We discuss a spectral summation formula which relates the product of four Fourier coefficients of half-integral weight cusp forms in Kohnen's subspace to arithmetic objects. The arithemetic objects occurring here are certain generalized class numbers of pairs of quadratic forms with integer coefficients.
We consider the question when central $L$-values generate the field generated by its Dirichlet coefficients. In case under consideration this leads to certain shifted convolutions that are solved by a mixture of methods from automorphic forms, algebraic geometry and sieve theory.
Let $R$ be an infinite ring that is finitely generated over $\mathbb{Z}$. Hilbert’s $10^{\text{th}}$ problem for $R$ asks: does there exist an algorithm that given as input a polynomial $f$ in $R[X_1, …, X_n]$ outputs yes if $f$ has a zero in $R$ and no otherwise. Mazur and Rubin proved that Hilbert’s $10^{\text{th}}$ problem is undecidable assuming finiteness of Sha. I will outline how to prove the same result without assuming finiteness of Sha. The key new input is a recent version of the Green-Tao theorem for number fields. This is joint work with Carlo Pagano.
I will discuss upcoming joint works with Brian Conrey, Yongxiao Lin, and Caroline Turnage-Butterbaugh, in which we establish an unconditional asymptotic formula for the moment described in the title, using the method of Asymptotic Large Sieve. I will also talk about applications to the statistics of critical zeros and related questions.
The original quantum ergodicity theorem of Snirelman, Zelditch, and Colin de Verdiere concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which converge in the sense of Benjamini-Schramm to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. The breakthrough work of Anantharaman-Le Masson established this type of quantum ergodicity for regular graphs. Subsequently a new proof was given by Brooks-Le Masson-Lindenstrauss using a sort of wave propagation on graphs with desirable spectral and geometric properties. This technique was adapted by Le Masson-Sahlsten to prove analogous results for hyperbolic surfaces, and by Abert-Bergeron-Le Masson for rank one locally symmetric spaces. Brumley-Matz initiated study of the higher rank setting, focusing on compact locally symmetric spaces associated to $\operatorname{SL}(n, \mathbb{R})$, and in my thesis I investigated compact quotients of the Bruhat-Tits building associated to $\operatorname{SL}(3)$ over a non-archimedean local field. In joint work with Farrell Brumley, Simon Marshall, and Jasmin Matz, we extend these results to nearly all compact locally symmetric spaces and in particular repair an error in the earlier work of Brumley-Matz.
We survey various known bounds on the error term in the hyperbolic circle problem. This includes pointwise bounds, second radial moments, second spatial moments, and more. We then go on to explain how we may obtain a logarithmic saving on Selberg’s pointwise bound in the case of the full modular group, and specific Heegner points related to different squarefree fundamental discriminants. This is joint work with Dimitrios Chatzakos, Giacomo Cherubini and Stephen Lester.
We consider the distribution of $L^2$-mass for the sequence $\psi_n$ of Hecke--Maass forms constructed by Pitale on a congruence quotient of the hyperbolic 4-space as lifts from half-integral weight forms (the $\psi_n$ are non-holomorphic analogues of the Saito--Kurokawa lifts). We show that this sequence satisfies the quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak.
This is progress report on an ongoing joint project with M. Harris and T. Kobayashi. Suppose that $H$, $H'$ are unitary groups of hermitian spaces over an imaginary quadratic field with $H(R) = G$, $H'(R) = G'$ and let $\Pi$ and $\Pi'$ be cuspidal automorphic representations of $H = U (V)$ and $H' = U (V')$, respectively. The goal of the project is obtain an expression for the central value of a certain global automorphic $L$-function $L(s, \Pi \otimes (\Pi')^{\vee})$ in terms of automorphic periods of automorphic forms in $\Pi ⊗ (\Pi')^{\vee}$. The tool is understanding the restriction of coherent cohomology of certain Shimura varieties to subvarieties. Here problems in the representation theory of the real Lie groups $G = U (V)$, $G' = U (V')$ need to be solved, where $V$ is an $n + 1$-dimensional complex vector space with a non-degenerate hermitian inner product and $V ′ \subset V$ a (non-degenerate) codimension one hermitian subspace; most of my lecture will discuss this aspect of the project.
I will discuss joint work with N. Arala, J. R. Getz, J. Hou, C.-H. Hsu, and H. Li, concerning a new, nonabelian circle method and its applications to counting problems of a classical flavor.
