**Eisenstein congruences and arithmetic**

We will begin with two disparate and highly influential questions in arithmetic. For

what odd primes p is it straightforward to prove that the Fermat equation xp + yp = zp

has no non-trivial solutions among the rational numbers? And considering all possible

elliptic curve equations, one particular example being y2 + y = x3 – x2, 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. These results rely on new techniques to interpolate the

reciprocity laws of the Langlands program, which we will describe to conclude.

The new results are joint work with Preston Wake.