Number Theory Seminar:Dong Quan Nguyen

We show that for each prime p congruent to 1 (mod 8), there exists a threefold X_p such that the existence of certain rational points on X_p produces families of generalized Mordell curves and of generalized Fermat curves that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction. We also introduce a notion of the descending chain condition (DCC) for sequences of curves, and prove that there are sequences of generalized Mordell curves and of generalized Fermat curves satisfying DCC.