Derong Qiu
In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of congruence properties of elliptic curves are also presented and discussed.
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We exhibit two non-isogenous rational elliptic curves with $17$-torsion subgroups isomorphic as Galois modules.
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil lattice and the geometric Picard number.
We put together some known theoretical results and the fact that certain computations can be done efficiently in SAGE to come up with a fast algorithm for calculating congruence primes linking a newform with integer coefficients (i.e. a newform associated to an elliptic curve) with the old space at the same level.
We describe a systematic investigation into the existence of congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod $p$ congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes $p\ge7$ to the elliptic curves in the LMFDB database, which currently includes all el…
In this work we prove congruences between special values of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analogue for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Grössencharacters, can be expressed as values of Hilbert-Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences f…
We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.
In this note we consider questions about parametrisations of elliptic curves defined over number fields by quotients of the upper half-plane by finite index subgroups of SL_2(Z). We ask if we can choose such a parametrisation of an elliptic curve to reflect the field over which it is defined: this can be viewed as a generalisation of the Shimura-Taniyama-Weil conjecture.
We prove two results concerning the generalized Fermat equation $x^4+y^4=z^p$. In particular we prove that the First Case is true if $p \neq 7$.
In the genus 1 case, mirror symmetry reduces to the statement that a certain family of generating functions, relating to an elliptic curve, are quasimodular. In their proof of this fact, Kaneko and Zagier used a related family of generating functions $A_n(τ)$, which they show to be quasimodular. We show that these $A_n$'s also satisfy Kummer-type congruences. Additionally, we show that for a prime $p$, the $p$th power coefficients of $A_n$ $p$-adically converge to zero, for specific values of $n$.
We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence formula when $p=2$ to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves $E$. We show that the number of twists of $E$ up to twisting discriminant $X$ of analytic rank zero (resp. one) is $\gg X/\log^{5/6}X$, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We also prove th…
We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p) is the p-th Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. Using the interplay between singular (a.k.a. CM) j-invariants in characteristic zero and supersingular ones in characteristic p, we obtain such congruences in which F is the minimal polynomial of a CM j-invariant, and give a sufficient condition for G to be a constant polynomial in these congruences.
We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs $(N, r)$ such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an …
Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1)) is -1 (respectively, +1), we prove a ``Jochnowitz congruence'' between the algebraic part of L'_K(E,1) (expressed in terms of Heegner points on Shimura curves) and the algebraic part of L_K(g,1). This establishes a relation between Zhang's formula of Gross-Zagier type for central derivatives of L-series and his formula of Gross type for special values. Our r…
Congruences of models of elliptic curves
本文证明了在离散赋值环上，当无穷小纤维层级足够大时，椭圆曲线的最小 Weierstrass 模型和最小正则模型的特异纤维可由该无穷小纤维及伽罗瓦群作用唯一确定。
The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible $p$-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves.
p-adic congruences in iterated derivatives of the Weierstrass elliptic function
本研究反向运用Ando–Hopkins–Rezk的$\mathbb E_\infty$定向机制，结合Tate不动点对象的定向存在性结果，证明Weierstrass椭圆函数迭代导数间的p进Kümmer同余。
For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.
This article computes the Galois groups of congruence covers arising in the context of certain hyperbolic triangle groups. As a consequence of this computation, the genera of the respective curves are deduced.
We prove formulas for power moments for point counts of elliptic curves over a finite field $k$ such that the groups of $k$-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$. As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$ which include as special cases the groups $Γ_1(N)$ and $Γ(N)$. Our formulas generalize results of Birch and Ihara (the case of the trivial s…
We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL_2(q) and PGL_2(q) regularly as Galois groups.
Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions
本文通过研究二次型与椭圆曲线点计数之间的关系，建立了一种利用theta级数和同余条件来分析整数解与椭圆曲线有理点数量的联系。
On the average congruence class bias for cyclicity and divisibility of the groups of $\mathbb{F}_p$-points of elliptic curves
本文改进了Banks和Shparlinski关于椭圆曲线模素数约化点群循环性与可除性的平均密度结果，将素数限制在算术级数中，并研究不同同余类间的统计偏差。
We present explicit equations of semi-stable elliptic surfaces (i.e., having only type $I_n$ singular fibers) which are associated to the torsion-free genus zero congruence subgroups of the modular group as classified by A. Sebbar.
On $12$-congruences of elliptic curves
本文通过计算模对角商曲面的显式双有理模型，构造了定义在Q上且12-挠子群作为Galois模同构的椭圆曲线对无限族，推广了Chen和Fisher的工作。证明关键在于以j不变量给出2、3或4-挠子群同构的代数条件。
In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show that for all but finitely many primes $p$, the group of Markoff automorphisms acts transitively on the nonzero $\mathbb{F}_p$-points of the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$. This yield…
We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes.
Let $p\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$. We consider supersingular elliptic curves with a basis of $2$-torsion over $\overline{\mathbf{F}}_p$, or equivalently supersingular Legendre $λ$-invariants. Let $F_p(X,Y) \in \mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $λ$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\frac{F_p(X,X^{p})}{p}$ is in $\mathbf{Z}[X]$. We give a formula for $R(λ)$ if $λ$ is a supersingular. This formula is related to the Manin--Dri…
A $p$-adic cohomological approach to congruences of meromorphic modular forms
本文提出了一种基于p进上同调的方法，用于研究解析模形式的同余关系，通过刚性上同调与晶格结构的相互作用，建立了弗罗贝尼乌斯作用与U_p算子之间的联系。
Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves
本文改编Zywina方法提出算术级数中Koblitz猜想，细化Jones定理，发现循环性与Koblitz常数在同余类上平均偏差相反，并提供Magma计算代码。
Let $p\ge 5$ be a prime. We show that the space of weight one Eisenstein series defines an embedding into $\PP^{(p-3)/2}$ of the modular curve $X_1(p)$ for the congruence group $Γ_1(p)$ that is scheme-theoretically cut out by explicit quadratic equations.
This work is a sequel of a previous work of one of the authors (Y.Ô), which treated certain congruence relation between an elliptic Gauss sum and a coefficient of power series expansion at the origin of the lemniscate sine function. We extend the previous result (in \cite{O}) which concerned only for non-vanishing elliptic Gauss sums. We give new congruence relations between power series coefficients of the lemniscate cosine function, which hold if and only if the corresponding elliptic Gauss sum vanishes.
Elliptic curves and Fourier coefficients of meromorphic modular forms
本文研究了模形式系数的同余关系，揭示其与椭圆曲线对称幂及$p$-adic Chowla--Selberg周期的联系，并通过超几何函数和Borcherds--Shimura提升进行证明。
In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension.
Congruence obstructions in the refined Koblitz conjecture
本文分类了非CM椭圆曲线的本原同余障碍，证明其仅出现在有限级别，并借助模210伽罗瓦像判定Koblitz–Zywina常数正性，同时在GRH下给出最小素因子的下界。
We consider character sums determined by isogenies of elliptic curves over finite fields. We prove a congruence condition for character sums attached to arbitrary cyclic isogenies, and produce explicit formulas for isogenies of small degree.
We find that the first Hurwitz triplet possesses two distinct arithmetic structures. As Shimura curves $X_1$, $X_2$, $X_3$, whose levels are with norm 13. As non-congruence modular curves $Y_1$, $Y_2$, $Y_3$, whose levels are 7. Both of them are defined over ${\Bbb Q}(\cos \frac{2 π}{7})$. However, for the third non-congruence modular curve $Y_3$, there exist an "exotic" duality between the associated non-congruence modular forms and the Hilbert modular forms, both of them are related to ${\Bbb Q}(e^{\frac{2 πi}{13}})$! Our results have relations and applications to modular equations of degree…
2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family
本文针对亏格为正的模曲线场景调整现有技术，研究2-拉长平面分拆菱形计数函数$d_2(n)$关于7的幂的亏格1同余族，对比相关方法并猜想另一适用类似技术的同余族。
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.
In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence of newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t o…
Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross-Koblitz formula. In this paper, we aim to prove four supercongruences between two truncated hypergeometric series ${}_2F_1$. This generalizes these four Rodriguez-Villegas supercongruences.
We introduce a procedure for constructing a generalized elliptic curve from a genus-one twisted curve, and we use this procedure to define an explicit, modular isomorphism between two compactifications of moduli stacks of elliptic curves with level structure.
Old Meets New: Connecting Two Infinite Families of Congruences Modulo Powers of 5 for Generalized Frobenius Partition Functions
本文在与层级 20 模曲线相关的两类广义 Frobenius 分拆函数同余族之间建立模函数环同构，并借助 Atkin–Lehner 对合提出研究复合层级同余的新策略。
Congruences modulo $23$ to $y^2=x^3-23$ are trivial
本文运用 Mazur 策略、Jacobian 的 Tate 模及 BSD 新实例确定扭曲模曲线有理点，证明与 $y^2=x^3-23$ 的模 $23$ 同余均平凡。
Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\mathbb{F}_{p}$ for primes $p \geq 5$. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. In particular, we find that \[ Z_p(A;t) = \frac{\widehat{Z}_p(A; t)}{\displaystyle \prod_{a \in \textrm{…
Some thoughts on the congruence D=0,1(mod 4) for the absolute discriminant D of a number field
We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.
For several congruence subgroups of low levels and their conjugates, we derive differential equations satisfied by the Eisenstein series of weight 4 and relate them to elliptic curves, whose associated new forms of weight 2 constitute the list of Martin and Ono of new forms given by eta-products/quotients.
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.
We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.
We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also show that, if we want such a curve to be just geometrically hyperelliptic, so having a degree two map to a conic, then there are some with genus 3.
Congruences of modular forms and modularity of Tate-Shafarevich classes
本文在适当假设下证明了有理数上椭圆曲线的$p$-挠Tate-Shafarevich类在模曲线Jacobi商群中的可见性，验证了Jetchev-Stein猜想，其核心方法是利用Bertolini-Darmon二部Kolyvagin系统的非平凡性来推导Selmer群模$p$的基。
In this paper, we find a power series expansion of the invariant differential $ω_E$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we introduce several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.
The coefficient series of the holomorphic Picard-Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These have been widely studied in the case of the torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider arithmetic properties of the Picard-Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups. We prove that elliptic families with rational parameters admit linear reparametrizations such that their as…
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。
For certain elliptic curves $E/\mathbb{Q}$ with $E(\mathbb{Q})[2]=\mathbb{Z}/2 \mathbb{Z}$, we prove a criterion for prime twists of $E$ to have analytic rank 0 or 1, based on a mod 4 congruence of 2-adic logarithms of Heegner points. As an application, we prove new cases of Silverman's conjecture that there exists a positive proposition of prime twists of $E$ of rank zero (resp. positive rank).
For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.
The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here we provide explicit representations of the modular forms associated to certain Legendre form elliptic curves 2_E_1(λ) as linear combinations of quotients of Dedekind's eta-function. We also give congruences for some of the modular forms' coefficients in terms of Gaussian hypergeometric functions.
In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular Abelian variety with an odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^α$ or $N=pq$ for some prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and…
We obtain distribution results for traces of Frobenius for various families of elliptic curves with respect to the Lang-Trotter conjecture, extremal primes, and the central limit theorem. This includes some generalisations and bounds related to the work of Sha-Shparlinski on the average Lang-Trotter conjecture for single-parametric families of elliptic curves and the work of various authors on the trace of Frobenius for primes in congruence classes. Some results are also obtained for modular forms.
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