Lecture Notes: Modular Forms and Q-Series

Introduction

• Lecture focuses on modular forms, inspired by the book "Problems in the Theory of Modular Forms."
• Aim to cover one chapter per lecture, not diving into every detail.
• The objective is to inspire further study and exploration of modular forms.
• Modular forms are central to mathematics, humorously referred to as a fundamental operation by Martin Eichler.

Background on Modular Forms

• Origins traced back to Q-series, starting with Euler's work.
• Euler’s partition function, P(n), as the number of ways to write n as a sum of integers.
• Euler's power series approach to study the partition function.
• Generating series introduced by Euler.

Q-Series and Their Importance

• Euler's power series: 1/(1-q)(1-q²)...
• Ignoring convergence issues.
• Euler's observation: packaging sequences into power series helps find closed formulas.
• Q-series led to interesting identities and recursion formulas.

Jacobi Triple Product Identity

• Introduced in 1829 by Jacobi.
• General formula: product of terms equals a series.
• Importance in combinatorial mathematics.
• Relations to partition function.

Q-Philosophy

• New paradigm: Q-philosophy, viewing natural numbers as limits of q-expressions.
• Consequences in physics, number theory, and combinatorics.
• Concept of finite field of one element.
• Construct Q-analogues of classical functions.

Q-Exponential Function

• Classical exponential function, e^x = Σ(x^n/n!), has a Q-analogue.
• Q-analogue defined using Q-analogues of factorial.
• Resulting Q-exponential function has a product representation.
• Importance of Q > 1 for convergence.

Applications of Q-Series

• Proving Jacobi’s Triple Product Identity using star identity and transformations.
• Application in representing numbers as sums of squares.
• Use in combinatorics to determine ways of writing numbers as sum of squares.

Advanced Topics

• Introduction to modular forms with focus on Ramanujan’s work.
• Tau function and conjectures related to the function.
• Multiplicative property of tau function.

Conclusion

• Emphasis on the beauty and utility of modular forms and Q-series.
• Encouragement to delve deeper into these mathematical concepts.
• Future lectures will explore further aspects of modular forms.