Philadelphia Area Number Theory Seminar
Wednesday, October 24, 2018 - 2:15pm
Raphael Steiner
Institute for Advanced Study
It is a classical theorem in the theory of modular forms that the points x/ N, where x ∈ Z runs over all the solutions to i=1 xi = N, equidistribute on Sn−1 for n 4 as N (odd) tends to infinity. The rate of equidistribution poses however a more challenging problem. Due to its Diophantine nature the points inherit a repulsion property, which opposes equidistribution on small sets. Sarnak conjectures that this Diophantine repulsion is the only obstruction to the rate of equidistribution. Using the smooth delta-symbol circle method, developed by Heath-Brown, Sardari was able to show that the conjecture is true for n 5 and recovering Sarnak’s progress towards the conjecture for n = 4. Building on Sardari’s work, Browning, Kumaraswamy, and myself were able to reduce the conjecture to correlation sums of Kloosterman sums of the following type: q≤Q 1/q S(m, n; q) exp(4πiα√mn/q). Assuming the twisted Linnik conjecture, which states that the above sum is O((Qmn)) for |α| 2, we are able to verify Sarnak’s Conjecture. I shall lose a few words on the unconditional progress towards this conjecture and how (unfortunately) it is insufficient to improve unconditionally what is known towards Sarnak’s conjecture. If time permits, I will talk about ongoing research of how the automorphic approach and the circle method approach may be combined to hopefully give better insight into Sarnak’s conjecture.