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Transcendental Brauer-Manin obstructions on singular K3 surfaces
Mohamed Alaa Tawfik1, Rachel Newton1
1Department of Mathematics, King's College London, Strand, London, WC2R 2LS UK.
Research in Number Theory
|December 23, 2024
Summary
This study investigates Brauer groups of specific elliptic surfaces with complex multiplication. Researchers found new transcendental Brauer-Manin obstructions, offering insights into weak approximation problems in number theory.
Area of Science:
- Number Theory
- Algebraic Geometry
Background:
- Elliptic curves with complex multiplication are fundamental objects in number theory.
- Understanding the Brauer group of algebraic varieties is crucial for studying their arithmetic properties.
- Weak approximation is a key concept in Diophantine geometry, relating rational and real points on varieties.
Purpose of the Study:
- To compute and analyze the Brauer groups of minimal desingularisations of quotients of elliptic curves by Heegner divisors.
- To construct new examples of transcendental Brauer-Manin obstructions.
Main Methods:
- Utilizing the theory of complex multiplication for elliptic curves.
- Employing techniques from algebraic geometry to study minimal desingularisations.
- Applying the theory of Brauer groups and the Brauer-Manin obstruction.
Main Results:
- The Brauer groups of the studied surfaces Y are explicitly analyzed.
- New examples of transcendental Brauer-Manin obstructions are provided.
- These obstructions demonstrate failures of weak approximation for certain varieties.
Conclusions:
- The Brauer group provides a powerful tool for detecting arithmetic obstructions.
- Transcendental Brauer-Manin obstructions offer a refined method for studying weak approximation.
- The results contribute to the understanding of the arithmetic of Shimura varieties and related moduli spaces.
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