arXiv subjects
Publications and source records attributed to Matthew P. Young.
We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number theory.
Using a cubic moment, we prove a Weyl-type subconvexity bound for the quadratic twists of a holomorphic newform of square-free level, trivial nebentypus, and arbitrary even weight. This generalizes work of Conrey and Iwaniec in that the newform that is being twisted may have arbitrary square-free level, and also that the quadratic character may have even conductor. One of the new tools developed in this paper is a more general Petersson formula for newforms of square-free level.
We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies restricted QUE. We prove a lower bound that matches this predicted asymptotic. We also prove an upper bound that nearly matches the lower bound assuming the Riemann Hypothesis (unconditionally, the sharp upper bound holds for almost all $T$). Finally, we show the restricted QUE conjecture for geodesics with rational endpoints.
We prove a sharp bound on the fifth moment of modular L-functions of fixed small weight, and large prime level.
We derive explicit formulas for some Kloosterman sums on $Γ_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.
We prove a Weyl-type subconvexity bound for the central value of the $L$-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 $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.
We analyze certain bilinear forms involving $GL_3$ Kloosterman sums. As an application, we obtain an improved estimate for the $GL_3$ spectral large sieve inequality.
We prove a quantitative lower bound on the number of nodal domains of the real-analytic Eisenstein series. The main tool in the proof is a quantitative restricted QUE theorem where the support of the test function is allowed to shrink with the Laplace eigenvalue.
We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_k(z) E_{\ell}(z) - E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. We moreover find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $x = \pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.
We prove a sup norm bound on the real-analytic Eisenstein series, of the form $E(z, 1/2 + iT) \ll T^{3/8 + \varepsilon}$, uniformly for $z$ in a fixed compact subset of $\mathbb{H}$.
We estimate the number of solutions to $|y^2 - x^3| \leq X$ with $N \leq y \leq 2N$, in terms of both $N$ and $X$.
We develop some of the finer details of the location of the zeros of the weight two Eisenstein series. These zeros are the same as the zeros of the derivative of the Ramanujan delta function.
We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or ``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 infinity inside the modular surface.
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^η for 0 < η< 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.
We study the third moment of quadratic Dirichlet L-functions, obtaining an error term of size $O(X^{3/4 + \varepsilon})$.
We study the analytic behavior of the restriction of a Siegel modular form to $\mathbb{H} \times \mathbb{H}$ in the case that the Siegel form is a Saito-Kurokawa lift. A formula of Ichino links this behavior to a family of $GL_3 \times GL_2$ $L$-functions.
We study the sum of additively twisted Fourier coefficients of a symmetric-square lift of a Maass form invariant under the full modular group. Our bounds are uniform in terms of the spectral parameter of the Maass form, as well as in terms of the additive twist.
We prove a sharp upper bound on the $L^2$ norm of a $GL_3$ Maass form restricted to $GL_2 \times \mathbb{R}^{+}$.