Moments, Monodromy, and Perversity: A Diophantine Perspective
In this book the author studies equidistribution properties of several classes of trigonometric sums in characteristic p depending on a sufficiently large number of parameters. As in his previous work, the key point is to identify the perverse sheaf on the parameter space responsible for the trigonometric sum in question and then determine its geometric monodromy groups G. In the first two chapters, the author develops a new, global method for determining G. He begins by proving approximate “orthogonality relations” for trace functions of irreducible perverse sheaves, which allow him to compute the dimensions of tensor invariants of G in degrees less than a bound depending on n. On the other hand, results of M. Larsen, suitably generalized, imply that G is, essentially, determined by its tensor invariants in degree 8. In subsequent chapters, this general method is applied to additive character sums, multiplicative character sums and sums related to L-functions of elliptic curves (even though other techniques have to be used to distinguish the cases G=SO(N) and O(N)). The penultimate chapter investigates equidistribution results when p is a variable and the last chapter is devoted to equidistribution results for analytic ranks in families of elliptic curves over function fields.