Counting low degree number fields with almost prescribed successive minima

Abstract: The successive minima of an order in a degree $n$ number field are $n$ real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for $n = 3,4,5$. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss a geometric analogue of this problem: scrollar invariants of covers of $\mathbb{P}^1$.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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