Matthew P. Young

Low-lying zeros of families of elliptic curves

The random matrix model predicts that many statistics associated to zeros of a family of L-functions can be modeled (or predicted) by the distribution of eigenval ues of large random matrices in one of the classical linear groups. If the statistics of a family of L-functions are modeled by the eigenvalues of the group G, then we say that G is the symmetry group (or symmetry type) associated to the family. The statistic of interest to us in this work is the density of zeros near the central point (also known as the 1-level density). The random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types G (unitary, symplectic, and orthogonal). All of the different groups G have distinct behavior in this regard. Therefore, computing the 1-level density gives a theoretical way to predict the symmetry type of a family. The 1-level density has been studied for a wide variety of families of L-functions; see [R], [KSI], [ILS], [Mil] for example. It is standard to assume the Generalized Riemann Hypothesis (GRH) to study the 1-level density and we do so throughout this work. In particular, it is necessary to use GRH for the application of obtaining a bound on the average analytic rank from a density theorem. In some cases the use of GRH improves the range of the density theorem, which translates to an improved bound on the average rank. Besides these crucial applications of GRH, we have freely assumed GRH even when it could be removed with extra work since it simplifies arguments in some non essential places. It is especially interesting to investigate the 1-level density for families of L functions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information (by the conjecture of Birch and Swinnerton Dyer). These investigations have been the main focus of this work.

More than 41% of the zeros of the zeta function are on the critical line

We prove that at least 41.05% of the zeros of the Riemann zeta function are on the critical line.

Distribution of mass of holomorphic cusp forms

We prove an upper bound for the L^4-norm and for the L^2-norm restricted to the vertical geodesic of a holomorphic Hecke cusp form of large weight. The method is based on Watsons formula and estimating a mean value of certain L-functions of degree 6. Further applications to restriction problems of Siegel modular forms and subconvexity bounds of degree 8 L-functions are given.

Lower-order terms of the 1-level density of families of elliptic curves

The Katz-Sarnak philosophy predicts that statistics of zeros of families of L-functions are strikingly universal. However, subtle arithmetical differences between families of the same symmetry type can be detected by calculating lower-order terms of the statistics of interest. In this paper we calculate lower-order terms of the 1-level density of some families of elliptic curves. We show that there are essentially two different effects on the distribution of low-lying zeros. First, low-lying zeros are more numerous in families of elliptic curves E with relatively large numbers of points (mod p). Second, and somewhat surprisingly, a family with a relatively large number of primes of bad reduction has relatively fewer low-lying zeros. We also show that the lower order term can grow arbitrarily large by taking a biased family with a relatively large number of points (mod p) for all small primes p.

The fourth moment of Dirichlet L-functions

We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.

The quantum unique ergodicity conjecture for thin sets

Abstract We consider some analogs of the quantum unique ergodicity conjecture for geodesics and “shrinking” families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite geodesic connecting 0 and ∞ inside the modular surface.

THE TWISTED FOURTH MOMENT OF THE RIEMANN ZETA FUNCTION

Abstract We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length .

Weyl-type hybrid subconvexity bounds for twisted

We prove a Weyl-type subconvexity bound for the central value of the

L

L

-functions and Heegner points on shrinking sets

-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet