On the arithmetic of certain modular curves

Computational Arithmetic of Modular Forms (Course notes)

These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided.

Fourier coefficients of half-integral weight modular forms in arithmetic progressions

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Salié sums.

Arithmetic of Eisenstein quotients

In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second section, we give an example of modular abelian varieties due to Gross and study its properties in some details. In the third section, we define Eisenstein quotients of the modular Jacobians in general and give a criterion of the non-triviality of Heegner points on such Eisenstein quotients. The last two sections return to the concrete examples when the level of…

Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups

After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize these arithmetic properties of traces of singular values of a weakly holomorphic modular function on the full modular group to those on a congruence subgroup $Γ_0(N)$.

Arithmetic properties of Fourier coefficients of meromorphic modular forms

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coeff…

Arithmetic statistics of modular symbols

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture…

Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions

Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when $n$ runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $n\le x$ such that $a(n)>n^{-α}$ where $x$ and $α$ are positive and $f$ is not necessarily a Hecke eigenform.

Arithmetic partial differential equations, II: modular curves

We classify ``arithmetic convection equations'' on modular curves, and describe their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions of the same modular forms; in this sense, our arithmetic convection equations can be seen as "unifying" the two types of expansions. The theory can be generalized to one of ``arithmetic heat equations'' on modular curves, but we prove that modular curves do not carry ``arithmetic wave equations.'' Finally, we prove an instability result…

On modular forms for some noncongruence arithmetic subgroups

In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup (with algebraic coefficients) have bounded denominators. It was observed by Atkin and Swinnerton-Dyer that this is no longer true for modular forms for noncongruence subgroups and they pointed out that unbounded denominator property is a clear distinction between modular forms for noncongruence and congruence modular forms. It is an open question whether genuine…

The arithmetic of modular grids

A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for e…

关于C-递归整数序列在模算术中项的另外两种表示

On other two representations of the C-recursive integer sequences by terms in modular arithmetic

本文研究了C-递归整数序列的模算术表示，提出了两种新表示形式：对取余运算结果做除法，以及两个取余运算的复合，扩展了已有工作中仅使用除法后取余的表示方法。

酉 Shimura 簇的模高 II: 除子的算术生成级数

Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors

本文通过定义并计算酉 Shimura 簇上除子的算术生成级数及其算术相交数，推导了酉 Shimura 曲线的模高公式及其上 CM 点的高度公式。

Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet-Langlands correspondence

We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic twisted Hilbert modular surface. This quantity is then related to the arithmetic volume of a Shimura curve, via the arithmetic Grothendieck-Riemann-Roch theorem and the Jacquet-Langlands correspondence.

Optimal bounds for Büchi's problem in modular arithmetic II

Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence $(f(1),\dots,f(N))$ is a sequence of squares, then $N$ is at most $M$. We obtain this result by reducing to the case where $f$ has an invertible dominant coefficient.

Arithmetic properties of Fredholm series for p-adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

算术 theta 级数的拉回及其在酉 Shimura 曲线上的模性

Pullback of arithmetic theta series and its modularity for unitary Shimura curves

本文通过将 $U(1,1)$ Shimura 曲线嵌入多个 $U(n-1,1)$ Shimura 簇，证明算术除子生成级数的拉回公式，并借助 BHKRY 模性结果与 theta 级数非消失性，证明了 $U(1,1)$ 情形下的模性。

Arithmetic conjectures suggested by the statistical behavior of modular symbols

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $χ$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,χ,1) = 0$. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell-Weil rank grows in abelian extensions of $\mathbb{Q}$. Using this heuristic we find a large class of infinite abelian extensions $F$ where we expect $E(F)$ to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, an…

算术模链接

Arithmetic modular links

本文在模曲面上构造测地线序列，其典范提升的补空间为算术三维流形，建立了模曲面与算术流形间的联系。

The Borel-Moore homology of an arithmetic quotient of the Bruhat-Tits building of PGL of a non-archimedean local field in positive characteristic and modular symbols

We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, th…

On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on Γ_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.

Arithmetic compactifications of the Hilbert modular varieties and Hilbert modular forms for $Γ_1(c,n)$

We construct local charts for the ramified cusps of the Hilbert modular variety of level $Γ_1(\mathfrak{c},\mathfrak{n})$. This allows us, following closely M. Rapoport and C.-L. Chai, to construct arithmetic toroidal and minimal compactifications of these modular varieties. Some applications to modular forms are given. See our joint work with J. Tilouine for other applications.

Structure of ordinary $Λ$-adic arithmetic cohomology groups

We study the $Λ$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $Λ$-adic Selmer groups coming from two different towers have "almost same" $Λ$-module structures. We also prove the cotorsionness of $Λ$-adic Tate-Shafarevich group under mild assumptions.

On two arithmetic theta lifts

Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type. These functions play a key role in the arithmetic geometry of the special cycles in the context of Kudla's program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a modular form with trivial holomorphic projection. Along the way, we construct a section of the Maass lowering operator for moderate growth modular forms valued in the Weil representation us…

New Residue Arithmetic Based Barrett Algorithms, Part I: Modular Integer Computations

In this paper, we derive new computational techniques for residue number systems (RNS) based Barrett algorithm (BA). The focus of the work is an algorithm that carries out the entire computation using only modular arithmetic without conversion to large integers via the Chinese Remainder Theorem (CRT). It also avoids the computationally expensive scaling-rounding operation required in the earlier work. There are two parts to this work. First, we set up a new BA using two constants other than powers of two. Second, a RNS based BA is described. A complete mathematical framework is described inclu…

The variance of arithmetic measures associated to closed geodesics on the modular surface

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.

The arithmetic of vector-valued modular forms on $Γ_{0}(2)$

Let $ρ$ denote an irreducible two-dimensional representation of $Γ_{0}(2)$. The collection of vector-valued modular forms for $ρ$, which we denote by $M(ρ)$, form a graded and free module of rank two over the ring of modular forms on $Γ_{0}(2)$, which we denote by $M(Γ_{0}(2))$. For a certain class of $ρ$, we prove that if Z is any vector-valued modular form for $ρ$ whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of Z is unbounded. Our methods involve computing an explicit basis for $M(ρ…

Twisted arithmetic Siegel Weil formula on X0(N)

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (Δ, r) where Δ>0 and Δ\equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function ϕ_{Δ, r}(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of ϕ_{Δ, r}(τ), along the way, we also identify the arithmetic pairing \langle ϕ_{Δ, r}(τ),\widehatω_N \rangle with special value of some Eisenstein series, where \widehatω_N is a normalized metric Hodge line bundle.

Arithmetic Hilbert modular varieties and forms for $Γ_1(c,n)$

We recall first the analytic theory of the Hilbert modular varieties of level $Γ_1(\mathfrak{c},\mathfrak{n})$ and their compactifications. We construct arithmetic toroidal compactifications of the universal Hilbert-Blumenthal abelian variety (and of its fiber products), over the compactifications of the Hilbert modular variety constructed by the first author. We give several applications related to Hodge theory, Hilbert-Jacobi modular forms and Hilbert modular forms of half-integer weight.

论高斯日历公式：模算术的一个应用

On Gauss Calendar Formula: An Application of Modular Arithmetic

本文以娱乐方式介绍高斯日历公式及其应用，并基于此给出确定特定月份星期几日期的公式，核心数学概念为模算术。

非分裂Cartan模形式上的算术交集

Arithmetic intersections on non-split Cartan modular curves

研究非分裂Cartan模曲线上的算术交集，当素数p不分解时，确定相应除子在所有有限素数处的交集数，关键在于分析p处的不良缩减。

算术子群的形式西格尔模形式

Formal Siegel modular forms for arithmetic subgroups

本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。

Analogy between arithmetic of elliptic curves and conics

In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function.

模曲线乘积上的算术Siegel–Weil公式

Arithmetic Siegel--Weil formula on the product of modular curves

该研究证明了两条模曲线乘积上带任意奇异系数的完整生成级数算术Siegel–Weil公式，对应Kudla纲领的GSpin(2,2)情形。

Arithmetic of p-irregular modular forms: families and p-adic L-functions

Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three…

构造算术西格尔模形式：实乘法阿贝尔曲面的西塔提升与显式方法

Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces

该研究提出显式可计算蓝图，利用酉对偶对$(\text{U}(2,2), \text{Sp}_4)$的西塔对应构造实乘法阿贝尔曲面的向量值西格尔模形式，通过局部施瓦茨函数、分歧素数测试向量构造及计算流程，为朗兰兹函子性实例提供路线图。

Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.

Cyclicity of Elliptic Curves Modulo Primes in arithmetic progressions

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of Serre's Cyclicity Conjecture.

纵坐标成等差数列的莫德尔曲线

Mordell Curves with Ordinates in Arithmetic Progression

研究纵坐标成等差数列的莫德尔曲线，通过证明未被先前工作排除，给出特定长度的非同构曲线族，还构造适度秩椭圆曲线参数族，其子族有理点成等差数列。

Arithmetic geometry of algebraic curves and their moduli space

We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories to the arithmetic of the moduli space of algebraic curves, especially to automorphic forms on this space.

Incongruences for modular forms and applications to partition functions

The study of arithmetic properties of coefficients of modular forms $f(τ) = \sum a(n)q^n$ has a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of C.-S. Radu, S. Ahlgren, B. Kim, N. Andersen, and S. Löbrich have employed the $q$-expansion principle of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we extend the method to give additional results for a large class of modular forms. We also give analogous results for generalized Frobenius partitions and the two mock theta functions $f(q)$ a…

Exotic arithmetic structure on the first Hurwitz triplet

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…

由等差数列生成的比例模数值半群

On proportionally modular numerical semigroups that are generated by arithmetic progressions

本文刻画了使得比例模数值半群和区间生成半群等于给定等差数列生成数值半群的参数条件。

Searching for simultaneous arithmetic progressions on elliptic curves

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

On the Arithmetic of Elliptic Curves and a Homotopy Limit Problem

In this note, I study a comparison map between a motivic and étale cohomology group of an elliptic curve over $\mathbb{Q}$ just outside the range of Voevodsky's isomorphism theorem. I show that the property of an appropriate version of the map being an isomorphism is equivalent to certain arithmetical properties of the elliptic curve.

On the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces

We study the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic self-intersection number of the dualizing sheaf on minimal regular models of the modular curves associated with congruence subgroups $Γ_0(N)$ with square free level, as well as for the modular curves X(N) and the Fermat curves with prime exponent.

模椭圆曲线与双曲均匀化

Modular elliptic curves and hyperbolic uniformization

研究将椭圆曲线双曲均匀化概念从有理数域扩展到奇数次完全实域，利用志村曲线，证明算术型双曲均匀化的存在意味着几何模性。

On the topological aspects of arithmetic elliptic curves

In this short note, we shall construct a certain topological family which contains all elliptic curves over Q and, as an application, show that this family provides some geometric interpretations of the Hasse-Weil L-function of an elliptic curve over Q whose Mordell-Weil group is of rank $\leq1$.

On simultaneous arithmetic progressions on elliptic curves

In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally point out open questions which focus on the most interesting aspects of the problem for us.

On the arithmetic of twists of superelliptic curves

In this paper, we consider a family of twists of a superelliptic curve over a global field, and obtain results on the distribution of the Mordell-Weil rank of these twists. Our results have applications to the distribution of the number of rational points.

On arithmetic progressions on genus two curves

We study arithmetic progression in the $x$-coordinate of rational points on genus two curves. As we know, there are two models for the curve $C$ of genus two: $C: y^2=f_{5}(x)$ or $C: y^2=f_{6}(x)$, where $f_{5}, f_{6}\in\Q[x]$, $\operatorname{deg}f_{5}=5, \operatorname{deg}f_{6}=6$ and the polynomials $f_{5}, f_{6}$ do not have multiple roots. First we prove that there exists an infinite family of curves of the form $y^2=f(x)$, where $f\in\Q[x]$ and $\operatorname{deg}f=5$ each containing 11 points in arithmetic progression. We also present an example of $F\in\Q[x]$ with $\operatorname{deg}F=…

阿基米德算术西格尔-韦尔公式：关于 Shimura 曲线上的一般权重特异除子的平均公式

Archimedean arithmetic Siegel--Weil formula for general weights over Shimura curves

本文基于阿基米德算术理论，研究Shimura曲线上的特异除子权重问题，通过Whittaker函数的导数推导出平均公式，揭示了特殊除子与算术高度配对的关系。

Perfectoid spaces arising from arithmetic jet spaces

Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $δ$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $δ$-morphisms appearing in the theory such as the $δ$-characters of elliptic curves and the $δ$-period map on modular curves.

Modular forms and arithmetic geometry

This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form ϕ(τ) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve over Q. The q-expansion of this function is an analogue of the Hirzebruch-Zagier generating function for the cohomology classes of curves on a Hilbert modular surface. This`arithmetic theta function' is used to define an `arithmetic theta lift' from modular forms of weight 3/2 to the arithmetic Chow group of M. For integers t_1 and t_2 with t_1t_2 not a square, the (…