Mini-course: Explicit Methods for Modular Forms and L-functions
Lecturers: John Cremona and Tim Dokchitser. TA: Martin Dickson.
Description
The first part of the course will explain what classical L-functions are and how to compute with them. Most L-functions in number theory, such as the Riemann and the Dedekind zeta-function, L-functions of Artin representations, L-functions of elliptic and higher genus curves, and of modular forms fall into one big framework of L-functions associated to Galois representations. The course will explain how the connection goes, what are the basic invariants of these L-functions and what is the shape of the functional equation. Most examples will be drawn from elliptic curves and (classical) modular forms. The second part of the course looks closely at classical modular forms, modular curves and the theory of modular symbols. The main motivation is the question how to determine all modular forms of a given weight and level. The lectures review what modular curves are, as quotients of the upper-half plane and as spaces parameterising elliptic curves, and their homology. The course then discusses modular symbols, how to compute them, and their relation to elliptic curves and the Birch-Swinnerton-Dyer conjecture The lectures will have a pronounced computational flavour.
Syllabus
- Galois representations;
- L-functions;
- invariants of L-functions;
- modular curves;
- modularity of elliptic curves;
- modular symbols.
Prerequisites
A first course on elliptic curves, basic algebraic number theory, Galois theory.
Notes and problems
John Cremona: lecture notes and exercises 1 and 2. Please see also these saved versions of Sage worksheets: the first doing the calculation of some of the genus 0 j-maps (including p=2) as was done in the lectures and exercises; the second which does the p=11 case from scratch in more detail than during the lectures.
Tim Dokchitser lecture notes and exercises.