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be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on
using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on
. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.
We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over
whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.
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