Reinier Broker, Andrew V. Sutherland
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.
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Explicit height bounds on modular polynomials for the elliptic $j$-invariant, cube root of $j$, and Weber modular function $\mathfrak f$
研究改进椭圆\(j\)不变量等相关模多项式显式上界，通过改进Breuer等人给出的界，还证明了\(j\)立方根及韦伯模函数\(\mathfrak f\)的模多项式高度新显式上界。
In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over $\mathbb Z$. As applications, we obtain the first `non-abelian' Chowla-Selberg formula, which is a special case of Colmez's conjecture; an explicit arithmetic intersection formula between arithmetic Humbert surfaces and CM cycles in the arithmetic Siegel modular variety of genus two; Lauter's conjecture about the denominators of CM values of Igusa invariants; and a result about bad reductions of CM genus two curves.
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。
Heights of Drinfeld modular polynomials and Hecke images
本文研究了 Drinfeld 模多项式与 Hecke 像的高度问题，获得了多项式系数大小的显式上下界，并证明了其主项的渐近最优性，同时给出了秩2 Drinfeld 模 Hecke 像高度的精确估计。
In this note, we propose the modular height of an abelian variety defined over a field of finite type over Q. Moreover, we prove its finiteness property.
We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.
Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem
通过比较混合 Eisenstein-theta 级数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。
Modular Heights of Unitary Shimura Varieties I: Derivatives of Eisenstein Series
本文通过比较混合 Eisenstein-theta 级数导数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。
We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman's inequality and isogeny estimates by Gaudron and Rémond. We give applications to the study of Vélu's formulas and of modular polynomials.
In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.
We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$ approaches infinity. We also show that, for a fixed degree, the density of Dumas polynomials in all irreducible integer coefficient polynomials is strictly less than 1.
We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height.
We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree $d$ and of height at most $H$, as $H\to \infty$. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most $H$.
Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number $α$ under certain assumptions on $α$. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. Our theorem contains, as corollaries, a slight generalization of the above results as well as some new lower bounds in other special cases.
Explicit bounds on the coefficients of the modular polynomials and the size of $X_0(N)$
本文给出模多项式系数的显式上下界，使渐近界显式化，并给出 Hecke 点估计的显式版本及模曲线高度与模多项式高度的渐近比较。
In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded discriminant. The main term of the obtained formula is the best possible for the considered range of polynomial discriminant values.
We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant…
In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.
Let $A_n$ denote the height of cyclotomic polynomial $Φ_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le ε_kϕ(n)^{k^{-1}2^{k-1}-1}$ with $-\logε_k\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $Ψ_n$. Additionally, we improve on a bound of Kaplan for the maximal height of divisors of $x^n-1$, denoted by $B_n$. We show that $B_n<η_k n^{(3^k-1)/(2k)-1}$, with $-\log η_k \sim c3^k$ and the same $c$.
A Note on Heights of Cyclotomic Polynomials
本文证明了对于任意正整数 $h$，$h$ 或 $h+1$ 必为某个由三个不同素数乘积决定的分圆多项式 $Φ_n$ 的高度。
We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring F_q[T].
Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a Weil height on the base by little-o of a Weil height. In the case where f is a family with an invariant hyperplane, whose restriction to this invariant hyperplane is isotrivial, we improve this by showing that the canonical height along a section differs from a Weil height on the base by a bounded amount.
We establish (Theorem 3.6) polynomial-growth estimates for the Fourier coefficients of holomorphic logarithmic vector-valued modular forms.
Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.
Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review well-known bounds obtained from interpolation algorithms given values at $d+1$ (resp. $2d+1$) points, and obtain tighter results when considering a larger number of evaluation points.
Modular Heights of Quaternionic Shimura Curves
本文证明了一个公式，将全实数域上四元数 Shimura 曲线的模高表示为 Dedekind zeta 函数对数导数，基于 Yuan-Zhang-Zhang 和 Yuan-Zhang 的工作，受 Gross-Zagier 和 Kudla 纲领启发。
We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.
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.
We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular polynomials attached to Drinfeld modules.
We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve's level. Our main technical tools come from effective Arakelov descriptions of modular curves and jacobians. We then fulfill this program in the following particular case: If $p$ is a not-too-small prime number, let $X_0 (p )$ be the classical modular curve of level $p$ over $\bf Q$. Assume Brumer's conjecture on the dimension of winding quotients of $J_0 (p)$. We prove that there is a function $b(p)=O(p^{5} \log p )$ (depend…
Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving the Weil height. The height appears as a solution to a certain extremal problem involving polynomials. We further generalize the above techniques to acquire both the projective height and the height on subspaces in the same way. We further obtain lower bounds on the heights of points on some subvarieties of $\mathbb P^{N-1}(\overline{\mathbb Q})$…
We provide an asymptotic expression for the probability that a randomly chosen polynomial with given degree, having integral coefficients bounded by some B, has a prescribed signature. We also give certain related formulas and numerical results along this line. Our theorems are closely related to earlier results of Akiyama and Pethő, and also yield extensions of recent results of Dubickas and Sha.
In this paper, we give an explicit bound for the height of integral points on $X_0(p)$ by using a very explicit version of the Chevalley-Weil principle. We improve the bound given by Sha in \cite{sha2014bounding1}.
Geometric construction of modular polynomials with level structures
本文提出了一种纯代数方法，用于构造具有更高层次结构的模多项式，展示了与椭圆曲线模型直接相关的不变量的模多项式存在性，并给出了计算这些多项式的算法。
On the height of polynomials that split completely over a fixed number field
本文证明了在固定数域上完全分裂的 $n$ 次多项式的最小高度随 $n$ 呈指数级增长，并精确确定了全实数域及特定复二次域情况下的隐含常数。
We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither $p$-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the `division polynomials' associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.
In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when the exceptional set is given by certain natural geodesics related to binary quadratic forms of (positive) discriminant $D$. We furthermore show that the dimension is the largest possible if and only if $D$ is an even square. Following this, we describe how to use the methods developped in this paper to establish an algorithm which explicitly determines the sp…
We show that all but 5 of the zeros of the period polynomial associated to a Hecke cusp form are on the unit circle.
In this paper, we study how small a box contains at least two points from a modular cubic polynomial $y \equiv a x^3 + b x^2 + c x + d \pmod p$ with $(a, p) = 1$. We prove that some square of side length $p^{1/6 + ε}$ contains two such points.
In a recent breakthrough, Dimitrov solved the Schinzel-Zassenhaus Conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ where $p$ is a prime number and where the orbit of $0$ is finite. For example, if $p=2$, and $0$ is periodic under $T^2+c$ with $c\in\mathbb{R}\smallsetminus\{-2\}$, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this we are able to deduce a lower bound for the canonical height of a wandering point that decays lik…
This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies ar…
In this paper, we derive a new computational algorithm for Barrett technique for modular polynomial multiplication, termed BA-P. BA-P is then applied to a new residue arithmetic based Barrett algorithm for modular polynomial multiplication (BA-MPM). The focus of the work is an algorithm that carries out the entire computation using only modular arithmetic without conversion to large degree polynomials. There are several parts to this work. First, we set up a new BA-P using polynomials other than u^alfa. Second, residue arithmetic based BA-MPM is described. A complete mathematical framework is …
On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$
研究\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度差，建立与次数\(d\)无关的精确界，以此确定\(\mathbb{Q}\)上高度为0的多项式映射的有理预周期点集并分类其画像。
We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.
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, …
The conjecture of Colmez and reciprocity laws for modular forms
本文提出改进科尔梅兹猜想，明确CM阿贝尔簇的Faltings高度与Artin L函数的对数导数关系，并证明其对椭圆曲线成立，揭示模形式在CM点的精细互反律。
Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves
该研究针对素数模下类多项式与模多项式的计算问题，借助同源密码学推动下发展的超奇异曲线Deuring对应研究成果，提出多种基于超奇异椭圆曲线的新启发式算法，打破了此前超奇异情形效率低的固有认知。
The height of a polynomial $f(x)$ is the largest coefficient of $f(x)$ in absolute value. Let B(n) be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. In this paper we investigate the maximal height of divisors of $x^{pq^b}-1$ and prove that some conjectures on the maximal height of divisors of $x^{pq^b}-1$ are true.
Drinfeld modular polynomials of level $T$
本文研究了参数化秩 $r\geq 2$ 的 Drinfeld 模间 $T$-同源的模多项式，通过显式分类同源，导出了系数 $T$-次数的显式界限，并获得了多项式高度的精确表达式。
On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials
本文证明了对于任意奇素数p和整数h，存在足够大的素数q和r，使得对应的循环多项式Φ_{pqr}的高度为h，并揭示了其直径的可能取值。
The height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic polynomials. One result, due to H. Maier, gives a best possible upper bound of n^{ψ(n)} for almost all n, where ψ(n) is any function that approaches infinity as n tends to infinity. We will discuss the related problem of bounding the maximal height over all polynomial divisors of x^n - 1 and give an analogue of Maier's result in this scenario.
For each of the groups PSL2(F25), PSL2(F32), PSL2(F49), PGL2(F25), and PGL2(F27), we display the first explicitly known polynomials over Q having that group as Galois group. Each polynomial is related to a Galois representation associated to a modular form. We indicate how computations with modular Galois representations were used to obtain these polynomials. For each polynomial, we also indicate how to use Serre's conjectures to determine the modular form giving rise to the related Galois representation.
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。
We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, …
The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that the set of monic polynomials g(z) of given degree with the property that there exists a d > 1 such that g(z^d) is post-critically finite, is also a set of bounded height. Moreover, we establish a lower bound on the critical height of g(z^d).
The relations satisfied by period polynomials associated to modular forms yield a way to count dimensions of spaces of cusp forms. After showing how these relations arise from those on the mapping class group $PSL(2, \mathbb{Z})$ of the moduli space $\mathcal{M}_{0,4}$ of genus 0 curves with 4 marked points, the author goes on to define period polynomials associated to Picard modular forms. Relations on these Picard period polynomials are then determined, and via an embedding of a monodromy representation of the moduli space $\mathcal{M}_{0,5}$ of genus 0 curves with 5 marked points in $PU(2,1…
Number of characteristic polynomials of matrices with bounded height
本文研究元素高度受 H 限制的整数 n×n 矩阵，给出其不同不可约特征多项式及特征值数量的上下界，并改进已有结果。
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.
Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant
本文研究椭圆 $j$-不变量模多项式 $Φ_N$ 的系数规模，给出适用于所有 $N\geq1$ 的显式上界，并证明其主项随 $N$ 趋于无穷时渐近最优。
Let p and r be two primes and n, m be two distinct divisors of pr. Consider the n-th and m-th cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of one of them modulo the other one. We mention an application to torus-based cryptography.
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