• Lecture notes for this course can be found HERE (Last update: 26.01.2026).

(Updates: Minor corrections & Newly added Section 5.1: Hilbert's 10th problem over number rings)

• The exam will be oral. It will contain both theory (definitions, proofs, statements of main theorems) and exercises very similar to the recommended exercises (without the "harder" ones). 

• Presentation slided for 13.01.2026 can be found HERE. 

Links: Magma online calculator, LMFDB (elliptic curves), Rank records

• Presentation slides for 23.12.2025 can be found HERE.

(Updates: uploaded sketch of proof for Exercises 3.47, 4.8, 4.27, 4.31, and 4.36.)

• A tentative syllabus for this course can be found HERE.

• The main references are: 

  • Silverman's book on "The Arithmetic of Elliptic Curves" (Link)

  • James Newton's "Lecture notes on Elliptic Curves" (Link).

• Prerequisites: 

  • Strongly recommended: Group theory, Ring theory, basic Galois theory

  • Recommended, but not required: Basic notions from algebraic number theory, algebraic geometry, specifically algebraic curves

Tentative Lectures Schedule:

1. 14th October: (Sun Woo + Diana) Introduction, Crash course on algebraic curves, Group law, & First examples I

Recommended exercises: 1.4, 1.8, 1.9, 1.24.

2. 21st October: (Diana) Group law, First examples II

Recommended exercises: 1.33, 1.34, 1.35, 1.36, 1.38.

3. 28th October: (Sun Woo) Crash course on function fields + Isogenies I (up to Example 1.51 & Remark 1.52)

Recommended exercises: 1.43, 1.47, 1.48

4. 4th November: (Sun Woo) Isogenies II (from Theorem 1.53)

Recommended exercises: 1.54, 1.59, 1.61, 1.65

5. 11th November: (Diana) Elliptic curves over finite fields + Crash course in p-adics (up to Remark 3.5)

Recommended exercises: 2.2, 2.3, 2.4, 2.6 (harder), 2.13 

6. 18th November: (Sun Woo) Crash course in p-adics + Elliptic curves over local fields I: Hasse Principle / Minimal Weierstrass equations (up to Proposition 3.29)

Recommended exercises: 3.8, 3.9, 3.10, 3.12, 3.17, 3.20, 3.22, 3.23, 3.24, 3.25 (harder)

7. 25th November: (Diana) Elliptic curves over local fields II: Minimal Weierstrass equations, points of finite order (up to Remark 3.41)

Recommended exercises: 3.32, 3.35, 3.37, 3.42 (discussed at the end of the lecture).

8. 2nd December: (Sun Woo) Elliptic curves over local fields III: Good & Bad reduction + Elliptic curves over global fields I: Torsion + Weak Mordell-Weil (up to statement of Definition 4.13 and Lemma 4.15) + Student Evaluations at the end of the course.

Recommended exercises: 3.17, 3.46, 3.47, 4.5, 4.12 (Important)

9. 9th December: (Diana) Elliptic curves over global fields II: Weak Mordell-Weil + Heights + Mordell-Weil Theorem

Recommended exercises: 4.13, 4.18, 4.23

10. 16th December: (Diana) Elliptic curves over global fields III: Explicit 2-Descent + Computing MW Groups

Recommended exercises: 4.8, 4.27, 4.28, 4.29

11. 23rd December: (Sun Woo) Exercises & Review (Tentative)

Recommended exercises: 4.30, 4.31 (harder), 4.35, 4.36

12. 13th January: (Diana) Elliptic curves over global fields IV: Explicit 2-Descent via Magma + Survey on ranks

Recommended exercises: 4.30, 4.31 (harder), 4.35, 4.36, 4.43, 4.44

13. 20th January: (Sun Woo) Selected topics I: Arithmetic Statistics of ranks of elliptic curves 

14. 27th January: (Sun Woo) Selected topics II: Hilbert's 10th problem via additive combinatorics

15. 3rd February: (Diana) Selected topics III: Fermat's last theorem