We start with two disparate and influential questions in arithmetic. For what odd primes p is it straightforward to prove that the Fermat equation has no non-trivial solutions among the rational numbers? And considering all possible elliptic curve equations, one particular example being , what are all of the possibilities for the structure of the rational solutions as an abelian group?

 

We will explain that both of these questions are closely related to the existence of congruences between two kinds of modular forms: cusp forms and Eisenstein series. We will discuss some new results that refine these links, quantifying these congruences in terms of arithmetic. The new results are joint work with Preston Wake.