Hasse Principle for Hyperelliptic Curves and Degree 4 del Pezzo Surfaces in Family
- A+
：Yongqi Liang (University of Science Technology of China)
： 2024-09-27 15:00
：Conference Room C610 at Administration Building at Haiyun Campus
Speaker：Yongqi Liang (University of Science and Technology of China)
Time：2024-9-27, 15:00
Location：Conference Room C610 at Administration Building at Haiyun Campus
Abstract:
Scharaschkin and Skorobogatov conjectured that when smooth projective curves violate the Hasse principle then the violations are explained by the Brauer-Manin obstruction. Colliot-Thelene and Sansuc have similar conjecture for degree 4 del Pezzo surfaces. We use this obstruction to construct explicit algebraic families of degree 4 del Pezzo surfaces and smooth projective curves of genus congruent to 1 modulo 4, such that they violate the Hasse principle. In particular, over every number field, we find explicit families of elliptic curves whose Tate-Shafarevich groups are non-trivial. This gives an explicit answer to a question of Colliot-Thelene and Poonen asked about 20 years ago, when they proved the existence of such a family of elliptic curves over the field \mathbb{Q} of rational numbers. This is a joint work with K. Huang.
