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Counting Number Fields -- Conjectures, Results, and Methods

  • Sameera Vemulapalli (Princeton University)
G3 10 (Lecture hall)

Abstract

For a fixed integer d, how many number fields are there of fixed degree d of absolute discriminant less than X? Denote this number N(d, X). This talk will be a friendly foray through the known results, proofs, and main conjectures regarding this question. In particular, it is known that if d = 2,3,4,5 then N(d, X) = O(X). We will discuss the key components of the proofs, which involve the parametrizations of number fields and geometry of numbers. For n >= 6, we will discuss Malle's conjecture, which predicts the asymptotics of N(d, X) for general degree d. We will also discuss classical upper bounds on N(d, X).

Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail

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