C3.7 Elliptic Curves - Material for the year 2020-2021

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Prof. Alan Lauder
General Prerequisites: 

It is helpful, but not essential, if students have already taken a standard introduction to algebraic curves and algebraic number theory.
For those students who may have gaps in their background, I have placed the file "Preliminary Reading" permanently on the Elliptic Curves webpage, which gives in detail (about 30 pages) the main prerequisite knowledge for the course.

Course Term: 
Course Lecture Information: 

16 lectures.

Course Weight: 
1.00 unit(s)
Course Level: 

Assessment type:

Course Overview: 

Elliptic curves give the simplest examples of many of the most interesting phenomena which can occur in algebraic curves; they have an incredibly rich structure and have been the testing ground for many developments in algebraic geometry whilst the theory is still full of deep unsolved conjectures, some of which are amongst the oldest unsolved problems in mathematics. The course will concentrate on arithmetic aspects of elliptic curves defined over the rationals, with the study of the group of rational points, and explicit determination of the rank, being the primary focus. Using elliptic curves over the rationals as an example, we will be able to introduce many of the basic tools for studying arithmetic properties of algebraic varieties.

Learning Outcomes: 

On completing the course, students should be able to understand and use properties of elliptic curves, such as the group law, the torsion group of rational points, and 2-isogenies between elliptic curves. They should be able to understand and apply the theory of fields with valuations, emphasising the $p$-adic numbers, and be able to prove and apply Hensel's Lemma in problem solving. They should be able to understand the proof of the Mordell-Weil Theorem for the case when an elliptic curve has a rational point of order 2, and compute ranks in such cases, for examples where all homogeneous spaces for descent-via-2-isogeny satisfy the Hasse principle. They should also be able to apply the elliptic curve method for the factorisation of integers.

Course Synopsis: 

Non-singular cubics and the group law; Weierstrass equations.

Elliptic curves over finite fields; Hasse estimate (stated without proof).

$p$-adic fields (basic definitions and properties).

1-dimensional formal groups (basic definitions and properties).

Curves over $p$-adic fields and reduction mod $p$.

Computation of torsion groups over $\mathbb{Q}$; the Nagell-Lutz theorem.

2-isogenies on elliptic curves defined over $\mathbb{Q}$, with a $\mathbb{Q}$-rational point of order 2.

Weak Mordell-Weil Theorem for elliptic curves defined over $\mathbb{Q}$, with a $\mathbb{Q}$-rational point of order 2.

Height functions on Abelian groups and basic properties.

Heights of points on elliptic curves defined over $\mathbb{Q}$; statement (without proof) that this gives a height function on the Mordell-Weil group.

Mordell-Weil Theorem for elliptic curves defined over $\mathbb{Q}$, with a $\mathbb{Q}$-rational point of order 2.

Explicit computation of rank using descent via 2-isogeny.

Public keys in cryptography; Pollard's $(p-1)$ method and the elliptic curve method of factorisation.

Reading List: 
  1. J.W.S. Cassels, Lectures on Elliptic Curves, LMS Student Texts 24 (Cambridge University Press, 1991).
  2. N. Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics 114 (Springer, 1987).
  3. J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics (Springer, 1992).
  4. J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106 (Springer, 1986).
Further Reading: 
  1. A. Knapp, Elliptic Curves, Mathematical Notes 40 (Princeton University Press, 1992).
  2. G, Cornell, J.H. Silverman and G. Stevans (editors), Modular Forms and Fermat's Last Theorem (Springer, 1997).
  3. J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151 (Springer, 1994).