Supervisor:
听
Working Thesis
Obstructions to rational points and zero cycles.
In number theory, a fundamental problem is to find rational solutions to Diophantine equations. Classically, an algebraic variety is the set of solutions to Diophantine equations over real or complex numbers; modern approaches generalize this concept. One essential technique in studying Diophantine equations is analyzing local solutions. However, many varieties do not satisfy the Local-Global principle, which means that even if it has all local points, it may still not have a global point. In most cases, this local-global failure is explained by the Brauer鈥揗anin obstruction, first introduced by Manin. When a rational point does not exist, a natural question is to ask whether the variety admits a zero-cycle of degree one. We will also try to answer whether the Brauer鈥揗anin obstruction is the only obstruction for zero-cycles on bielliptic surfaces. Furthermore, we will examine additional varieties to identify zero-cycles of degree one and construct further examples, explicitly exhibiting zero cycles of degree one. This will provide additional evidence in support of Colliot-Th茅l猫ne's conjecture. We will also study conditions under which the existence of a zero cycle of degree one guarantees the existence of a rational point.
听
Research interests:
Number theory
听
Academic History:听
Master of Science in mathematics, Indian Institute of Technology Kanpur, India.
Bachelor of Science in mathematics, Scottish church college, India.
