Modular forms: Classification

This lecture is part of an online graduate course on modular forms. We first show that the number of zeros of a (level 1 holomorphic) modular form in a fundamental domain is weight/12, and use this to show that the graded ring of modular forms is the ring of polynomials in E4 and E6. For the other lectures in the course see    • Modular forms  

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Modular forms: Modular functions
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Modular forms: Modular functions

Modular forms: Eisenstein series
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Modular forms: Eisenstein series

We're 99.9% sure this pattern is true, but no one can prove it
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We're 99.9% sure this pattern is true, but no one can prove it

Modular forms: Introduction
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Modular forms: Introduction

Counting points on the E8 lattice with modular forms (theta functions) | #SoME2
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Counting points on the E8 lattice with modular forms (theta functions) | #SoME2

Modular forms: Discriminant and E2
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Modular forms: Discriminant and E2

What is... an elliptic curve?
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What is... an elliptic curve?

Modular forms: Fourier coefficients of Eisenstein series
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Modular forms: Fourier coefficients of Eisenstein series

The Story of Information Theory: from Morse to Shannon to ENTROPY
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The Story of Information Theory: from Morse to Shannon to ENTROPY

Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem
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Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem

Modular forms: Theta functions in higher dimensions
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Modular forms: Theta functions in higher dimensions

China’s Secret | The Most Unbelievable Megaprojects in China | 4K Travel Documentary
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China’s Secret | The Most Unbelievable Megaprojects in China | 4K Travel Documentary

Portugal – Demokratische Republik Kongo Highlights | Gruppe K, FIFA WM 2026 | sportstudio
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Portugal – Demokratische Republik Kongo Highlights | Gruppe K, FIFA WM 2026 | sportstudio

Categories 1 Introduction
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Categories 1 Introduction

a formula for the "circumference" of an ellipse.
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a formula for the "circumference" of an ellipse.

Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations
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Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

The Actual Reason Semiconductors Are Different From Conductors and Insulators.
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The Actual Reason Semiconductors Are Different From Conductors and Insulators.

[ENG] PRANK That’s NOT a student…A World-Class Violinist Secretly Entered a Real Korean Competition
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[ENG] PRANK That’s NOT a student…A World-Class Violinist Secretly Entered a Real Korean Competition

Modular forms: Fundamental domain
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Modular forms: Fundamental domain