[Paper Review] Class numbers and representations by ternary quadratic forms with congruence conditions

Kathrin Bringmann, Ben Kane|arXiv (Cornell University)|Feb 5, 2020

Advanced Algebra and Geometry10 references4 citations

TL;DR

This paper establishes uniform formulas linking representations of integers by ternary quadratic forms with congruence conditions to Hurwitz class numbers. It proves that for integral ternary quadratic forms with genus class number one, the number of representations satisfying specific congruence conditions on the variables is proportional to the total number of representations, with the proportionality constant depending only on the residue class of the integer modulo a fixed modulus.

ABSTRACT

In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.

Motivation & Objective

• To extend classical formulas relating sums of squares to class numbers by incorporating congruence conditions on the variables.
• To identify all ternary quadratic forms with genus class number one for which such congruence-restricted representation counts are proportional to total representation counts.
• To systematically determine pairs $(\bm{h}, N)$ such that the number of representations $r_{\bm{a},\bm{h},N}(n)$ is proportional to $r_{\bm{a}}(n)$, with the constant depending only on $n \mod M$.
• To provide explicit formulas for these proportionality constants across all such forms and congruence classes, using Siegel-Weil averages and genus theory.
• To resolve a question posed by Petersson regarding specific congruence conditions on ternary forms, particularly for $\bm{a} = (1,1,1)^T$ and $N=4$.

Proposed method

• Utilizes the Siegel-Weil average formula to relate representation numbers with congruence conditions to local densities and class numbers.
• Applies Jones' classification of ternary quadratic forms with genus class number one, restricting attention to forms $Q_{\bm{a}}$ for which the genus class number is one.
• Employs the valence formula for modular forms on congruence subgroups $\Gamma_0(N)$ to bound the dimension of spaces of cusp forms, enabling verification of identities.
• Derives explicit formulas by comparing coefficients of modular forms associated with $r_{\bm{a}}(n)$ and $r_{\bm{a},\bm{h},N}(n)$, using known results on Eisenstein series and cusp forms.
• Uses the fact that for class number one forms, the genus average collapses to the total representation number, simplifying the analysis.
• Constructs tables of proportionality constants $d_{\bm{a},\bm{h},N}(n)$ that depend only on $n \mod M$, based on $p$-adic properties and local densities.

Experimental results

Research questions

• RQ1 For which integral ternary quadratic forms $Q_{\bm{a}}$ with genus class number one do the representation numbers $r_{\bm{a},\bm{h},N}(n)$ with congruence conditions on $\bm{x}$ satisfy a uniform proportionality to $r_{\bm{a}}(n)$?
• RQ2 What are the explicit values of the proportionality constants $d_{\bm{a},\bm{h},N}(n)$ that relate $r_{\bm{a},\bm{h},N}(n)$ to $r_{\bm{a}}(n)$, and how do they depend on $n \mod M$?
• RQ3 Can the identities observed by Petersson and Ebel for specific $\bm{h}$ and $N$ be generalized to all forms with genus class number one?
• RQ4 To what extent do congruence conditions on $\bm{x}$ force the representation to lie in a fixed residue class modulo $N$, and when does this fail to hold?
• RQ5 How can the local densities $\beta_{(Q,\bm{h},N),p}(n)$ be used to derive global formulas linking representation numbers to Hurwitz class numbers?

Key findings

• For each $\bm{a} \in \mathcal{C}$, the set of $\bm{a}$ for which $Q_{\bm{a}}$ has genus class number one, and for each $ (\bm{h}, N) \in \mathcal{S}_{\bm{a}} $, the identity $ r_{\bm{a},\bm{h},N}(n) = d_{\bm{a},\bm{h},N}(n) r_{\bm{a}}(n) $ holds, where $ d_{\bm{a},\bm{h},N}(n) $ depends only on $ n \mod M $.
• For $\bm{a} = (1,1,1)^T$, the identity $ r_{\bm{1},\bm{h},4}(n) = \frac{1}{12} \delta_{n \equiv 2 \pmod{8}} r_{\bm{1}}(n) = \delta_{n \equiv 2 \pmod{8}} H(4n) $ holds for $\bm{h} = (0,1,1)^T$, linking representations to Hurwitz class numbers.
• For $\bm{a} = (1,2,2)^T$ and $\bm{h} \in \{(1,0,3)^T, (3,1,2)^T\}$, $ r_{\bm{a},\bm{h},6}(n) = \delta_{n \equiv 19 \pmod{24}} (H(4n) - 2H(n)) $, showing non-trivial dependence on congruence classes.
• The proportionality constants $ d_{\bm{a},\bm{h},N}(n) $ are explicitly tabulated in Appendix A, with values zero if the congruence class does not occur.
• The method confirms that such identities arise only when $Q_{\bm{a}}$ has class number one, as shown via the Siegel-Weil average and local density analysis.
• The paper provides bounds on the dimensions of spaces of cusp forms via the valence formula, enabling verification of modular identities for all $\bm{a} \in \mathcal{C}$, with coefficients listed in Appendix D.