Open conditions for infinite multiplicity eigenvalues on elliptic curves

Bounds on the multiplicity of the Hecke eigenvalues

Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.

利用数论研究可表示为两平方和的 Laplacian 特征值的重数

Multiplicity of Laplacian eigenvalues that can be represented by sum of two squares using number theory

该研究借助数论证明了二维椭圆 PDE 中关于两平方和型 Laplacian 特征值任意指定奇偶重数的猜想，并给出构造公式。

关于夹紧圆板特征值重数界限的讨论

On multiplicity bounds for eigenvalues of the clamped round plate

研究夹紧圆板谱特征值重数是否仅源于球面等距同构，通过Siegel-Shidlovskii理论证明任意特征函数至多由两个可分函数组成，并得出二维下特征值重数不超过四。

On exceptional eigenvalues of the Laplacian for $Γ_0(N)$

An explicit Dirichlet series is obtained, which represents an analytic function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at points $s_j$ that correspond to exceptional eigenvalues $λ_j$ of the non-Euclidean Laplacian for Hecke congruence subgroups $Γ_0(N)$ by the relation $λ_j=s_j(1-s_j)$ for $j=1,2,..., S$. Coefficients of the Dirichlet series involve all class numbers $h_d$ of real quadratic number fields. But, only the terms with $h_d\gg d^{1/2-ε}$ for sufficiently large discriminants $d$ contribute to the residues $m_j/2$ of the Dirichlet series at the poles $s_j…

A refinement of strong multiplicity one for spectra of hyperbolic manifolds

Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively, multiplicities of lengths of closed geodesics in $\calM_1$ and $\calM_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.