**The International Center for Mathematical Sciences – Sofia (ICMS-Sofia)**

**presents**

## ICMS Seminar

30.01.2024, 14:00, Sofia time

ICMS-Sofia, Room 403

# Density of Hasse failures for diagonal affine cubic surfaces

## Vladimir Mitankin, IMI-BAS

А question of Mordell asks which integers are the sum of three cubes provided that there are no local obstructions to that. While Mordell’s question seems out of reach with the current machinery, it is widely believed that every integer not congruent to 4 or 5 mod 9 is a sum of three integral cubes. If, instead of the sum of three cubes, a general ternary cubic is considered, local conditions no longer suffice for solubility, i.e. the integral Hasse principle may fail. In this talk we shall apply the integral version of the Brauer-Manin obstruction to construct the first examples of such failures not explained by local conditions in the setting of affine diagonal ternary cubics. We will then explore in three different natural ways how such failures are distributed across the family of affine diagonal ternary cubics.

This talk is based on a joint work with Harkaran Uppal and Julian Lyczak.