A generalization of Watson transformation and representations of ternary quadratic forms

TL;DR

This paper generalizes Watson transformations for ternary quadratic forms, providing recursive formulas to compute representation numbers across related genera, with explicit criteria and calculations for additional correction terms.

Contribution

It introduces a generalized Watson transformation for ternary lattices, establishing recursive relations for representation numbers across genera with explicit correction terms.

Findings

Recursive formulas for representation numbers in related genera

Explicit criteria for when correction terms are needed

Calculation of correction terms in specific cases

Abstract

Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$ be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzero isotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let $\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\cdot dL$ on the quadratic space $L^{p^m}\otimes \q$ such that for each lattice $T \in \mathcal G_{L,p}(m)$, a $\frac 12\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. Let $r(n,M)$ be the number of representations of an integer $n$ by a $\z$-lattice $M$. In this article, we show that if $m \le 2$ and $n$ is divisible by $p$ only when $m=2$, then for any $T \in \mathcal G_{L,p}(m)$, $r(n,T)$ can be written as a linear summation of $r(pn,S_i)$ and $r(p^3n,S_i)$ for $S_i \in \mathcal G_{L,p}(m+1)$ with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute $r(n,T)$, for any $T \in \mathcal G_{L,p}(m)$, by using the number of representations of some integers by lattices in $\mathcal G_{L,p}(m+1)$ for an arbitrary integer $m$.