On the sparsity of positive-definite automorphic forms
Heilbronn Number Theory Seminar
22nd May 2019, 4:00 pm – 5:00 pm
Howard House, 4th Floor Seminar Room
We say that an automorphic form or an L-function is positive-definite if the corresponding completed L-function (times s^k(1-s)^k, where k is the order of pole at s=1) is a positive-definite function when restricted to the critical line. It is fairly straightforward to prove that a positive-definite L-function does not have any zero on (1/2,1), hence no Siegel zero. It is also known that most automorphic forms with small conductor are positive-definite, including Riemann zeta function. For instance, 145 among first 168 quadratic Dirichlet characters with prime conductor are positive-definite. Also, among 319 rank zero elliptic curves of conductor less than 1000, 283 of them are positive-definite.
A natural question is to understand the abundance of such forms. It is remarkable that in 1990, Baker and Montgomery proved that positive-definite Dirichlet characters are, in fact, very sparse. In particular, they showed that when quadratic Dirichlet characters are ordered by their conductor, then the natural density of positive-definite characters is zero.
In this talk, I'm going to present a soft proof of their result, and generalize it to various families of automorphic forms. In particular, I will explain how such a proof can be applied to deduce that the natural density of positive-definite automorphic forms within the following families is zero:
(1) family of primitive holomorphic cusp forms of fixed weight,
(2) family of modular forms of weight 1 associated to a Hilbert class
character of an imaginary quadratic field, and
(3) one-parameter family of elliptic curves.