Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers

TL;DR

The paper proves asymptotic formulae for small, weighted solutions to diagonal quadratic congruences modulo prime powers $q=p^m$, addressing both inhomogeneous ($Q({\bf x})=\lambda_1x_1^2+\cdots+\lambda_nx_n^2-\lambda_{n+1}$) and homogeneous cases (when $\lambda_{n+1}=0$). The authors develop a unified Poisson-summation framework and perform explicit evaluations of Gauss sums, Kloosterman sums, and Salié sums, complemented by a Jon-type weighted representation analysis, to derive main terms $T_0 = B_p(Q)\hat{\Phi}(0)^n N^n/p^m$ (inhomogeneous) and $T_0 = A_p(Q)\hat{\Phi}(0)^n N^n/p^m$ (homogeneous) with error terms smaller by a power of $p^m$. The results require $N\ge p^{(1/2+\varepsilon)m}$ and hold for $n\ge6$ in the inhomogeneous case and $n\ge4$ in the homogeneous case with coefficients allowed to vary with $m$, extending prior work of BaBaHa and collaborators. The work provides quantitative, high-accuracy counts for small representations by quadratic forms modulo prime powers, with potential implications for distribution of modular square roots and short-interval representations.

Abstract

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $λ_1x_1^2+\cdots +λ_nx_n^2\equiv λ_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $λ_1,...,λ_{n+1}$ are integer coefficients such that $(λ_1\cdots λ_{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $λ_1,...,λ_n>0$ and $(λ_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\varepsilon)m}$, centered at the origin. If $n\ge 4$ and $λ_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $λ_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag.

Theorems & Definitions (26)

• Theorem 1.1: Theorem 1 in BaHa
• Theorem 1.2: Theorem 2 in BaBaHa for $(x_{0,1},...,x_{0,n})=(0,...,0)$)
• Theorem 1.3: Theorem 2 in BaHa
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 2.1
• proof
• Proposition 2.2
• proof