Counting integral points on indefinite ternary quadratic equations over number fields.

The article focuses on deriving a precise asymptotic formula for counting integral points on affine varieties defined by non-degenerate indefinite integral ternary quadratic forms \( f \) representing a non-zero integer \( a \), particularly when \(-a \cdot \det(f)\) is a square in the number field \( k \). It establishes that the leading term in the asymptotic count is given by the product over all finite primes \( v \) of local densities weighted by factors \( (1 - q_v^{-1}) \), multiplied by a singular integral over archimedean places and a logarithmic growth factor in the norm bound \( T \). The work extends previous results from the rational field \( \mathbb{Q} \) to general number fields, employing harmonic analysis on adelic groups, strong approximation, and the Brauer–Manin obstruction to handle convergence issues of local factors. An explicit example is provided for the quadratic form \( f(x,y,z) = x^2 + y^2 - \delta z^2 \) with positive integer \( \delta \), where the asymptotic count involves Dirichlet \( L \)-functions and local densities expressed via Legendre symbols, illustrating the interplay between arithmetic invariants and counting integral solutions.