Simple zeros of modular L-functions
Micah B. Milinovich, Nathan Ng
在广义黎曼假设下，我们证明了经典全纯新形式下L函数临界线上单零点个数的定量估计。
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.
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On the asymptotic distribution of zeros of modular forms
研究了全纯模形式的零点分布。假设广义黎曼假设，我们证明了随着权的增加，模群的Hecke本征形的零点关于模域上的双曲测度是等分布的.
On the zeros of weakly holomorphic modular forms
本文研究了弱全纯模形式的零点性质。特别是，我们证明了超越零点的模形式的更高的水平，并为某些扩展Kohnen的工作Sparke群。进一步，我们研究了弱全纯模形式的值的代数独立性。
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。
Interlacing of zeros of weakly holomorphic modular forms
我们证明了极值模形式族的零点是交错的，解决了Nozaki的一个问题。此外，我们表明，几乎所有形式的零点在一个基础上的空间的弱全纯模形式的重量$k$为$\SL_2(\mathbb{Z})$交错的基本区的大部分下边界。
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.
On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels I
本文研究了亏格为零且水平至多为12的同余子群的正规化子所对应的Hecke型Faber多项式的Eisenstein级数的零点和模函数的位置.在第一部分，我们将考虑规范化子的模函数的一般理论。
The nontrivial zeros of period polynomials of modular forms lie on the unit circle
我们发现，但5个与Hecke尖点形式相关的周期多项式的零点都在单位圆上。
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.
On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels II
本文研究了亏格为零且水平至多为12的同余子群的正规化子所对应的Hecke型Faber多项式的Eisenstein级数的零点和模函数的位置.在第二部分中，我们将通过数值计算观察上述函数零点的位置。
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…
Zeros of weakly holomorphic modular forms of levels 2 and 3
设$M_k^\sharp(N)$是$Γ_0(N)$的弱全纯模形式的空间，这些模形式在除了可能在$\infty$之外的所有尖点处都是全纯的。本文研究了$M_k^\sharp(2)$和$M_k^\sharp(3)$的一个典范基，证明了在这个典范基中几乎所有的模形式都具有这样的性质：它们在基本区中的零点的大部分位于基本区的下边界弧上.
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…
Genus Zero Modular Functions
该项目由布兰迪斯大学的希夫奖学金计划赞助。这个项目涉及使用幂级数方法来构造一个三阶非线性常微分方程，Schwarzian方程，为每个“亏格零”模函数函数，在康威-诺顿论文中描述。我们首先使用Borcherd递归公式来生成，在每种情况下，一个模函数到任何程度，我们的愿望，然后使用的事实，有一个施瓦茨方程，由一个单一的有理函数，我们称之为Q值。通过类似的幂级数方法，我们计算我们的有理函数的系数，因此拥有所有必要的数据来为每个模函数创建Schwarzian微分方程。这个方程又可以用来恢复模函数本身。
Zeros of weakly holomorphic modular forms of level 4
设$M_k^\sharp(4)$是权$k$和水平$4$的弱全纯模形式的空间，它们在$\infty$的尖点处是全纯的。我们定义了一个规范的基础，这个空间，并表明，几乎所有的基础元素，他们的零点在一个基本域的$Γ_0(4)$谎言的下边界的基本区。此外，我们还证明了基元的傅里叶系数满足一个有趣的对偶性质。
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。
The exceptional zero conjecture for symmetric powers of CM modular forms: the ordinary case
证明了CM尖点特征形在常素数处的对称幂的例外零猜想。换句话说，我们确定了相关的p-adic L-函数的平凡零点，计算了L-不变量，并证明了它们与Greenberg的L-不变量一致。
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。
Divisibility Properties of Coefficients of Level $p$ Modular Functions for Genus Zero Primes
Lehner在1949年关于$j$ -不变量的结果显示了函数系数被素数$p\in\{2,3,5,7\}$的高可分性。扩展他的结果，我们研究了一个典型的基础上的空间水平$p$模函数全纯的尖点0。我们表明，这些功能的傅里叶系数往往是高度可分的这些相同的素数。
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.
On special zeros of $p$-adic $L$-functions of Hilbert modular forms
设$E$是全实数域$F$上的模椭圆曲线.我们证明了弱例外零猜想，它把附着在$E$上的$p$ -adic $L$ -函数的一个（高）导数与附着在相应Hilbert模形式上的某些$p$ -adic周期联系起来，在$p$之上的地方，$E$有分裂的乘性约化.在对$p$和$E$的导子的适当限制下，我们导出了强形式的例外零猜想（即其中，自守$p$ -adic周期被根据Tate周期定义的$E$的$\cL$ -不变量所取代）。我们的方法的关键是一个新的构造的$p$ -adic $L$ -函数的$E$在本地数据。
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.
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.
Explicit lower bounds on the modular degree of an elliptic curve
我们得到了椭圆曲线的对称平方L-函数的一个显式无零区域，并利用它得到了有理椭圆曲线的模次数的一个显式下界。这些技术类似于经典的Dirichlet L-函数的无零区域的推导，但是在这里，由于Goldfield-Hoffstein-Lieman的工作，我们知道没有Siegel零点，这导致了一个加强的结果。
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 study the algebras of modular forms on type IV symmetric domains for simple lattices; that is, lattices for which every Heegner divisor occurs as the divisor of a Borcherds product. For every simple lattice $L$ of signature $(n,2)$ with $3\leq n \leq 10$, we prove that the graded algebra of modular forms for the maximal reflection subgroup of the orthogonal group of $L$ is freely generated. We also show that, with five exceptions, the graded algebra of modular forms for the maximal reflection subgroup of the discriminant kernel of $L$ is also freely generated.
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.
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。
Quadratic minima and modular forms II
我们给出了整个二级模形式的一个非零常数项和下一个非零傅里叶系数之间的间隙大小的上界。给出了二阶偶正定二次型表示的最小正整数的上界。这些界推广了第I部分的部分结果。
In this paper, we introduce modular polynomials for the congruence subgroup $Γ_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respec…
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。
Weighted low-lying zeros of L-functions attached to Siegel modular forms
本文研究 2 次 Siegel 模形式的旋量与标准 L-函数的加权低矮零点，证明二者均具有辛对称类型，并通过按权平均扩大检验函数 Fourier 变换的支集范围，进而讨论中心值非消失性。
Partial Hasse invariants for genus zero curves in Hilbert modular varieties
本文研究了Hilbert模形式中零 genus 曲线的部分Hassé不变量的特征零提升，并通过计算Teichmüller曲线得到类似Deuring的公式。
Integral traces of weak Maass forms of genus zero odd prime level
杜克大学和第二作者定义了一个家庭的线性映射从空间的弱全纯模形式的负积分权和水平1到空间的弱全纯模形式的半积分权和水平4，并表明，这些升降机保持完整性的傅里叶系数。我们证明了这些提升到亏格为0奇素数水平的模形式的推广也保持了傅里叶系数的完整性。
This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or mo…
The rank of the modular multiplication matrix
该研究证明了布埃诺等人关于正整数模乘法矩阵秩的猜想公式，通过加法傅里叶变换和乘法特征分解进行证明，且删除零行零列后公式仍成立。
Modular Inverse and Reciprocity Formula
证明了非零整数模逆的一个互易公式，并给出了互易公式在计算或验证某些特定形式的模逆中的应用，包括高斯整数模另一高斯整数的模逆.
Cycle integrals of a sesqui-harmonic Maass form of weight zero
权为1/2和3/2的弱全纯模形式空间的Borcherds-Zagier基共享作为奇异模迹的傅里叶系数。最近，Duke，Imamoglu和Tóth构造了权1/2模模形式空间的一个基，其中每个成员都有Zagier生成的奇异模迹序列作为其影子.他们还表明，傅里叶系数的模拟模形式是总和循环积分的$j$ -功能，这是真正的二次类似物的奇异模。本文证明了权为3/2的模形式空间的基的傅里叶系数是权为零的倍半调和Maass形式的循环积分之和，其在双曲Laplacian下的像是$j$ -函数.进一步地，我们将这些和表示为权为1/2的弱全纯模形式的正则化内积。
On modular forms of characteristic p>0
我们比较了特征$p>0$的模形式（即Drinfeld的模形式）和自守形式。我们证明了这些模形式的空间（它们具有特征$p$）可以用特征零函数空间来描述，接近于自守形式的特征零函数空间.
Critical points of modular forms
本文研究具有实傅里叶系数的模形式在标准基本域平移区域内的临界点计数，利用边界零点、$γ^{-1}(\infty)$的值与权给出闭式公式，并推广讨论拟模形式零点数。
We report on a systematic computation of weight one cuspidal eigenforms for the group $Γ_1(N)$ in characteristic zero and in characteristic $p>2$. Perhaps the most surprising result was the existence of a mod 199 weight~1 cusp form of level 82 which does not lift to characteristic zero.
We construct a $(\mathfrak{gl}_2, B(\mathbb{Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb{P}^1$, landing in the compactly supported completed $\mathbb{C}_p$-cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$. For classical weight $k\geq 2$, the Verma has an algebraic quotient $H^1(\mathbb{P}^1, \mathcal{O}(-k))$, and on …
Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values
该研究针对权为$\textbf{k}$、素水平为$\mathfrak{q}$的本原希尔伯特模形式集，分析以中心L值幂为权的L函数低零点单级密度，证明r=1,2,3时分布与随机矩阵理论预测一致，并给出一般r≥1时的猜想公式。
Rational points on modular curves via maps to elliptic curves with rank zero
本文提出一种构造模曲线到秩为零的椭圆曲线的映射的方法，并系统确定了水平不超过70的模曲线中超过99%的有理点。
Modular case of Levinson's theorem
本文计算了本原尖点形式的L-函数的积分软化二阶矩，并得到了这类L-函数位于临界线上的零点的正比例。
There have been a number of recent works on the theory of period polynomials and their zeros. In particular, zeros of period polynomials have been shown to satisfy a "Riemann Hypothesis" in both classical settings and for cohomological versions extending the classical setting to the case of higher derivatives of $L$-functions. There thus appears to be a general phenomenon behind these phenomena. In this paper, we explore further generalizations by defining a natural analogue for Hilbert modular forms. We then prove that similar Riemann Hypotheses hold in this situation as well.
Some applications of modular units
证明了弱全纯模函数可以表示为高阶模单位的和。进一步得到了当$g$次Siegel模函数被限制在Siegel上半空间$\mathbb{H}_g$的某个子集上时，其在定义域上既无零点也无极点的充要条件.
On generalized modular forms supported on cuspidal and elliptic points
本文推广了已有的结果，证明了具有有理Fourier展开式的广义模形式（其因子仅在尖点和上半平面的某些点处得到支持）实际上是经典的模形式.我们讨论了可能的限制，这种扩展和可能的零素水平的模形式提出的问题。
We consider specific linear combinations of two loop modular graph functions on the toroidal worldsheet with $2s$ links for $s=2, 3$ and $4$. In each case, it satisfies an eigenvalue equation with source terms involving $E_{2s}$ and $E_s^2$ only. On removing certain combinations of $E_{2s}$ and $E_s^2$ from it, we express the resulting expression as an absolutely convergent Poincare series. This is used to calculate the power behaved terms in the asymptotic expansion of the zero mode of the Fourier expansion of these graphs in a simple manner.
Non vanishing of Central values of modular L-functions for Hecke eigenforms of level one
设F（z）是一个新的具有平凡特征标的权为2k的水平1的形式，并假定F（z）是所有Hecke算子的非零本征形.本文研究了F的扭模L-函数的中心值的非零性。
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