Quasimodular forms and modular differential equations which are not apparent at cusps: I

关于素数检测拟模形式的一个猜想

On a conjecture about prime-detecting quasimodular forms

本文证明了Craig等人关于素数检测拟模形式的分类猜想，关键方法是证明拟模尖形式的傅里叶系数有无穷多次符号变化。

随机矩阵的超几何函数与拟模形式

Hypergeometric Functions of Random Matrices and Quasimodular Forms

该研究将多元统计学中的复矩阵超几何函数用作圆酉系综的全纯可观测量，分析其期望导数的高维行为并证明渐近展开可用拟模形式描述，建立了CUE与数论的新关联。

The second moment of $GL(3) \times GL(2)$ $L$-functions at special points

For a fixed SL(3, Z) Maass form g, we consider the family of L-functions L(g \times u_j, s) where u_j runs over the family of Hecke-Maass cusp forms on SL(2,Z). We obtain an estimate for the second moment of this family of L-functions at the special points \half + it_j consistent with the Lindelöf Hypothesis. We also obtain a similar upper bound on the sixth moment of the family of Hecke-Maass cusp forms at these special points; this is apparently the first occurrence of a Lindelöf-consistent estimate for a sixth power moment of a family of GL(2) L-functions.

A construction of $\mathfrak v$-adic modular forms

The classical theory of $p$-adic (elliptic) modular forms arose in the 1970's from the work of J.-P.\ Serre \cite{se1} who took $p$-adic limits of the $q$-expansions of these forms. It was soon expanded by N.\ Katz \cite{ka1} with a more functorial approach. Since then the theory has grown in a variety of directions. In the late 1970's, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms \cite{go1}, \cite{go2}. The associated expansions at $\infty$ are quite complicated and no obvious limits at finite primes ${\mathfrak v}$ were apparent. …

On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture

This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level $Γ_0(4)$ and half-integral weights. Based on substantial calculations, the question is raised whether the distribution of normalised Fourier coefficients with bounded indices can be approximated by a generalised Gaussian distribution. Moreover, it is argued that the apparent symmetry around zero of the data lends strong evidence to the Bruinier-Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute va…