Vinay Viswanathan

Counting points on quadrics with arithmetic weights

The averages of arithmetic functions along thin sequences have been widely examined in analytic number theory. In this talk, I will attempt to contribute to this body of research by discussing a problem that has hitherto not been analysed systematically: given a nonsingular diagonal quadratic form $F$ in 4 variables with integer coefficients, can we count integer solutions in bounded but expanding boxes, to $F=0$, where one, or several, of the coordinates are weighted by Fourier coefficients of automorphic forms? This problem can be solved when only one of the coordinates is weighted, and I will sketch a proof of this result using the smooth $\delta$-method of Duke, Friedlander and Iwaniec.