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Roots of sextics mod p: Serre’s dream

Consider your favourite monic polynomial f(x) with integer coefficients. Is it irreducible? If not, then maybe consider another of your favourite such polynomials...

Question: How many roots does f(x) have mod p? For a fixed p this is easy to compute.

Tougher question: Is there an object that explains these counts for all p simultaneously?

(Btw: if you chose a linear polynomial, that was clearly a boring choice!)

In a beautiful 2003 paper titled, "On a theorem of Jordan", Serre investigates this problem for the specific family of polynomials f_n(x) = x^n - x - 1 (these are generic enough to reflect the behaviour of the general question). He takes the reader on an exhilarating adventure involving Legendre symbols, Galois theory, modular forms and representation theory...culminating in a full solution of the problem for n =< 4.

Serre ends his paper by throwing down a gauntlet and challenging the reader to solve the n = 5 case. This was only solved recently!

In this talk I will discuss all of the above and my own ongoing efforts to understand the n = 6 case.