Heekyoung Hahn

Integrable systems and modular forms of level 2

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup

On zeros of Eisenstein series for genus zero Fuchsian groups

\Gamma_0(2)

A general simple relative trace formula

of the modular group

Convolution Sums of Some Functions on Divisors

SL_2(\mathbb{Z})

Septic analogues of the Rogers–Ramanujan functions

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

Eisenstein series associated with Γ0(2)

S(0,0,1/2; z)

A Simple Twisted Relative Trace Formula

.

New Ramanujan-Kolberg type partition identities

Let r < SL 2 (R) be a genus zero Fuchsian group of the first kind with ∞ as a cusp, and let E Γ 2k be the holomorphic Eisenstein series of weight 2k on r that is nonvanishing at ∞ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on Γ, and on a choice of a fundamental domain F, we prove that all but possibly c(Γ, F) of the nontrivial zeros of E Γ 2k lie on a certain subset of {z ∈ h: j Γ (z) ∈ R}. Here c(Γ, F) is a constant that does not depend on the weight, S) is the upper half-plane, and j Γ is the canonical hauptmodul for Γ.

On tensor third -functions of automorphic representations of _{}(_{})

In this paper we prove a relative trace formula for all pairs of connected algebraic groups H G G, with G a reductive group and H the direct product of a reductive group and a unipotent group, given that the test function satisfies simplifying hypotheses. As an application, we prove a relative analogue of the Weyl law, giving an asymptotic formula for the number of eigenfunctions of the Laplacian on a locally symmetric space associated to G weighted by their L 2 -restriction norm over a locally symmetric subspace associated to H0 G.