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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.