Rational functions and modular forms

Modular Functions and Modular Forms

These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions and corrections. Please send comments and corrections to me at [email protected].

Trigonometric functions, elliptic functions, elliptic modular forms

Mock Theta Functions and Quantum Modular Forms

Ramanujan’s last letter to Hardy concerns the asymptotic properties of modular forms and his ‘mock theta functions’. For the mock theta function f (q), Ramanujan claims that as q approaches an even-order 2k root of unity, we have f (q)− (−1)(1− q)(1− q)(1− q) · · · (1− 2q+ 2q − · · ·)= O(1). We prove Ramanujan’s claim as a special case of a more general result. The implied constants in Ramanuja...

Higher Green’s functions for modular forms

Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation ∆f = 0 we have equation ∆f = k(1 − k)f . Here k is a positive integer. Properties of these functions are related to the space of modular forms of weight 2k. I...

Galois characterization of Endoscopy for rational Siegel modular forms

We establish a relation between Galois reducibility and Endoscopy for genus 2 Siegel cusp forms which have rational eigenvalues and are unramified at 3.