Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian

2024/10/06 by Satadal Ganguly, Ganguly, Satadal, Rachita Guria +1 · 1 citation
Mathematics · #11F30 #11F72 #11L07 #11N45 #Advanced Algebra and Geometry #FOS: Mathematics #Graph theory and applications #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 11P21 #Secondary 11N36

paper · pdf · doi:10.48550/arxiv.2410.04637

openalex publication_date 2024/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use classical Fourier analysis along with tools from the spectral theory of Automorphic forms to derive an asymptotic formula with a strong error term for the number of integer solutions (a, b, c, d) inside the expanding box [-X,X]⁴ to the determinant equation ad-bc=r, where r ≠ 0 is a fixed integer. Furthermore, we apply our method to study sums over these solutions where the variables are weighted by periodic arithmetical functions in two of the variables in one case, and by an arbitrary sequence of complex numbers in another.

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• Counting 2 × 2 integer matrices with a given determinant

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