Elliptic Curves with Complex Multiplication and Supersingular Primes
- Degree Grantor:
- University of California, Santa Barbara. Mathematics
- Degree Supervisor:
- Adebisi Agboola
- Place of Publication:
- [Santa Barbara, Calif.]
- Publisher:
- University of California, Santa Barbara
- Creation Date:
- 2012
- Issued Date:
- 2012
- Topics:
- Mathematics
- Keywords:
- Complex multiplication,
Supersingular - Genres:
- Online resources and Dissertations, Academic
- Dissertation:
- M.A.--University of California, Santa Barbara, 2012
- Description:
This thesis explores the theory of elliptic curves with complex multiplication needed to prove Noam Elkies result that there are infinitely many supersingular primes for every elliptic curve defined over Q . The first chapter contains a brief overview of the basic theory of elliptic curves, including the topics of the j-invariant, isogenies, the endomorphism ring, the Tate module, elliptic curves over finite fields, good and bad reduction, and elliptic curves over C . In the second chapter we discuss elliptic curves defined over C with complex multiplication focusing on the action of the ring class group. In Chapter 3 we discuss the properties of the j-function and modular polynomials in order to prove that the j-invariants of elliptic curves with complex multiplication are algebraic integers and that these j-invariants generate ring class fields. In Chapter 4 we discuss ray class fields and give a brief summary of idelic class field theory in order to prove the main theorem of complex multiplication and some important results of Deuring. Finally in Chapter 5 we conclude with a proof that every elliptic curve defined over Q has infinitely many supersingular primes.
- Physical Description:
- 1 online resource (108 pages)
- Format:
- Text
- Collection(s):
- UCSB electronic theses and dissertations
- Other Versions:
- http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:1530106
- ARK:
- ark:/48907/f32j68st
- ISBN:
- 9781267768513
- Catalog System Number:
- 990039147820203776
- Copyright:
- Megan Maguire, 2012
- Rights:
- In Copyright
- Copyright Holder:
- Megan Maguire
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