Solutions of x12+x22−x32=D with small x3: a spectral approach
Abstract
We count the integral solutions of x12+x22−x32=D weighted by Φ(x3/M), for M smaller than √D. Such solutions correspond to binary quadratic forms of discriminant D, and the count becomes a sum, over the geodesics of discriminant D, of a function of their distance to the point i. We evaluate it by the spectral theory of automorphic forms. Throughout, θ=7/64. For square D=n2 the geodesics run from cusp to cusp, and Hecke theory reduces them to the single geodesic from $0$ to ∞. For every n and both parities of x3, we obtain an asymptotic formula with error O(n1+θ+εM−1/2), using the Kim-Sarnak bound and the large sieve of Deshouillers and Iwaniec (and the theta group for the parity of x3). It is non-trivial for M≥ n71/96+ε. This improves the range n5/6+ε of the elementary method, and confirms the corrected main term of the author's earlier work. For non-square D=f2D0 the geodesics are closed. A Hecke relation for their traces reduces everything to the fundamental discriminant D0. We obtain an error O(Dεf1+θD02/3M−1/2), using the formulas of Duke-Imamoglu-Toth and Baruch-Mao together with the cubic moment bound of Young and of Andersen-Wu. For D≡ 5 bmod 8 this counts all solutions, and the range M≥ D1/2−1/1332 of Friedlander and Iwaniec improves to M≥ D4/9+ε. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.
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BibTeX
@misc{baier2026solutions,
title = {{Solutions of $x_1^2+x_2^2-x_3^2=D$ with small $x_3$: a spectral approach}},
author = {Stephan Baier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15706},
eprint = {2610.15706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15706},
}
RIS
TY - UNPB
TI - Solutions of x_1^2+x_2^2−x_3^2=D with small x_3: a spectral approach
AU - Stephan Baier
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.15706
UR - https://arxiv.org/abs/2610.15706
AB - We count the integral solutions of x_1^2+x_2^2−x_3^2=D weighted by Φ(x_3/M), for M smaller than √D. Such solutions correspond to binary quadratic forms of discriminant D, and the count becomes a sum, over the geodesics of discriminant D, of a function of their distance to the point i. We evaluate it by the spectral theory of automorphic forms. Throughout, θ=7/64. For square D=n^2 the geodesics run from cusp to cusp, and Hecke theory reduces them to the single geodesic from $0$ to ∞. For every n and both parities of x_3, we obtain an asymptotic formula with error O(n^1+θ+εM^−1/2), using the Kim-Sarnak bound and the large sieve of Deshouillers and Iwaniec (and the theta group for the parity of x_3). It is non-trivial for M≥ n^71/96+ε. This improves the range n^5/6+ε of the elementary method, and confirms the corrected main term of the author's earlier work. For non-square D=f^2D_0 the geodesics are closed. A Hecke relation for their traces reduces everything to the fundamental discriminant D_0. We obtain an error O(D^εf^1+θD_0^2/3M^−1/2), using the formulas of Duke-Imamoglu-Toth and Baruch-Mao together with the cubic moment bound of Young and of Andersen-Wu. For D≡ 5 bmod 8 this counts all solutions, and the range M≥ D^1/2−1/1332 of Friedlander and Iwaniec improves to M≥ D^4/9+ε. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.
ER -