Introduction to Elliptic Curves
Instructor: Michele Serra, Prof. Dr. Salma Kuhlmann
Time: Lecture: Wednesdays 10:00 - 11:30, room D404
Tutorial: Wednesdays 13:30 - 15:00, room M627
First lecture: Wednesday, 12 April 2023
Target group: from 5th semester on
Prerequisites: Linear Algebra I and II and Algebra I and II. Algorithmic Algebraic Geometry (B5) will be very helpful.
Languages: English / German
Goal
The aim of this course is to give a general introduction to arithmetic geometry, using elliptic curves as a Leitmotiv. We will cover the basics concerning rational points on elliptic curves aiming at a proof of Mordell's theorem.
Topics covered
12.04.2023: Introduction. Reminders on on affine and projective planes and plane curves.
19.04.2023: Rational points on lines, conics and singular (Weierstraß) cubics; Definition of elliptic curve. Resultants.
References: [1, Ch. 1].
26.04.2023: Discriminants; Bézout's theorem; Definition and first properties of the group law on an elliptic curve.
References: [1, Ch. 1], the proof of Bézout's theorem was roughly based on these notes of Pawel Gladki.
03.05.2023: Cayley-Bacharach Theorem, associativity of the group law, isomorphisms of elliptic curves.
References: [1, Ch. 1], [4, Ch.2, Sec.3]
17.05.2023: j-invariant; Endomorphisms of elliptic curves. [3, Ch. 3]
24.05.2023: Degree of an endomorphism, the multiplication-by-n-map, separability, the n-torsion subgroup [2, Ch. III]; Review of valuations
24.05.2023: Integral Weierstraß equations; the p-adic filtration [3, Ch. 5]
31.05.2023: The torsion group; the Nagell-Lutz theorem [2, Sec. VIII.7]; the reduction-mod-p map; Siegel's theorem on rational points and Mazur's theorem on the torsion subgrop (no proof)
14.06.2023: Heights: height on the projective space, logarithmic height, canonical height on elliptic curves. [2, Sec. VIII.9 - note that they use a more general notion: we restrict to the case where the function hf is just the x-coordinate]
28.06.2023: Properties of the canonical height; isogenies; construction of an isogenous curve to an elliptic curve with a 2-torsion point.
05.07.2023: The weak Mordell-Weil theorem (proof for an elliptic curve with a 2-torsion point); The Mordell-Weil theorem (over ℚ).
12.07.2023: Examples of computation of the group of rational points of an elliptic curve.
19.07.2023: The group of rational point of a singular cubic [1, Ch. 3]; the action of the Galois group on an Elliptic Curve [1, Ch 7].
Exercises
There will be biweekly exercise sheets and Tutorials. Tutorials take place Wednesdays, every two weeks, 13:30 - 15:00, in room M627, starting on 03.05.23.
Literature
[1] J. H. Silverman , J. T. Tate - Rational points on elliptic curves
[2] J. H. Silverman - The arithmetic of elliptic curves
[3] D. Husemöller - Elliptic curves
[4] J. S. Milne - Elliptic curves
We will follow chapters 1-3 of [1] as a road map and use the other references to fill in proofs and details.
The university library has several copies of [1], [2] and [3].
[4] is available online from the author's webpage.