Theta Functions Course
Dive into theta functions, progressing from series convergence and basic properties to modular transformations and their role in elliptic functions. Construct explicit models via theta quotients, explore connections to the Weierstrass ℘-function, and discover applications in elliptic curves, modular forms, and physics, with rigorous proofs and practical examples throughout.

4 to 360h flexible workload
certificate valid in your country
What will I learn?
This course provides a structured journey through theta functions, starting from fundamental definitions and convergence properties, advancing to essential functional equations, transformation laws, and quasi-periodicity. Participants will learn to encode lattices, build elliptic functions using theta quotients, connect them to the Weierstrass ℘-function, and leverage uniqueness and dimension theorems, supported by detailed proofs, examples, and tips for rigorous writing and proper citations.
Elevify advantages
Develop skills
- Master theta series convergence, including control over domains and holomorphic properties.
- Derive key theta identities such as the heat equation, quasi-periodicity, and modular S-transformation.
- Construct elliptic functions using theta quotients to achieve specified divisors.
- Analyze relations between theta functions and Weierstrass ℘-function via poles, zeros, and fields.
- Develop skills in writing precise mathematical notes, normalizing lattices, and citing sources accurately.
Suggested summary
Before starting, you can change the chapters and the workload. Choose which chapter to start with. Add or remove chapters. Increase or decrease the course workload.What our students say
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