Arithmetic structure of L2-norms of SL2(ℤ) matrices
Abstract
For a matrix γ∈SL2(ℤ), we define ℛ(γ)=a12+a22+a32+a42, where γ=a1&a2 a3&a4, and let Ssq(X) count the number of matrices γ with |γ|∞ = max{|a1|, |a2|,|a3|,|a4|} ≤ X and such that ℛ(γ) is squarefree. We prove that Ssq(X) = SℛsqN(X) +O(X19/10+o(1)), as X→ ∞, where N(X)=#{γ∈SL2(ℤ):|γ|∞≤ X} and Sℛsq is an explicit positive Euler product of local p2-densities. The proof combines the δ-method for small moduli with a sum-of-two-squares estimate for large square divisors. This complements a result of J. B. Friedlander and H. Iwaniec (2009) on prime values of ℛ(γ), which, however, is conditional on a very strong form of the Elliott--Halberstam conjecture. We also show that ℛ(γ) is squarefree and has at most $9$ prime divisors for at least cN(X)/log X matrices γ∈SL2(ℤ) with |γ|∞≤ X, where c>0 is an absolute constant.
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BibTeX
@misc{shparlinski2026arithmetic,
title = {{Arithmetic structure of $L_2$-norms of $\mathrmSL_2(\mathbbZ)$ matrices}},
author = {Igor E. Shparlinski and Yixiu Xiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21460},
eprint = {2609.21460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21460},
}
RIS
TY - UNPB
TI - Arithmetic structure of L_2-norms of SL_2(ℤ) matrices
AU - Igor E. Shparlinski
AU - Yixiu Xiao
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21460
UR - https://arxiv.org/abs/2609.21460
AB - For a matrix γ∈SL_2(ℤ), we define ℛ(γ)=a_1^2+a_2^2+a_3^2+a_4^2, where γ=a_1&a_2 a_3&a_4, and let S_sq(X) count the number of matrices γ with |γ|_∞ = max{|a_1|, |a_2|,|a_3|,|a_4|} ≤ X and such that ℛ(γ) is squarefree. We prove that S_sq(X) = S_ℛ^sqN(X) +O(X^19/10+o(1)), as X→ ∞, where N(X)=#{γ∈SL_2(ℤ):|γ|_∞≤ X} and S_ℛ^sq is an explicit positive Euler product of local p^2-densities. The proof combines the δ-method for small moduli with a sum-of-two-squares estimate for large square divisors. This complements a result of J. B. Friedlander and H. Iwaniec (2009) on prime values of ℛ(γ), which, however, is conditional on a very strong form of the Elliott--Halberstam conjecture. We also show that ℛ(γ) is squarefree and has at most $9$ prime divisors for at least cN(X)/log X matrices γ∈SL_2(ℤ) with |γ|_∞≤ X, where c>0 is an absolute constant.
ER -