Eisenstein series associated with Γ0(2)\Gamma_0(2)

In this paper, we define the normalized Eisenstein series $\mathcal{P}$, $e$, and $\mathcal{Q}$ associated with $\Gamma_0(2),$ and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on $\Gamma_0(2)$ and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.Comment: This is an old paper uploaded for archival purpose

On Classical groups detected by the tensor third representation

Motivated by the Langlands' beyond endoscopy proposal for establishing functoriality, we study the representation $\otimes^3$ in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $G$ where $G$ is either $\mathrm{SO}(2n+1)$, $\mathrm{Sp}(2n)$ and $\mathrm{SO}(2n)$. In particular, under what conditions on partitions $\lambda$, we examine whether or not $\otimes^3$ detects the subgroups $\mathbb{S}_{[\lambda]}(G)$ for $G$ with type $B_n$ and $D_{2n}$ or $\mathbb{S}_{\langle\lambda\rangle}(G)$ for $G$ with type $C_n$. Here $\mathbb{S}_{[\lambda]}$ and $\mathbb{S}_{\langle\lambda\rangle}$ are the usual Schur functors associated to the partition $\lambda$

A simple twisted relative trace formula

In this article we derive a simple twisted relative trace formula.Comment: This is an old paper uploaded for archival purpose

On zeros of Eisenstein series for genus zero Fuchsian groups

Let \GN\leq\SLR be a genus zero Fuchsian group of the first kind with $\infty$ as a cusp, and let \Ek be the holomorphic Eisenstein series of weight $2k$ on \GN that is nonvanishing at $\infty$ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on \GN, and on a choice of a fundamental domain \F, we prove that all but possibly c(\GN,\F) of the non-trivial zeros of \Ek lie on a certain subset of \{z\in\mathfrak{H} : \JN(z)\in\mathbb{R}\}. Here c(\GN,\F) is a constant that does not depend on the weight $2k$ and \JN is the canonical hauptmodul for $\GN.

Convolution sums of some functions on divisors

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.Comment: This is an old paper uploaded for archival purpose

On tensor third LL-functions of automorphic representations of GLn(AF)\mathrm{GL}_n(\mathbb{A}_F)

Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ where $F$ is a global field

Poles of triple product LL-functions involving monomial representations

In this paper, we study the order of the pole of the triple tensor product $L$-functions $L(s,\pi_1\times\pi_2\times\pi_3,\otimes^3)$ for cuspidal automorphic representations $\pi_i$ of $\mathrm{GL}_{n_i}(\mathbb{A}_F)$ in the setting where one of the $\pi_i$ is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands' beyond endoscopy proposal

Algebraic cycles and Tate classes on Hilbert modular varieties

Let $E/\mathbb{Q}$ be a totally real number field that is Galois over $\mathbb{Q}$, and let $\pi$ be a cuspidal, nondihedral automorphic representation of $\mathrm{GL}_2(\mathbb{A}_E)$ that is in the lowest weight discrete series at every real place of $E$. The representation $\pi$ cuts out a "motive" $M_\mathrm{et}(\pi^{\infty})$ from the $\ell$-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If $\ell$ is sufficiently large in a sense that depends on $\pi$ we compute the dimension of the space of Tate classes in $M_\mathrm{et}(\pi^{\infty})$. Moreover if the space of Tate classes on this motive over all finite abelian extensions $k/E$ is at most of rank one as a Hecke module, we prove that the space of Tate classes in $M_\mathrm{et}(\pi^{\infty})$ is spanned by algebraic cycles.Comment: This is an old paper posted for archival purpose

A general simple relative trace formula and a relative Weyl law

In this paper, we prove a general simple relative trace formula. As an application, we prove a relative analogue of the Weyl law.Comment: To be published in the Pacific Journal of Mathematics, 19 pages. Previous version contains a more detailed proof of the relative Weyl la

Integrable systems and modular forms of level 2

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear equations are analogues of the well known Ramanujan equations, as well as the Chazy and Darboux-Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz trianle function $S(0,0,1/2; z)$.Comment: PACS numbers: 02.30.Ik, 02.30.Hq, 02.10.De, 02.30.G