Zeev Rudnick
We study the distribution of zeros of holomorphic modular forms. Assuming the Generalized Riemann Hypothesis we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows.
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The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further study.
In 2010 Zagier introduced the notion of a quantum modular form. One of his first examples was the "strange" function $F(q)$ of Kontsevich. Here we produce a new example of a quantum modular form by making use of some of Ramanujan's mock theta functions. Using these functions and their transformation behaviour, we also compute asymptotic expansions similar to expansions of $F(q)$.
An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function
本文基于概率方法，利用具有简单无穷乘积的模函数表示，证明了椭圆模 $j$-函数系数的渐近公式。
In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen. Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.
Asymptotic distribution of CM points on the reduction of the Drinfeld modular curve
研究全球函数域上CMDrinfeld模在Drinfeld模曲线分析约化不可约分量中的渐近分布，利用建筑映射和布鲁hat-蒂茨树商的邻接算子谱分解性质。
We give an asymptotic formula for the number of non-zero coefficients of modular forms (mod p).
We prove under GRH that zeros of $L$-functions of modular forms of level $N$ and weight $k$ become uniformly distributed on the critical line when $N+k\to\infty.$
In this paper we study the exceptional zero phenomenon for Hilbert modular forms in the anticyclotomic setting. We prove a formula expressing the leading term of the p-adic L-functions via arithmetic L-invariants.
The Serre Derivatives and Zeros of Modular Forms
本文研究模形式的塞尔雷导数零点，证明若弱解析模形式在标准基本域内所有零点位于下边界，则其塞尔雷导数亦具此性质。
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight $k$ for $\SL_2(\mathbb{Z})$ interlace on most of the lower boundary of the fundamental domain.
In this article, we construct differential modular forms for compact Shimura curves over totally real fields bigger than rational of non-zero integral weights that is not classical (of order zero) generalizing the construction of Buium [8].
We determine the 1-level density of families of Hilbert modular forms, and show the answer agrees only with orthogonal random matrix ensembles.
We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part I, we will consider the general theory of modular functions for the normalizers.
We show that all but 5 of the zeros of the period polynomial associated to a Hecke cusp form are on the unit circle.
In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(ζ_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.
We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part II, we will observe the location of the zeros of the above functions by numerical calculation.
Assuming the generalized Riemann hypothesis, we prove quantitative estimates for the number of simple zeros on the critical line for the L-functions attached to classical holomorphic newforms.
For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain…
We study the second moment of the central values of quadratic twists of a modular $L$-function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.
Let $M_k^\sharp(N)$ be the space of weakly holomorphic modular forms for $Γ_0(N)$ that are holomorphic at all cusps except possibly at $\infty$. We study a canonical basis for $M_k^\sharp(2)$ and $M_k^\sharp(3)$ and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.
Values of quaternionic modular forms are related to twisted central $L$-values via periods and a theorem of Waldspurger. In particular, certain twisted $L$-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture that almost all forms have no nontrivial zeroes. In particular, almost all forms with appropriate local signs should have no zeroes. We show these conjectures follow from a conjecture on the…
Let $M_k^!(Γ_0^+(3))$ be the space of weakly holomorphic modular forms of weight $k$ for the Fricke group of level $3$. We introduce a natural basis for $M_k^!(Γ_0^+(3))$ and prove that for almost all basis elements, all of their zeros in a fundamental domain lie on the circle centered at 0 with radius $\frac{1}{\sqrt{3}}$.
We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $\mathbb{Z}+\frac{1}{2}$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $Γ_0(4)$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincaré series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simult…
We prove divisibility results for the Fourier coefficients of canonical basis elements for the spaces of weakly holomorphic modular forms of weight $0$ and levels $6, 10, 12, 18$ with poles only at the cusp at infinity. In addition, we show that these Fourier coefficients satisfy Zagier duality in all weights, and give a general formula for the generating functions of such canonical bases for all genus zero levels.
In this paper, we study common zeros of the iterated derivatives of the Eisenstein series for $Γ_0^+(N)$ of level $N=1,2$ and $3$, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms, and prove that all the zeros of the iterated derivatives of the Eisenstein series $\frac{d^m E_k^{(N)}(τ)}{dτ^m}$ of weight $k=2,4,6$ for $Γ_0^+(N)$ of level $N=2,3$ are simple by generalizaing the results of Meher \cite{MEH} and Gun and Oesterlé \cite{SJ20} for SL$_2(\mathbb{Z})$.
Fourier Coefficients of Weight Zero Mixed False Modular Forms
本文利用并推广圆法及Bringmann、Nazaroglu、Rademacher的技巧，结合Kloosterman和的模$k$上界，给出无穷族权零混合假模形式傅里叶系数的精确公式。
We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy \[ f_k(A, τ) \colon = q^{-k}(1+a(1)q+a(2)q^2+...) + O(q),\] where $a(n)$ are numbers satisfying a certain analytic condition. We show that the zeros of such $f_k(τ)$ in the fundamental domain of $SL_2(\mathbb{Z})$ lie on $|τ|=1$ and are transcendental. We recover as a special case earlier work of Witten on extremal "partition" functions $Z_k(τ)$. These functions were originally conceived as possible generalizations of constructions in thr…
This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we com…
Let $M_k^\sharp(4)$ be the space of weakly holomorphic modular forms of weight $k$ and level $4$ that are holomorphic away from the cusp at $\infty$. We define a canonical basis for this space and show that for almost all of the basis elements, the majority of their zeros in a fundamental domain for $Γ_0(4)$ lie on the lower boundary of the fundamental domain. Additionally, we show that the Fourier coefficients of the basis elements satisfy an interesting duality property.
We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for $ζ_K(s)$. New elegant transformations are obtained when $K$ is a quadratic extension, one of which involves the modified Bessel function of the second kind.
The effect of the trace operator on the duality of modular grids in genus zero levels
本文研究了迹算子对亏格零级$Γ_0(N)$模形式典范基系数对偶性的影响，精确确定了应用迹算子后对偶性依然成立的条件。
We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic L-functions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.
Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions
本文通过发展非尖点Siegel模形式的Hida理论，利用Greenberg-Stevens方法构造改进的p进L-函数，研究了P-普通Siegel模形式的标准p进L-函数在半稳定平凡零点处的导数，从而部分证明了Greenberg关于p进L-函数平凡零点的阶数与首项系数的猜想。
Lehner's 1949 results on the $j$-invariant showed high divisibility of the function's coefficients by the primes $p\in\{2,3,5,7\}$. Expanding his results, we examine a canonical basis for the space of level $p$ modular functions holomorphic at the cusp 0. We show that the Fourier coefficients of these functions are often highly divisible by these same primes.
We show that the values of a certain family of weakly holomorphic modular functions at points in the divisors of any meromorphic modular form with algebraic Fourier coefficients are algebraic. We use this to extend the classical result of Schneider by proving that zeros or poles of any non-zero meromorphic modular form with algebraic Fourier coefficients are either transcendental or imaginary quadratic irrational.
Restricted Overpartitions and concave compositions: their modularity and asymptotics
本文研究受限超分拆与凹分解，其生成函数涉及模形式、拟theta函数、拟Maass theta函数和假theta函数，揭示了受限分拆问题中的混合模结构，并获得其渐近主项。
Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic $p$-adic periods are replaced by the $\cL$-invariants of $E$ defined in terms of Tate periods) from a…
We determine the asymptotic distribution of Manin's iterated integrals of length at most 2. For all lengths we compute all the asymptotic moments. We show that if the length is at least 3 these moments do in general not determine a unique distribution.
Considering the family of $L$-functions $\{L(s,f)\}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp forms for $SL_2(\mathbb{Z})$, we prove a zero density estimate near the central point, valid as the weight $k \to \infty$. This is an ingredient in the author's related paper, which gives an unconditional upper bound on the distribution of the central values.
A modular relation of the form $F(α, w)=F(β, iw)$, where $i=\sqrt{-1}$ and $αβ=1$, is obtained. It involves the generalized digamma function $ψ_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $ζ_w(s, a)$. The limiting case $w\to0$ of this modular relation is a famous result of Ramanujan on page $220$ of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function $Ξ(t)$ as $α\to\infty$. Not only does it give the asymptotic estimate of the integral occurring in our modul…
We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function $p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the $j$-invariant, is odd.
We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result.
We prove that the modular symbols appropriately normalized and ordered have an asymptotical normal distribution for all cocompact subgroups of SL_2(R). We introduce hyperbolic Eisenstein series in order to calculate the moments of the modular symbols.
Genus formulas for dormant modular curves and asymptotic behavior of their function fields
本文研究了基于高阶水平dormant PGL_2-opers的模曲线的亏格公式，并分析了其函数域的渐进行为，与经典模曲线和Drinfeld模曲线进行比较。
Let $k=12 m(k)+s \ge 12$ for $s\in \{0,4,6,8,10,14\}$, be an even integer and $f$ be a normalised modular form of weight $k$ with real Fourier coefficients, written as $$ f=E_k+\sum_{j=1}^{m(k)}a_jE_{k-12j}Δ^j. $$ Under suitable conditions on $a_j$ (rectifying an earlier result of Getz), we show that all the zeros of $f$, in the standard fundamental domain for the action of ${\bf SL}(2,\mathbb Z)$ on the upper half plane, lies on the arc $A:= \left\{ e^{i θ} : \fracπ{2} \le θ\le \frac{2π}{3} \right\}$. Further, extending a result of Nozaki, we show that for certain family $\{f_k\}_k$ of normal…
We prove an asymptotic for the second moment of quadratic twists of a modular $L$-function. This was previously known conditionally on GRH by the work of Soundararajan and Young.
Explicit bounds on the coefficients of the modular polynomials and the size of $X_0(N)$
本文给出模多项式系数的显式上下界，使渐近界显式化，并给出 Hecke 点估计的显式版本及模曲线高度与模多项式高度的渐近比较。
Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group.
Let $M_k^\sharp(N)$ be the space of weight $k$, level $N$ weakly holomorphic modular forms with poles only at the cusp at $\infty$. We explicitly construct a canonical basis for $M_k^\sharp(N)$ for $N\in\{8,9,16,25\}$, and show that many of the Fourier coefficients of the basis elements in $M_0^\sharp(N)$ are divisible by high powers of the prime dividing the level $N$. Additionally, we show that these basis elements satisfy a Zagier duality property, and extend Griffin's results on congruences in level 1 to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25.
The second moment of derivatives of quadratic twists of modular $L$-functions
本文证明了二次模形式 $L$-函数二次扭曲导数的二阶矩渐进行式，包含三个主导项，改进了 Kumar 等人的先前结果，方法基于 Li 建立的大筛不等式及大位移扭曲 $L$-函数渐进行式计算。
On Modular Approach to Diophantine Equation $x^4-y^4=nz^p$ over Number Fields
本文利用模方法证明了丢番图方程 $x^4-y^4=nz^p$ 在各类数域上解的渐近结果，并证明了方程 $x^4-y^4=2^αz^p$ 的渐近广义费马定理在无限多个二次数域上成立。
Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime $\mathfrak{p}$ of F above p we prove the existence of an L-invariant $\mathcal{L}_{\mathfrak{p}}$, depending only on $\mathfrak{p}$ and f, such that when the p-adic L-function of f has an exceptional zero at $\mathfrak{p}$, its derivative can be related to the classical L-value multiplied by $\mathcal{L}_{\mathfrak{p}}$. The proof uses cohomological method…
Zeros of modular forms and Faber polynomials
本文利用 Faber 多项式方法，研究了模群上大权重且在无穷处高消失阶的尖点形式零点分布，发现其零点聚集在 D 条垂直线附近，不同于 Eisenstein 级数或 Hecke 特征形式。
We give upper bounds on the size of the gap between a non-zero constant term and the next non-zero Fourier coefficient of an entire level two modular form. We give upper bounds for the minimum positive integer represented by a level two even positive-definite quadratic form. These bounds extend partial results in part I.
In this note we study $SL(2,\mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincaré series. In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/a…
We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of Res_{K/Q_p} GL_2 for K a finite extension of Q_p. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.
Asymptotic expansion for the Fourier coefficients associated with the inverse of the modular discriminant function $Δ$
本文研究了模判别式函数 $Δ$ 之逆的 Fourier 系数渐近展开及误差界，证明了 $24$-色划分 $p_{24}(n)$ 满足多种不等式，且该方法可推广至具有 Rademacher 型精确公式的 Dedekind-eta 商。
Low-Lying Zeros of $L$-functions of Adélic Hilbert Modular Forms and their Convolutions
本文在广义黎曼猜想下，研究 Adélic Hilbert 模形式 $L$-函数及其卷积的 Katz-Sarnak 密度猜想，推广了前人工作，并得到 $L$-函数在 $s=1/2$ 处平均阶上界及 Rankin-Selberg $L$-函数非消失正比例。
We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+ε})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions.
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