Counting square-free numbers in arithmetic geometry

Abstract: We delve into an innocuous question about counting "square-free numbers" in various forms, following the philosophy of Weil's three columns.

Example 1. We count square-free integers.

Example 2. We count square-free polynomials of a fixed degree over a finite field.

Example 3. We compute the Betti numbers of the space of square-free polynomials of a fixed degree over complex numbers by quoting a theorem of Arnol'd in topology.

By viewing Example 3 as counting square-free 0-cycles on the affine line over complex numbers, we add one more example to this list, using our main result.

Example 4. We compute the Betti numbers of the space of square-free 0-cycles of a fixed degree on a punctured elliptic curve over complex numbers.

We briefly explain how Examples 3 and 4 can be obtained by showing that the mixed Hodge structure of the i-th singular cohomology group with rational coefficients is pure of some weight different from i. This is joint work with Yifeng Huang.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar