Speaker: Nadir Hajouji

Title: The Weil Conjectures & Prescribed Quadratic Residues

 

While reading* about the Weil conjectures, I came across the following exercise:

 

Let n be a positive integer, and choose e_1, ..., e_n in the set {1, -1}.  Prove that for all p sufficiently large (relative to n), there exists an integer x such that x+k is a quadratic residue mod p if e_k=1, and x+k is a nonresidue if e_k = -1. 

 

After giving some background on the Weil conjectures, I will present a solution to the exercise.

 

*cf. www-math.mit.edu/~poonen/papers/Qpoints.pdf, Exercise 7.7