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Heekyoung Hahn

Publications and source records attributed to Heekyoung Hahn.

13 recordsLinked to original sources
arXiv2025

Triple product $L$-functions and the Ramanujan conjecture

We prove that the Ramanujan conjecture is true under the assumption that the expected analytic properties of triple product $L$-functions hold. Further, we explain how these analytic properties imply certain reduction steps in the construction of functorial transfers in the sense of Langlands. Roughly, at the level of stably automorphic representations, they allow one to reduce any functorial transfer from a given reductive group $G$ to a general linear group to a finite family of transfers depending on $G.$

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arXiv2019

Poles of triple product $L$-functions involving monomial representations

In this paper, we study the order of the pole of the triple tensor product $L$-functions $L(s,π_1\timesπ_2\timesπ_3,\otimes^3)$ for cuspidal automorphic representations $π_i$ of $\mathrm{GL}_{n_i}(\mathbb{A}_F)$ in the setting where one of the $π_i$ 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.

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arXiv2017

From partition identities to a combinatorial approach to explicit Satake inversion

In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands' beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of $\mathrm{Sym}^j(\mathrm{Sym}^kV)$ for $\mathrm{GL}_2$ in the case $k=3$. We compute the plethysm explicitly for the cases $k=3, 4$. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power $L$-function of $\mathrm{GL}_2$ for these two cases.

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arXiv2015

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 $L(s,π,\,\otimes^3),$ where $π$ 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 $λ$, we examine whether or not $\otimes^3$ detects the subgroups $\mathbb{S}_{[λ]}(G)$ for $G$ with type $B_n$ and $D_{2n}$ or $\mathbb{S}_{\langleλ\rangle}(G)$ for $G$ with type $C_n$. Here $\mathbb{S}_{[λ]}$ and $\mathbb{S}_{\langleλ\rangle}$ are the usual Schur functors associated to the partition $λ$.

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arXiv2015

On tensor third $L$-functions of automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$

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 $L(s,π,\,\otimes^3),$ where $π$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ where $F$ is a global field.

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arXiv2015

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 $π$ 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 $π$ cuts out a "motive" $M_\mathrm{et}(π^{\infty})$ 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 $π$ we compute the dimension of the space of Tate classes in $M_\mathrm{et}(π^{\infty})$. 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 $M_\mathrm{et}(π^{\infty})$ is spanned by algebraic cycles.

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arXiv2015

Eisenstein series associated with $Γ_0(2)$

In this paper, we define the normalized Eisenstein series $\mathcal{P}$, $e$, and $\mathcal{Q}$ associated with $Γ_0(2),$ 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 $Γ_0(2)$ 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.

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arXiv2015

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 $\widetildeσ_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehatσ_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, 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 $\widetildeσ(n)$, $\widehatσ(n)$, and $\widetildeσ_3(n)$, 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 $δ_s(n)$, $s=4,$ $8$, in terms of $\widetildeσ(n)$, $\widehatσ(n)$, and $\widetildeσ_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $δ_s(n)$ 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.

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arXiv2013

Distribution of squarefree values of sequences associated with elliptic curves

Let E be a non-CM elliptic curve defined over Q. For each prime p of good reduction, E reduces to a curve E_p over the finite field F_p. For a given squarefree polynomial f(x,y), we examine the sequences f_p(E) := f(a_p(E), p), whose values are associated with the reduction of E over F_p. We are particularly interested in two sequences: f_p(E) =p + 1 - a_p(E) and f_p(E) = a_p(E)^2 - 4p. We present two results towards the goal of determining how often the values in a given sequence are squarefree. First, for any fixed curve E, we give an upper bound for the number of primes p up to X for which f_p(E) is squarefree. Moreover, we show that the conjectural asymptotic for the prime counting function π_{E,f}^{SF}(X) := #{p \leq X: f_p(E) is squarefree} is consistent with the asymptotic for the average over curves E in a suitable box.

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arXiv2006

Integrable systems and modular forms of level 2

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $Γ_0(2)$ 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 $S(0,0,1/2; z)$.

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arXiv2006

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 $\GN.$

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