arXiv subjects

Min Lee

Publications and source records attributed to Min Lee.

30 records · Page 2Linked to original sources

arXiv2019

Test vectors for Rankin-Selberg $L$-functions

We study the local zeta integrals attached to a pair of generic representations $(π,τ)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $π$ and $τ$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.

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arXiv2018

Values of Harmonic Weak Maass forms on Hecke orbits

Let $q:=e^{2 πiz}$, where $z \in \mathbb{H}$. For an even integer $k$, let $f(z):=q^h\prod_{m=1}^{\infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Γ_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$ of a modular form and the points in the support of $T_m.D$. In this paper, we extend this result to general $Γ_0(N)$ in terms of values of harmonic weak Maass forms of weight $0$. By the distribution of Hecke points, this applies to obtain an asymptotic behaviour of convolutions of sums of divisors of an integer and sums of exponents of a modular form.

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arXiv2018

New integral representations for Rankin-Selberg L-functions

We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test vector at finite places, irrespective of ramification. This enables a new proof of the functional equation for GL(3) x GL(2) Rankin-Selberg L-functions in many cases.

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arXiv2017

A conjectural extension of Hecke's converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.

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arXiv2016

Effective equidistribution of rational points on expanding horospheres

Einsiedler, Mozes, Shah and Shapira [Compos. Math. 152 (2016), 667-692] prove an equidistribution theorem for rational points on expanding horospheres in the space of d-dimensional Euclidean lattices, with d>2. Their proof exploits measure classification results, but provides no insight into the rate of convergence. We pursue here an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums, which in dimension d=3 yields an effective estimate on the rate of convergence.

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arXiv2016

The Selberg trace formula as a Dirichlet series

We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N>1.

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arXiv2014

Shifted Multiple Dirichlet Series

We develop certain aspects of the theory of shifted multiple Dirichlet series and study their meromorphic continuations. These continuations are used to obtain explicit spectral first and second moments of Rankin-Selberg convolutions. One consequence is a Weyl type estimate for the Rankin-Selberg convolution of a holomorphic cusp form and a Maass form with spectral parameter $|t_j|\le T$, namely: $$ \left| L\left(\frac{1}{2}+ir, f\times u_j\right) \right| \ll_N T^{2/3+ε}, $$ uniformly, for $|r| \le T^{2/3}$, with the implied constant depending only on $f$ and the level $N$.

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arXiv2013

Second Moments and simultaneous non-vanishing of GL(2) automorphic L-series

We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q, with an error term that has a close to optimal power savings in the exponent. However, one of the contributions to the main term is a special value of a shifted double Dirichlet series. We show that this special value is small on average, and obtain a corresponding estimate for a mean value of the second moment over Q. This mean value is non-zero even when applied to a product of two distinct L-series, leading to a simultaneous non-vanishing result. Our approach uses the theory of shifted multiple Dirichlet series to obtain some refined estimates for double shifted sums. These estimates are the key ingredient in the second moment estimate.

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arXiv2012

Effective Circle Count for Apollonian packings and Closed horospheres

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fundamental group has critical exponent bigger than 1. We also discuss applications to Affine sieves. Analogous results for surfaces are treated as well.

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arXiv2012

Approximate Converse Theorem

We present an approximate converse theorem which measures how close a given set of irreducible admissible unramified unitary generic local representations of GL(n) is to a genuine cuspidal representation. To get a formula for the measure, we introduce a quasi-Maass form on the generalized upper half plane for a given set of local representations. We also construct an annihilating operator which enables us to write down an explicit cuspidal automorphic function.

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arXiv2012

Effective Equidistribution of Closed horocycles for Geometrically finite surfaces

For a complete hyperbolic surface whose fundamental group is finitely generated and has critical exponent bigger than 1/2, we obtain an effective equidistribution of closed horocycles in its unit tangent bundle. This extends a result of Sarnak in 1981 for surfaces of finite area. We also discuss its applications in Affine sieves.

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arXiv2010

Fourier expansions of GL(2) newforms at various cusps

This paper studies the Fourier expansion of Hecke-Maass eigenforms for $GL(2, \mathbb Q)$ of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some cases 8 times) an odd squarefree number then there are multiplicative relations at every cusp. We also show that a local representation of $GL(2, \mathbb Q_p)$ which is isomorphic to a local factor of a global cuspidal automorphic representation generated by the adelic lift of a newform of arbitrary weight, level $N$, and character $χ\pmod{N}$ cannot be supercuspidal if $χ$ is primitive. Furthermore, it is supercuspidal if and only if at every cusp (of width $m$ and cusp parameter = 0) the $mp^\ell$ Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers $\ell$. In the last part of this paper a three term identity involving the Fourier expansion at three different cusps is derived.

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