Syllabus Application
MATH 543
Elliptic Curves
Faculty
Faculty of Engineering and Natural Sciences
Semester
Fall 2026-2027
Course
MATH 543 -
Elliptic Curves
Time/Place
Time
Week Day
Place
Date
13:40-15:30
Mon
FENS-L058
Sep 28-Dec 31, 2026
12:40-13:30
Tue
FENS-L065
Sep 28-Dec 31, 2026
Level of course
Masters
Course Credits
SU Credit:3, ECTS:10
Prerequisites
-
Corequisites
-
Course Type
Lecture
Instructor(s) Information
Mohammad Sadek
- Email: mmsadek@sabanciuniv.edu
Course Information
Catalog Course Description
Weierstrass equations, group law, isogenies, Tate module, Weil pairing, endomorphism ring. Zeta function of an elliptic curve, Weil conjectures. Uniformization theorem. Elliptic curves over local fields. Mordell-Weil theorem. Siegel's theorem. Modular curves and L-series.
Learning Outcomes:
| 1. | Understand Elliptic Curves: Define elliptic curves over arbitrary fields via Weierstrass equations, compute the discriminant and j-invariant, and characterize isomorphism classes over the algebraic closure. |
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| 2. | Master the Group Law: Derive the geometric (chord?tangent) and algebraic group law, prove that it forms an abelian group, and write explicit formulas for addition in affine and projective coordinates. |
| 3. | Analyze Torsion and Structure over Finite Fields: Determine the group order of an elliptic curve over finite fields, apply Hasse?s bound, classify possible torsion subgroups over the rational field (Mazur?s Theorem). |
| 4. | Apply Height Functions: Define the naive Weil height and the canonical Neron?Tate height, prove their quadraticity and non?degeneracy. |
| 5. | Prove the Mordell?Weil Theorem: Understand the weak Mordell?Weil theorem, perform a descent argument, and synthesize the complete proof that E (K) is finitely generated for any number field K. |
| 6. | Perform Computations: Compute invariants such as rank, torsion, and reduction types. |
Course Objective
To provide an introduction to the basic arithmetic properties of elliptic curves.
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