Balea, Briana-Mihaela (2025) Non-Existence of Elliptic Curves over Q with Good Reduction Everywhere over Quadratic Fields. Bachelor's Thesis, Mathematics.
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Abstract
This thesis investigates the minimal field extensions over which elliptic curves defined over Q acquire good reduction everywhere. While it is a classical result due to Tate that no elliptic curve over Q admits good reduction at all primes, a natural question arises: Can good reduction every- where be achieved over a quadratic extension of Q? We review and present in detail a theorem of Kida, which gives a negative answer to this question. To this end, we develop the necessary arithmetic background, including valuations, ramification theory, and the study of elliptic curves in both local and global settings. We analyze how reduction behavior is influenced by field extensions, and establish criteria for when good reduction occurs. Finally, we provide explicit examples of elliptic curves that achieve good reduction everywhere over number fields of degrees 3, 4 and 6, illustrating the connection between local invariants and the specific nature of the required field extensions.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Kilicer, P. and Ozman, E. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 25 Jul 2025 06:49 |
| Last Modified: | 28 Jul 2025 08:26 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/36387 |
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