相关论文: Explicit Construction of Maass Wave Forms and Thei…

In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic…

We construct two families of harmonic Maass Hecke eigenforms. This construction answers a question of Mazur about the existence of an "eigencurve-type" object in the world of harmonic Maass forms. Using these families, we construct $p$-adic…

We prove that Cohen's Maass wave form and Li-Ngo-Rhoades' Maass wave form are Hecke eigenforms with respect to certain Hecke operators. As a corollary, we find new identities of the $p$th coefficients of these Maass wave forms in terms of…

In this article, we compute the Petersson norm of a family of weight one cusp forms constructed by Hecke and express it in terms of the Rademacher symbol and the regulator of real quadratic field.

We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the…

Using special polynomials introduced by Hikami and the second author in their study of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new families of quantum modular forms, and…

We construct Hecke operators acting on Maass waveforms of integer non-zero weight and transforming according to a non-trivial multiplier system on the modular group. Using these Hecke operators we obtain multiplicativity relations for the…

We generalize the notions of locally and polar harmonic Maass forms to general orthogonal groups of signature $(2, n)$ with singularities along real analytic and algebraic cycles. We prove a current equation for locally harmonic Maass forms…

Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is…

In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of…

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_{\lambda}(N)$ of Maass forms with eigenvalue $1/4-\lambda^2$ on a congruence subgroup $\Gamma_1(N)$. We introduce the…

In this article, we derive a sub convexity estimate of Hecke eigen cusp forms associated to certain cocompact arithmetic subgroups of SL(2,R). The main result can be considered as the holomorphic version of the estimate of Hecke eigen Maass…

We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level $1$…

We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindel\"of hypothesis and a slightly…

Duke, Imamoglu, and Toth constructed a polyharmonic Maass form of level 4 whose Fourier coefficients encode real quadratic class numbers. A more general construction of such forms was subsequently given by Bruinier, Funke, and Imamoglu.…

In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions…

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients…

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and…

Matrix representations of Hecke operators on classical holomorphical cusp forms and corresponding period polynomials are well known. In this article we define Hecke operators on period functions and show that they correspond to the Hecke…

We explicitly construct non-tempered cusp forms on the orthogonal group O(1,5) of signature (1+,5-). Given a definite quaternion algebra B over $\mathbb{Q}$, the orthogonal group is attached to the indefinite quadratic space of rank 6 with…