Campana points on wonderful compactifications
Dylon Chow, Daniel Loughran, Ramin Takloo-Bighash, Sho Tanimoto
TL;DR
This paper proves a variant of Manin's conjecture for Campana points on wonderful compactifications of semisimple adjoint groups, establishing an asymptotic count ${\mathsf N}({\mathsf G}(F)_{\epsilon},\mathcal{L},B) \sim c B^{a} \log(B)^{b-1}$ under a rank-2, klt assumption and explicit geometric-automorphic data for the invariants $a$ and $b$. The authors deploy the height zeta function method, decompose the spectral contributions, and show the rightmost pole is governed by one-dimensional automorphic representations, with the leading constant interpreted via a corrected Campana Brauer-group framework and Tamagawa-type integrals. They propose a refined conjecture for the leading constant, incorporating sums over the Campana Brauer group and local invariants, and verify it in the setting of wonderful compactifications, while discussing compatibility with other group compactifications and known examples (toric, squareful counts, etc.). A key novelty is the introduction and utilization of the Campana Brauer group to capture obstructions and densities that alter the leading constant relative to the classical Manin-Peyre picture, providing evidence for a richer, Brauer-weighted leading term in the Campana context. The work broadens the scope of Manin-type counting to interpolate between rational and integral points, offering a concrete, testable conjectural framework for leading constants in Campana problems. Overall, it solidifies the height-zeta machinery in a nontrivial Campana setting and supplies a robust blueprint for analyzing leading constants via automorphic and Brauer-theoretic data.
Abstract
We prove a variant of Manin's conjecture for Campana points on wonderful compactifications of semi-simple algebraic groups of adjoint type. We use this to provide evidence for a new conjecture on the leading constant in Manin's conjecture for Campana points.