Over the summer of 2024, I was funded by UCL to undertake a research project under the supervision of Assoc. Prof. Cecilia Busuioc. I am very grateful to UCL for funding this and especially thankful for all the work Cecilia did in helping me with the project. She was a brilliant supervisor and I am very glad to have been able to work with her.
The project was on elliptic curves with complex multiplication. Below is attached a set of explanatory notes on the topic and I have pasted the introduction here for convenience.
The study of algebraic number theory is primarily the study of algebraic number fields, that is finite extensions of the rationals. When studying field extensions, one of the most indispensable tools is that of the Galois group of the extension and the famous result of Kronecker-Weber tells us that every finite extension of (\mathbb{Q}) with an abelian Galois group is contained in a field of the form ( \mathbb{Q}(\zeta_n) ) where ( \zeta_n ) is a primitive ( n )th root of unity. One may then ask, given some number field ( K ), does there exist a field ( L ) such that every abelian extension of ( K ) is a subfield of ( L )? This explanatory note aims to partially answer that question in the case where ( K = \mathbb{Q}(\sqrt{-m}) ) for some square-free ( m \in \mathbb{Z}_{>0} ), that is an imaginary quadratic field. To do so, we will develop the theory of elliptic curves with complex multiplication which provides us with a correspondence between certain sets of elliptic curves over ( \mathbb{C} ) and the class group of an order in an imaginary quadratic field. This correspondence relies on values of the miraculous ( j )-invariant, and it is ( j(\tau) ), for some ( \tau ) in the upper half plane that will generate the field ( L ) we are after. This result requires the machinery of class field theory, which we will not go into detail on. We will, however, build up enough understanding to motivate and grasp the statement of the theorems. Along the way, we will give a detailed account of the uniformisation theorem; delve into the study of modular functions; and end by proving some remarkable facts about the ( j )-invariant, in particular that, if ( \tau ) is an algebraic integer in an imaginary quadratic field, then ( j(\tau) ) is also an algebraic integer.
The structure of this note is as follows. We open with 2. Elliptic Curves which very briefly recaps key definitions relating to elliptic curves, mainly to cement definitions and conventions. With 3. Elliptic Functions, we study elliptic functions, in particular, the Weierstrass ( \wp )-function, which we will show can be used to parameterise an elliptic curve isomorphic to a complex torus. The content of 4. Uniformisation Theorem then shows that every elliptic curve is one of this form and hence isomorphic to a complex torus. To prove this, we introduce modular functions and the ( j )-invariant. We then take a brief aside in 5. Some Ideas from Class Field Theory to recap basic ideas of algebraic number theory and then cover some of the main results of class field theory, including the answer to the problem of abelian extensions of imaginary quadratic fields. These results are very useful in 6. Complex Multiplication where we introduce the notion of elliptic curves with complex multiplication and use them to show that ( j(\tau) ) is an algebraic integer when ( \tau ) is an algebraic integer in an imaginary quadratic field. 7. Calculating Examples is then devoted to finding explicit values of ( j(\tau) ) in these cases.
Whilst we aim to explain the majority of the theory needed, there are a few prerequisites required for understanding this text. Primarily, we recommend the reader be comfortable with the fundamentals of Galois theory, specifically the fundamental theorem. Also, it would be helpful if the reader were somewhat already familiar with the basic ideas of elliptic curves and algebraic number theory as we make extensive use of both and the recaps given do not go into a lot of detail.