### Integral points on diagonal affine cubic surfaces

Linfoot Number Theory Seminar

31st May 2023, 11:00 am – 12:00 pm

Fry Building, 2.04

The sum of three cubes conjecture state if n is an integer not congruent to 4 or 5 mod 9 then the affine surface

U_n: x^3+y^3+z^3=n in A^3_Z

always has an integral point. Colliot-Thélène and Wittenberg showed that U has no “Cohomological obstruction” to the existence of integral points (more formally there is no integral Brauer-Manin obstruction to the integral Hasse principle). In this talk we study natural generalisations of the surfaces U_n, namely the diagonal affine cubics surfaces:

ax^3+by^3+cz^3=n in A^3_Z

and we will show in this more general setting there are instances of the integral Hasse principle failing.

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