Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

This paper investigates the distribution of rational and algebraic points of bounded weighted height in weighted projective spaces over number fields. For a weighted projective space with weights q over a number field k of degree m, we derive an asymptotic formula for the count of such points, featuring a leading term D times X raised to m e Q, plus an error term, where e is the extension degree and Q is the sum of the weights. The constant D combines geometric aspects of the weights with an arithmetic obstruction given by the reciprocal of the gcd of the least common multiple of the weights and Euler's totient of m e. This obstruction stems from the non-surjectivity of the natural morphism from the weighted space to ordinary projective space on rational points, linked to nontrivial torsors under groups of roots of unity. We provide a cohomological interpretation, analogous to the Brauer-Manin obstruction. These findings refine a weighted version of the Batyrev-Manin conjecture and open avenues for applications in moduli theory and arithmetic geometry.