**Title:** Distribution of orders in number fields

**Speaker:** Ramin Takloo-Bighash

**Speaker Info:** UIC

**Brief Description:**

**Special Note**:

**Abstract:**

Let K/Q be a number field of degree n. Define N_K(B) = #{ R order in K; disc R < B }. The asymptotic behavior of N(B) has been investigated by Davenport--Heilbronn for n=3, and Nakagawa for n=4, as well as Bhargava who showed \sum_{K quintic} N_K(B) ~ cB for a constant c. In this talk I will explain the proof of the following recent result.Theorem. There is a natural number r that depends only on the Galois group of the normal closure of K such that (1) if n \leq 5, N(B) ~ C B^{1/2} (log B)^{r-1} for a positive constant C; (2) if n > 5, then B^{1/2} (log B)^{r - 1} << N(B) << B^{n/4-7/12}.

Here, r is the multiplicity of the trivial representation in a certain explicitly constructed representation of the Galois group. The proof of the theorem uses p-adic (and motivic) integration. This is joint work with Nathan Kaplan and Jake Marcinek.

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