Barry Brent
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.
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We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic p-adic L-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p-adic L-functions to more general Shimura curves.
In 2008, M. Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of M. Kaneko for a general modular function.
We prove that all elliptic curves defined over real quadratic fields are modular.
Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.
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.
Let $K$ be a composite field of some real quadratic fields. We give a sufficient condition on $K$ such that all elliptic curves over $K$ is modular.
We define "values" of the elliptic modular j-function at real quadratic irrationalities by using Hecke's hyperbolic Fourier expansions, and present some observations based on numerical experiments.
We construct an Euler system attached to a weight 2 modular form twisted by a Groessencharacter of an imaginary quadratic field, and apply this to bounding Selmer groups.
We develop an elementary divisor theory for the unimodular and the modular group over quadratic field extensions and quaternion algebras. In particular, we investigate which sets of elementary divisors can occur. Under an additional hypothesis we establish a correspondence of unimodular and modular double cosets.
Bayer and Blanco-Chacón have recently defined quadratic modular symbols for the Shimura curves X(D,N) attached to Eichler orders of level N of an indefinite quaternion rational algebra of discriminant D. In this paper, we give a cohomological interpretation of these quadratic modular symbols. Explicit computations for the homology of some Shimura curves are also provided.
We present theoretical and numerical evidence for a random matrix theoretic approach to a conjecture about vanishings of quadratic twists of certain L-functions
In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.
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.
We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.
We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.
Bounds for integral moments of quadratic twists of Fourier coefficients of modular forms
本文在广义黎曼猜想下给出模形式傅里叶系数二次扭特征和的积分矩上界，并证明其最优性。
Quadratic Chabauty Experiments on Genus 2 Bielliptic Modular Curves in the LMFDB
本文对LMFDB中亏格2双椭圆模曲线进行二次Chabauty实验，在有理数和二次虚域上应用方法，发现代数无理点，并提出模曲线水平与潜在数域关系的猜想。
Quantum Modular Forms from Real Quadratic Double Sums
本文研究Lovejoy和Osburn发现的十二个$q$-超几何级数，证明其傅里叶系数构成群$Γ_0(2)$上的三个向量值量子模形式。
Moment of Derivatives of Quadratic Twists of Modular $L$-Functions
本文研究了两个不同模$L$-函数二次扭导数矩的渐进行为，首次在不依赖GRH条件下给出证明。
Lower bounds for moments of quadratic twists of modular $L$-functions
该论文证明了模$L$-函数二次扭曲在中心点的$2k$-阶矩（$k \geq 1/2$）的尖锐下界。
Ratios conjecture for quadratic twist of modular $L$-functions
本文在广义黎曼假设下，利用多重狄利克雷级数，为模形式 $L$-函数的二次扭曲族在特定范围内建立了含单平移的 $L$-函数比值猜想。
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by the binary quadratic form $x^2 + 3xy + y^2$. The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.
In this paper, we study how small a box contains at least two points from a modular hyperbola $x y \equiv c \pmod p$. There are two such points in a square of side length $p^{1/4 + ε}$. Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6 + ε}$ or the least quadratic nonresidue is less than $p^{1/(6 \sqrt{e}) + ε}$.
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is prove…
In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^+(7)$. Using similar techniques for $p=11$, and employing a version of Chabauty for symmetric powers of curves for $p=13$, we show that the same holds for $X_{ns}(11)$ and $X_{ns}(13)$. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.
In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.
In this paper we determine the quadratic points on the modular curves X_0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81. As well as determining the non-cuspidal quadratic points, we give the j-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic \Q-curves.
The first moment of quadratic twists of modular $L$-functions
本文基于 Young 的工作，研究了模 $L$ 函数二次扭曲及其一阶导数的平滑第一矩，并给出了误差项为 $O(X^{\frac{1}{2} + \varepsilon})$ 的渐近公式。
In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-ad…
Let $D$ be an odd square-free positive integer and $C$ a divisor of $D$. For any quadratic character $χ$ modulo $C$, we prove that the $χ$-part of the group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the $χ$-part of its cuspidal subgroup, away from those primes of bad reduction or where possible congruences between oldforms and newforms occur.
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。
Computing points on bielliptic modular curves over fixed quadratic fields
本文提出 Mordell-Weil 筛，用于计算固定二次域上若干双椭圆模曲线 $X_0(N)$ 的点，并考察八个水平及 $|d|<100$ 的二次域。
Mean squares of quadratic twists of the Fourier coefficients of modular forms
本文渐近求值了给定模形式傅里叶系数在二次扭曲下的平滑求和版本，涉及Kronecker符号及正奇无平方整数求和。
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular.
Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary f…
Bruin and Najman, Ozman and Siksek, and Box described all the quadratic points on the modular curves of genus $2\leq g(X_0(n)) \leq 5$. Since all the hyperelliptic curves $X_0(n)$ are of genus $\leq 5$ and as a curve can have infinitely many quadratic points only if it is either of genus $\leq 1$, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves $X_0(n)$ naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves $X…
A modular framework for functions of Knopp and indefinite binary quadratic forms
本文提出了一种模块化框架，用于研究Knopp函数与不定二元二次型相关的非全纯双模形式，并定义了局部调和Maass形式的模类比负权重。
In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at $0,$ and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}$.
We investigate Siegel theta series for quadratic forms of signature $(m-1,1)$. On the one hand, we construct a holomorphic series that does not transform like a modular form. On the other hand, we construct a non-holomorphic series that transforms like a Siegel modular form of weight $m/2$. Moreover, the holomorphic series describes almost everywhere the holomorphic part of the modular series.
Let f be a newform of weight two, prime level p. If D is a fundamental discriminant, define the twisted L-function L(f,D,s) to be the L-function associated to the twist of f by the quadratic character of conductor D. In this paper we consider the question of computing the family of twisted central values {L(f,D,1) : |D| <= x} for some x, by using an explicit version of Waldspurger's formula relating the central values L(f,D,1) to the |D|-th Fourier coefficient of weigth 3/2 modular forms in Shimura correspondence with f.
Modular curves of prime-power level with infinitely many quadratic points
该研究确定了1085个素数幂水平的模曲线子群，这些子群对应的模曲线具有无限多个二次点。
The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.
In this paper we study certain real functions defined in a very simple way by Zagier as sums of infinite powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular forms which are the coefficients in Fourier expansion of the kernel function for Shimura-Shintani correspondence. We prove two conjectures of Zagier showing that the sums converge exponentially. We also prove unexpected results on the representation of these functions as sums over simple or reduced quadratic forms and the positive or negative continued frac…
We study the splitting of primes in number fields generated by points on modular curves. Momose was the first to notice that quadratic points on $X_1(n)$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm. We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $X_0(n)$ which are hyperelliptic (except for $n=37$) and in cubic fields generated by points on $X_1(2, 14)$.
Quadratic points on modular curves $X_0(N)$ for $N\leq 100$
针对$N\leq100$中未解决的11个$N$值，研究通过改进“下降法”（利用$N$的因子$n$对应的$X_0(n)$二次点模描述），确定了模曲线$X_0(N)$上的二次点。
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
Let $ F$ be an imaginary quadratic field and $\mathcal{O}$ its ring of integers. Let $ \mathfrak{n} \subset \mathcal{O} $ be a non-zero ideal and let $ p> 5$ be a rational inert prime in $F$ and coprime with $\mathfrak{n}$. Let $ V$ be an irreducible finite dimensional representation of $ \bar{\mathbb{F}}_{p}[{\rm GL}_2(\mathbb{F}_{p^2})].$ We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $ V$ already lives in the cohomology with coefficients in $ \bar{\mathbb{F}}_{p}\otimes det^e$ for some $ e \geq 0$; except possibly in some few cases.
We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for Γ_0(2). Numerical evidence indicates that a sharper bound holds for the weights h \equiv 2 . We derive upper bounds for the minimum positive integer represented by level two even positive-definite quadratic forms. Our data suggest that, for certain meromorphic modular forms and p=2,3, the p-order of the constant term is related to the base-p expansion of the order of the pole at infinity, and they suggest a connection between divisibility…
We study a modular function $Λ_{k,\ell}$ which is one of generalized $λ$ functions. We show $Λ_{k,\ell}$ and the modular invariant function $j$ generate the modular function field with respect to the modular subgroup $Γ_1(N)$. Further we prove that $Λ_{k,\ell}$ is integral over $\mathbf Z[j]$. From these results, we obtain that the value of $Λ_{k,\ell}$ at an imaginary quadratic point is an algebraic integer and generates a ray class field over the Hilbert class field.
Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations
本研究针对素级亏格≥2的Atkin-Lehner商模曲线，提出用弱全纯模形式计算二次Chabauty方法中关键的Hodge滤过的算法，提升了计算效率，并在$X_0^+(193)$上发现了权2尖点形式积分的同余关系。
Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equivλ\pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals of length $M$, under the assumption that $Q(x,y)-λ$ is an absolutely irreducible polynomial modulo $p$. In particular we prove that the number of solutions to this congruence is $M^{o(1)}$ when $M\ll p^{1/4}$. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence $xy\equiv λ\pmod{p}$.
A Quadratic Curve Analogue of the Taniyama-Shimura Conjecture
本文研究有限域$F_p$上二次曲线的解数问题，提出谷山-志村猜想中模形式的二次曲线类比版本，该模形式可导出高斯和且具备模变换结构。
We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature $(n-1,1)$. We include homogeneous and spherical polynomials in the definition of the theta series (generalizing a construction of the second author) to obtain holomorphic, almost holomorphic and modular theta series. We give a criterion for these series to coincide, enabling us to construct almost holomorphic and holomorphic cusp forms on congruence subgroups of the modular group. Further, we provide numerous explicit examples.
Let $k$ be a positive integer such that $k\equiv3\mod4$, and let $N$ be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace $S_{\frac{k}{2}}(Γ_{0}(4N),F)$ of half-integral weight modular forms associated, via the Shimura correspondence, to a newform $F\in S_{k-1}(Γ_{0}(N))$, which satisfies $L(F,\frac{1}{2})\neq0$. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given $F$ via local considerations, once a form in the Kohnen space has been determined
Real quadratic singular moduli and $p$-adic families of modular forms
本文综述了Darmon和Vonk基于p-进方法提出的实乘法猜想理论，重点讨论刚性亚纯上闭链在实二次点处的取值作为奇异模类比，并强调模形式族在复乘法与实乘法两种情形中的核心作用。
Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. Only for certain types of Bianchi modular forms, which we will call non-genuine, it is possible at present to develop dimension formulas: They are (twists of) those forms which arise from elliptic cuspidal modular forms via the Langlands Base-Change procedure, or arise from a quadratic extension of the imaginary quadratic field vi…
Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell--Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this …
We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.
Modular invariants for real quadratic fields and Kloosterman sums
本文研究了实二次域中理想类相关 $j$-函数积分的渐近分布，通过证明对所有参数一致的半整数权 Kloosterman 和的界来估计误差项，并利用 Kuznetsov 公式变体及 Young 的次凸性估计建立该结果。
For a fixed odd prime p and a representation ρof the absolute Galois group of Q into the projective group PGL(2,p), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that realize ρthrough the Galois action on their p-torsion modules. The modular curve to twist is either the fiber product of the modular curves X_0(N) and X(p) or a certain quotient of Atkin-Lehner type, depending on the value of N mod p. For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphis…
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