Hasse Principle for Hyperelliptic Curves and Degree 4 del Pezzo Surfaces in Family

：梁永祺（中国科技大学）
：2024-09-27 15:00
：海韵园行政楼C610

报告人：梁永祺（中国科技大学）

时  间：2024年9月27日15:00

地  点：海韵园行政楼C610

内容摘要:

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. 

个人简介：

梁永祺，中国科学技术大学特任教授。博士毕业于法国巴黎第十一大学，师从David Harari，此后曾在巴黎第七大学任教。研究领域为算术代数几何，主要关注定义在数域上的代数簇的有理点、0-cycle的局部整体原则和弱逼近、强逼近性质，研究成果发表在《Adv. Math.》、《Math. Res. Lett.》、《Int. Math. Res. Not.》、《Ann. Sci. Ec Norm. Super》、《J. Math. Pures Appl.》、《Math. Ann.》和《J. Algebra》等多个国际知名学术期刊上。

 

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