Riad Masri

Subconvexity and equidistribution of Heegner points in the level aspect

Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group character of Q(\sqrt{-D}), we establish a hybrid subconvexity bound for L(f \times h,s) at the central point when q = D^{\eta} for 0 < \eta < 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D become large with q < D^{1/20-\varepsilon}. Our approach to these problems is connected to estimating the L^2-restriction norm of a Maass form of large level when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke-Maass L-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet L-functions in certain ranges.

Analytic continuation of multiple Hurwitz zeta functions

We use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta function to , to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of .

Convergence rates for uniform B-spline density estimators on bounded and semi-infinite domains

Convergence rates for B-spline nonparametric density estimators on bounded and semi-infinite domains are discussed. We show how B-spline estimators can be adjusted to account for edge effects and then determine the mean integrated squared error for the adjusted estimator and its derivatives.

The asymptotic distribution of traces of cycle integrals of the

We establish an asymptotic formula with a power savings in the error term for traces of cycle integrals of the classical modular j-function j(z) = q−1 + 744 + 196884q + 21493760q + · · · . This implies a conjecture of Duke, Imamoḡlu, and Tóth [DIT, eq. (1.25)]. Duke Mathematical Journal, vol. 161 (2012), 1971–2000.

j

In a recent paper, Bruinier and Ono proved that certain harmonic weak Maass forms have the property that the Fourier coefficients of their holomorphic parts are algebraic traces of weak Maass forms evaluated on Heegner points. As a special case they obtained a remarkable finite algebraic formula for the Hardy-Ramanujan partition function

-function

p(n)

Fourier coefficients of harmonic weak Maass forms and the partition function

, which counts the number of partitions of a positive integer

Singular moduli and the distribution of partition ranks modulo 2

n

The Chowla–Selberg formula for abelian CM fields and Faltings heights

. We establish an asymptotic formula with a power saving error term for the Fourier coefficients in the Bruinier-Ono formula. As a consequence, we obtain a new asymptotic formula for

The scattering matrix for the Hilbert modular group

p(n)