Heegner Points, Hilbert's Twelfth Problem, and the Birch and Swinnerton-Dyer Conjecture

• Author / Creator

  Kostiuk, Jordan Allan

• Heegner points on modular curves play a key role in the solution of Hilbert’s twelfth problem for qua- dratic imaginary fields, as well as the proof of the Birch and Swinnerton-Dyer conjecture for the case ords=1 L(E, s) ≤ 1. The relationship between Heegner points and Hilbert’s twelfth is classically described by the j-function; we supply evidence that suggests that this relationship is one that transcends the j-function and should be able to be recast in terms of other suitable modular functions. The proof of the Birch and Swinnerton-Dyer conjecture for the case ords=1 L(E, s) ≤ 1 is examined and made concrete by using sage to illustrate, very explicitly, the role played by the Heegner points. Both of these results suggest a deep connection between geometry and arithmetic that we hope to see in other contexts.

• Subjects / Keywords

  • Number Theory
  • Heegner Points
  • Modular Forms

• Graduation date

  Fall 2013

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R38P5VJ4K

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.