
• Applications to Discrepancy (Corollaries 3.2 & 3.3):
  • Corollary 3.2: Under the condition that the narrow class number of $F$ is odd and specific ideals are principal generated by elements with trivial solutions, the paper establishes a lower bound for the discrepancy between the trace formula's main term and the actual sum. This improves upon previous results by BDS23.
  • Corollary 3.3: The refined asymptotic is used to derive a lower bound analogue to Theorem 1.6 of Jung and Sardari (JS20) regarding the discrepancy between the Sato-Tate measure and the discrete measure of eigenvalues for Hilbert cusp forms over multi-quadratic fields.

Significance and Claims
The paper claims to provide a necessary algebraic tool for analyzing lattice points in totally real multi-quadratic fields, which were previously lacking in explicit form. By linking the geometry of numbers (shortest lattice vectors) to the arithmetic of Diophantine equations, the author refines the error terms in the Petersson trace formula.

The significance lies in the ability to convert a complex triple sum in the trace formula into a double sum (or eliminate it entirely) based on the solvability of a specific Diophantine equation. This refinement allows for sharper lower bounds on the discrepancy of eigenvalue distributions, extending results known for classical cusp forms ($F=\mathbb{Q}$) to the setting of Hilbert cusp forms over multi-quadratic extensions. The author notes that these results serve as improvements to existing theorems in [BDS23] and provide an analogue to [JS20] in the multi-quadratic context. The work does not propose new experimental applications but rather strengthens the theoretical foundation for studying the distribution of eigenvalues in these specific number fields.