For every global function field K and finite abelian p-group G, the multivariate generating function that counts sub-G-extensions by successive higher-ramification heights is rational.
Equidistribution for abelian extensions of global fields
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abstract
We establish asymptotic formulas for abelian extensions of global function fields ordered by conductor and subject to prescribed local conditions. Our proof combines harmonic analysis with a theory of frobenian functions over global function fields developed in this paper. We interpret our result via equidistribution on algebraic stacks.
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Multivariate counting of wild abelian extensions
For every global function field K and finite abelian p-group G, the multivariate generating function that counts sub-G-extensions by successive higher-ramification heights is rational.