Torsors for finite group schemes of bounded height

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  • Ratko Darda
    Department of Mathematics and Computer Science University of Basel Basel Switzerland
  • Takehiko Yasuda
    Department of Mathematics Graduate School of Sciences Osaka University Toyonaka Japan

説明

<jats:title>Abstract</jats:title><jats:p>Let  be a nontrivial finite étale tame group scheme over a global field . We define height functions on the set of ‐torsors over , which generalize the usual heights such as the discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of ‐torsors over  of bounded height. This is a special case of our more general “stacky Batyrev–Manin” conjecture [18]. The conjectured asymptotic is proven for the case when  is commutative. When  is a number field, the leading constant is expressed as a product of certain arithmetic invariants of  and the volume of a space attached to . Moreover, an equidistribution property of ‐torsors in the space is established.</jats:p>

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