Steven J. Miller (Williams College)
Low-lying zeros of GL(2) L-functions
Physicists developed Random Matrix Theory (RMT) in the 1950s to
explain the energy levels of heavy nuclei. A fortuitous meeting over tea
at the Institute in the 1970s revealed that similar answers are found for
zeros of L-functions, and since then RMT has been used to model their
behavior. The distribution of these zeros is intimately connected to many
problems in number theory, from how rapidly the number of primes less than
X grows to the class number problem to the bias of primes to be congruent
to 3 mod 4 and not 1 mod 4. The Katz and Sarnak density conjectures state
that in the limit the behavior of scaled zeros in a family of L-functions
agrees with the scaling limits of eigenvalues of a classical compact
group.
We report on recent progress on the n-level densities of low-lying zeros
of GL(2) L-functions. We derive an alternate formula for the Katz-Sarnak
determinant expansions for test functions with large support that
facilitates comparisons between number theory and random matrix theory in
orthogonal, symplectic, and unitary settings. Using combinatorics,
generating functions, and analysis, we prove these formulas hold and
increase the region where number theory and random matrix theory can be
shown to agree for holomorphic cuspidal newforms. If time permits we will
discuss extensions to other GL(2) families. This is joint work with
Nicholas Triantafillou (Michigan) and Levent Alpoge (Harvard).