05 Mai 2022
17:00  - 18:00

Zoom: register here: https://www.ntwebseminar.org/

Gastvorlesung / Vortrag

## Number Theory Web Seminar: Levent Alpöge (Harvard University)

On integers which are(n't) the sum of two rational cubes

It's easy that 0% of integers are the sum of two integral cubes (allowing opposite signs!).

I will explain joint work with Bhargava and Shnidman in which we show:

1. At least a sixth of integers are not the sum of two rational cubes,

and

2. At least a sixth of odd integers are the sum of two rational cubes!

(--- with 2. relying on new 2-converse results of Burungale-Skinner.)

The basic principle is that "there aren't even enough 2-Selmer elements to go around" to contradict e.g. 1., and we show this by using the circle method "inside" the usual geometry of numbers argument applied to a particular coregular representation. Even then the resulting constant isn't small enough to conclude 1., so we use the clean form of root numbers in the family x^3 + y^3 = n and the p-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.