Computing classical modular forms

A quasi-linear time algorithm for computing modular polynomials in dimension 2

We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove under some heuristics that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.

Computing bases of modular forms using the graded algebra structure

We develop a new algorithm to compute a basis for $M_k(Γ_0(N))$, the space of weight $k$ holomorphic modular forms on $Γ_0(N)$, in the case when the graded algebra of modular forms over $Γ_0(N)$ is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.

Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.

Computing Special $L$-Values of Certain Modular Forms with Complex Multiplication

In this expository paper, we illustrate two explicit methods which lead to special $L$-values of certain modular forms admitting complex multiplication (CM), motivated in part by properties of $L$-functions obtained from Calabi-Yau manifolds defined over $\mathbb Q$.

Computing the level of a modular rigid Calabi-Yau threefold

In a previous article (a joint work with J. Manoharmayum) the modularity of a large class of rigid Calabi-Yau threefolds was established. To make that result more explicit, we recall (and re-prove) a result of Serre giving a bound for the conductor of an "integral" 2-dimensional compatible family of Galois representations and apply this result to give an algorithm that determines the level of a modular rigid Calabi-Yau threefold. We apply the algorithm to three examples.

计算任意同余子群的经典模形式

Computing Classical Modular Forms for Arbitrary Congruence Subgroups

本文提出了一种高效算法，用于计算任意同余子群的模形式q展开式，并讨论了相关理论和实际应用。

A Dual-Radix Modular Division Algorithm for Computing Periodic Orbits within Syracuse Dynamical Systems

This article analyzes the periodic orbits of Syracuse dynamical systems in a novel algebraic setting: the commutative ring of graded $n$-adic integers. Within this context, this article introduces a dual-radix modular division algorithm for computing the graded canonical expansions and graded quotients for a certain class of rational expressions that arise from periodic orbits within these dynamical systems. This division algorithm yields two novel methods for testing the integrality of the Böhm-Sontacchi numbers.

Computing weight one modular forms over $\C$ and $\Fpbar$

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.

Computing points on modular curves over finite fields

In this paper, we present a probabilistic algorithm to compute the number of $\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes $\textrm{O}(n^{56+δ+ε}\log^{9+ε} p)$ bit operations, where $δ$ is an absolute constant and $ε$ is any positive real number. As an application, we can compute $#X_1(17)(\mathbb{F}_p)\textrm{mod} 17$ for huge primes $p$. For example, we have $#X_1(17)(\mathbb{F}_{10^{1000}+1357})\textrm{mod} 17=3$.

计算固定二次域上双椭圆模曲线的点

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。

Computing the Charlap-Coley-Robbins modular polynomials

Let $\mathcal{E}$ be an elliptic curve over a field $K$ and $\ell$ a prime. There exists an elliptic curve $\mathcal{E}^*$ related to $\mathcal{E}$ by anisogeny (rational map that is also a group homomorphisms) of degree $\ell$ if and only $Φ_\ell(X, j(\mathcal{E})) = 0$, where $Φ_\ell(X, Y)$ is the traditional modular polynomial. Moreover, the modular polynomial gives the coefficients of $\mathcal{E}^*$, together with parameters needed to build the isogeny explicitly. Since the traditional modular polynomial has large coefficients, many families with smaller coefficients can be used instead, …

Computing Jacobi Forms

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows to generate a Jacobi eigenform directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

Computing intersections of closed geodesics on the modular curve

In a recent work of Duke, Imamoglu, and Tóth, the linking number of certain links on the space $\text{SL}(2,\mathbb{Z})\backslash\text{SL}(2,\mathbb{R})$ is investigated. This linking number has an alternative interpretation as the intersection number of closed geodesics on the modular curve, which is the focus of this paper. By relating the intersection number to a combinatorial computation involving rivers of Conway topographs, an efficient algorithm for computing intersection numbers is produced. A formula for the total intersection of a pair of positive coprime fundamental discriminants is…

Computing Modular Polynomials

We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between elliptic curves via Velu's formulas.

Computing isolated coefficients of the $j$-function

We consider the problem of efficiently computing isolated coefficients $c_n$ in the Fourier series of the elliptic modular function $j(τ)$. We show that a hybrid numerical-modular method with complexity $n^{1+o(1)}$ is efficient in practice. As an application, we locate the first few values of $c_n$ that are prime, the first occurring at $n = 457871$.

计算扭曲L-函数的经典模形式的欧拉乘积与系数

Computing Euler products and coefficients of classical modular forms for twisted L-functions

提出一种快速计算经典模形式系数及欧拉乘积的算法，并在大导子三重积L-函数上验证其效率。

Computing Puiseux Expansions at Cusps of the Modular Curve X0(N)

The goal in this preprint is to give an efficient algorithm to compute Puiseux expansions at cusps of X0(N). It is based on a relation with a hypergeometric function that holds for any N.

Computing the Satake p-parameters of Siegel modular forms

Hecke eigenvalues of classical modular forms often encode a wealth of arithmetic data. The Satake $p$-parameters of a Siegel modular form play a role analogous to the one played by Hecke eigenvalues in the characterization of classical modular forms. In this paper we present an algorithm by which we can compute the Satake $p$-parameters of a Siegel modular form $F$ if we are given the Hecke eigenvalues of $F$ with respect to the generators of the local Hecke algebra. We compute explicit examples and relate our computations to the Ramanujan-Petersson Conjecture.

Using the Charlap-Coley-Robbins polynomials for computing isogenies

The SEA algorithm for computing the cardinality of elliptic curves over finite fields in many characteristic uses modular polynomials. These polynomials come into different flavors, and methods to compute them flourished. Once equipped with some modular polynomials for prime $\ell$, algebraic formulas are used to compute a curve $\mathcal{E}^*/\mathbb{F}_q$ that is $\ell$-isogenous to the curve of interest $\mathcal{E}$. These formulas involve derivatives of the modular polynomial that may sometime vanish. One way to overcome this problem is to use alternative trivariate polynomials $U_\ell$, …

通过形变计算模多项式

Computing modular polynomials by deformation

本研究提出一种无条件CRT算法，通过将ℓ-迷向嵌入高维光滑次数迷向、计算迷向m阶形变，以拟线性时间计算模多项式Φ_ℓ(X,Y)及其模p约化，启发式实现验证了实用性。

Lattice methods for algebraic modular forms on classical groups

We use Kneser's neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to definite forms of classical reductive algebraic groups.

Computing modular polynomials in quasi-linear time

We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially (up to logarithmic factors) linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomials $Φ_\ell$ of prime level $\ell$ in time O (\ell^3 \log^4 \ell \log \log \ell). Besides treating modular polynomials for $Γ^0 (\ell)$, which are an important ingredient in m…

Computing Hilbert modular forms over fields with nontrivial class group

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over $\Q(\sqrt{10})$ and $\Q(\sqrt{85})$ and their Hilbert class fields, we present some new instances of the conjectural Eichler-Shimura construction for totally real fields, and in particular find new examples of modular abelian varieties with everywhere good reduction.

Partial Classicality of Hilbert Modular Forms

Let F be a totally real field and p a rational prime unramified in F. We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to certain but not all weights, an overconvergent form is partially classical. We use the method of analytic continuation.

偏经典Hilbert模形式的特征簇

Eigenvariety for partially classical Hilbert modular forms

本文提出偏经典Hilbert模形式的概念，通过对经典模层进行$p$-adic插值构造了权空间中变化的模形式族及相应特征簇，推广了前人的构造方法。

计算特征簇的切空间

Computing Tangent Spaces to Eigenvarieties

本文提出一种局部有效算法，计算Bianchi模形式在p进族中关于权空间的导数，并数值验证特征簇光滑性及族的方向猜想，同时应用于椭圆模形式的$\mathcal{L}$-不变量研究。

Computing genus 2 Hilbert-Siegel modular forms over $\Q(\sqrt{5})$ via the Jacquet-Langlands correspondence

In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.

Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation

We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.

Analytic evaluation of Hecke eigenvalues for classical modular forms

We propose a method for computing approximations to the Hecke eigenvalues of a classical modular eigenform $f$, based on the analytic evaluation of $f$ at points in the upper half plane. Our approach works with arbitrary precision, allows for a strict control of the error in the approximation, and outperforms current exact computation methods.

计算模多项式奇异点处的同源

Computing Isogenies at Singular Points of the Modular Polynomial

本文提出一种方法，在模曲线奇异点处，给定椭圆曲线及其 j-不变量，构造并高效计算归一化同源，利用已知算法实现。

Class polynomials for nonholomorphic modular functions

We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we obtain CRT-based algorithms that compute the class polynomials H_D(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form F_p(z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under t…

Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.

Realization of Modular Galois Representations in Jacobians of modular curves

In this paper, we propose an improved algorithm for computing mod $\ell$ Galois representations associated to eigenforms of arbitrary levels prime to $\ell$. Precisely, we present a method to find the Jacobians of modular curves which have the smallest possible dimensions in a well-defined sense to realize the modular Galois representations. This algorithm also works without the assumption $\ell \ge k-1$, where $k$ are the weights of the eigenforms.

经典模多项式系数的同余性质

Congruence properties of the coefficients of the classical modular polynomials

本文针对素数 $\ell$，以模不变函数 $j(z)$ 的傅里叶系数形式给出了经典模多项式 $\ell$ 个非平凡系数的闭式公式，并据此推导出同余式，支持了关于这些系数满足素数 $2,3$ 和 $5$ 的幂的同余猜想。

Models for quotients of modular curves

We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.

Slope classicality via completed cohomology

We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.

Modular forms and elliptic curves over the cubic field of discriminant -23

Let F be the cubic field of discriminant -23 and let O be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL(2,O), we computationally investigate modularity of elliptic curves over F.

Efficiently generated spaces of classical Siegel modular forms and the Boecherer conjecture

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set we verify the Boecherer conjec- ture for non-rational eigenforms. As one further application we verify another conjectures for weights up to 150 and investigate an analogue of the Victor-Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

计算模曲线上点的纤维的层数

Computing the Level of a Fiber for Points on Modular Curves

本文类比伽罗瓦表示层数定义构造模曲线上闭点纤维的层数，证明在特定条件下，$X_1(\u2113^{k+1})$上次数最大的点在更高层$X_1(\u2113^n)$上的提升次数也保持最大，技术受Lang和Trotter工作启发。

Nonvanishing of twists of $L$-functions attached to Hilbert modular forms

We describe algorithms for computing central values of twists of $L$-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and predictions from random matrix theory.

Computing an order complete basis for $M^{\infty}(N)$ and Applications

This paper gives a quick way to construct all modular functions for the group $Γ_0(N)$ having only a pole at $τ= i \infty$. We assume that we are given two modular functions $f,g$ for $Γ_0(N)$ with poles only at $i \infty$ and coprime pole orders. As an application we obtain two new identities from which one can derive that $p(11n+6)\equiv 0\pmod{11}$, here $p(n)$ is the usual partition function.

An explicit height bound for the classical modular polynomial

For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.

从亏格2的模方程计算同源

Computing isogenies from modular equations in genus two

提出一种算法，利用模方程和显式Kodaira-Spencer同构，在特征足够大且曲线足够一般的条件下，显式计算亏格2曲线上Jacobian的l-同源或循环核同源。

Modular forms invariant under non-split Cartan subgroups

In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$. As an application we compute, for certain small values of $p$, explicit equations over $\bf Q$ for the canonical embeddings of the associated modular curves.

计算模曲线 $X_0(N)$ 上的二次点

Computing quadratic points on modular curves $X_0(N)$

本文改进模曲线二次点计算方法，确定亏格至多 8（素数水平时至多 10）的一批 $X_0(N)$ 上全部二次点，并给出相应椭圆曲线的 $j$-不变量及 CM 点自同态环。

On the computation of algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$

In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for $\mathrm{GL}_2$, this algorithm in fact computes Hecke eigensystems of Hilbert-Siegel modular forms of genus 2. We give some examples of such eigensystems over $\Q(\sqrt{2})$.

Computing Genus 1 Jacobi Forms

We develop an algorithm to compute Fourier expansions of vector valued modular for Weil representations. As an application, we compute explicit linear equivalences of special divisors on modular varieties of orthogonal type. We define three families of Hecke operators for Jacobi forms, and analyze the induced action on vector valued modular forms. The newspaces attached to one of these families are used to give a more memory efficient version of our algorithm.

计算模雅可比簇 $J_{H}\left( p \right)$ 的尖点子群

Computing the Cuspidal Subgroup of the Modular Jacobian $J_{H}\left( p \right)$

本文针对模4余1的素数p对应的模曲线X_H(p)，计算其雅可比簇中由四个尖点生成的有限子群，覆盖亏格2至10的全部六种情形，并与相应数域上的有理挠子群进行比较。

通过弱全纯模形式的模曲线Atkin-Lehner商的二次Chabauty方法：Hodge滤过

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。

模函数特殊值上Galois作用的简化算法实现

A simplified algorithmic realization of Galois actions on special values of modular functions

本研究基于扩展形式类群理论，提出一种显式实用的算法，用于计算虚二次序相关CM点处模函数特殊值的Galois共轭元与不可约多项式，改进了Stevenhagen和Cho的早期方法。

A construction of antisymmetric modular forms for Weil representations

We give coefficient formulas for antisymmetric vector-valued cusp forms with rational Fourier coefficients for the Weil representation associated to a finite quadratic module. The forms we construct always span all cusp forms in weight at least three. These formulas are useful for computing explicitly with theta lifts.

New algorithms for modular inversion and representation by binary quadratic forms arising from structure in the Euclidean algorithm

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.

Classical weight one forms in Hida families: Hilbert modular case

The purpose of this paper is to investigate the number of classical weight one specializations of a non-CM ordinary Hida family of parallel weight Hilbert cusp forms. It is known that a specialization of a primitive ordinary Hida family at any arithmetic points of weight at least two is a classical holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. Balasubramanyam, Ghate and Vatsal proved that such a Hida family admits infinitely many classical weight one specializations if and only if it is of CM type. We give an explicit upper bound on the num…

The trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.