Related papers: Twist-minimal trace formula for holomorphic cusp f…
Let $\{\lambda_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, \chi)$, where…
We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is…
The Helton-Howe measure associated with an almost normal operator was constructed by Helton and Howe. It provides a trace formula that allows us to calculate the trace of commutators that would otherwise be incalculable. We will investigate…
For a given cusp form $\phi$ of even integral weight satisfying certain hypotheses, Waldspurger's Theorem relates the critical value of the $\mathrm{L}$-function of the $n$-th quadratic twist of $\phi$ to the $n$-th coefficient of a certain…
In this paper, we use Schur elements to derive semisimplicity criteria for (super)symmetric superalgebras. We obtain a closed formula for the Schur elements of cyclotomic Hecke-Clifford superalgebras $\mathcal{H}^{f}_{\mathbb{K}}$. As…
In this paper, we prove a converse theorem for half-integral weight modular forms assuming functional equations for $L$-series with additive twists. This result is an extension of Booker, Farmer, and Lee's result in [BFL22] to the…
The local trace formula gives strong relations between two types of invariant distributions on a reductive group defined over a local field: orbital integrals and characters of representations. For connected reductive groups, the formula…
Inspired by Borcherds' questions, Guerzhoy constructed a new type of Hecke operators $\mathcal{T}(p)$, called the multiplicative Hecke operators, which acts on the space of meromorphic modular forms on the full modular group ${\rm SL}(\Z)$.…
We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo-Jacquet trace formulae and our methods are…
The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in…
We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. Since the twisting of the commutators turns the…
In this paper we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residue characteristic under a…
We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of…
In this paper we extend the results in [Ra] on the representation of the Hecke algebra, determined by the matrix coefficients of a projective, unitary representation, in the discrete series of representations of the ambient group, to a more…
Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local…
In this paper we study the analytic torsion for a complete oriented hyperbolic manifold of finite volume. This requires the definition of a regularized trace of heat operators. We use the Selberg trace formula to study the asymptotic…
Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the…
Let $q \in \mathbb{Z} [i]$ be prime and $\chi $ be the primitive quadratic Hecke character modulo $q$. Let $\pi$ be a self-dual Hecke automorphic cusp form for $\mathrm{SL}_3 (\mathbb{Z} [i] )$ and $f$ be a Hecke cusp form for $\Gamma_0 (q)…
We introduce an alternate set of generators for the Hecka algebra, and give an explicit formula for the action of these operators on Fourier coefficients. With this, we compute the eigenvalues of Hecke operators acting on average Siegel…
Twistor forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We study twistor forms on compact…