Seminars in Number Theory and Forms

Ben Green: Quadratic forms in 8 prime variables

17th November, 2021

I will discuss a recent paper of mine, the aim of which is to count the number of prime solutions to Q(p1,..,p8) = N, for a fixed quadratic form Q and varying N. The traditional approach to problems of this type, the Hardy-Littlewood circle method, does not quite suffice. The main new idea is to involve the Weil representation of the symplectic groups Sp8(ℤ/qℤ). I will explain what this is, and what it has to do with the original problem. I hope to make the talk accessible to a fairly general audience.

Cameron L. Stewart: On integers represented by binary forms

15th October, 2020

We shall discuss the following results which are joint work with Stanley Xiao. Let F(x,y) be a binary form with integer coefficients, degree d (greater than 2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z2/d. Let k be an integer greater than 1 and suppose that there is no prime p such that pk divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z2/d.

Michel Waldschmidt: Representation of integers by cyclotomic binary forms

12th May, 2020

The representation of positive integers as a sum of two squares is a classical problem studied by Landau and Ramanujan. A similar result has been obtained by Bernays for positive definite binary form. In joint works with Claude Levesque and Etienne Fouvry, we consider the representation of integers by the binary forms which are deduced from the cyclotomic polynomials. One main tool is a recent result of Stewart and Xiao which generalizes the theorem of Bernays to binary forms of higher degree.