Geometric Methods in Algebra and Number Theory

Volume 235 of the series Progress in Mathematics pp 151-168

Counting extensions of function fields with bounded discriminant and specified Galois group

  • Jordan S. EllenbergAffiliated withDepartment of Mathematics, Princeton University
  • , Akshay VenkateshAffiliated withDepartment of Mathematics, Massachusetts Institute of Technology

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We discuss the enumeration of function fields and number fields by discriminant. We show that Malle’s conjectures agree with heuristics arising naturally from geometric computations on Hurwitz schemes. These heuristics also suggest further questions in the number field setting.