w notable applications of modular forms. These are only for illustrative purposes, so if you don't understand the details of these (and you likely won't until we are much further in the class

It is easy to define modular functions and forms, but less easy to say why they are important, especially to number theorists. Thus I shall begin with a rather long overview of the subject.

Now, we can write down tables for modular arithmetic. For example, here are the tables for arithmetic modulo 4 and modulo 5. ... The table for addition is rather boring, and it changes in …

Goal of this course: define and study spaces of modular forms ⇝ Mk(Γ), C-vector spaces of modular forms, finite-dimensional equipped with Hecke operators. 1. Theory of elliptic …

New notion of “sameness” or “equivalence” that will help us understand modular arithmetic. This is a predicate (T/F values) on integers . It does not produce numbers as output. There is really a …

We call b the residue of a modulo n and write b = (a mod n). 2 ( 2) = 4 is not a multiple of 9. This is the source of many mistakes. Suppose a b and c d (mod n). Then. 6 36 (mod 10), but …

Modular forms arise as di erentials on classifying spaces for elliptic curves, and there are natural ways to produce Galois representations from both elliptic curves and modular forms.