Michael Stoll
We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x |-> x^2 + c. This extends earlier results by Morton for N = 4 and by Flynn, Poonen and Schaefer for N = 5.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the p-rationality of real quadratic fields in terms of generalized Fibonacci numbers and their periods modulo positive integers.
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 refine a result of R. Benedetto in p-adic analysis in order to exhibit infinitely many quadratic polynomials with rational coefficients having a specified graph of rational preperiodic points.
We determine the possible degrees of cyclic isogenies defined over quadratic fields for non-CM elliptic curves with rational $j$-invariant.
We find all quadratic post-critically finite (PCF) rational maps defined over the rationals. We describe an algorithm to search for possibly PCF maps. Using the algorithm, we eliminate all but twelve rational maps, all of which are verifiably PCF. We also give a complete description of possible rational preperiodic structures for quadratic PCF maps defined over Q.
Representing rational integers by generalized quadratic forms over quadratic fields
研究二次域上值域为有理整数的广义二次型，证明实二次域中存在正定二元广义二次型可表示所有正整数，并证明仅有有限多个二次域存在三元广义二次型具有此类性质。
For each odd prime $p$, we prove the existence of infinitely many real quadratic fields which are $p$-rational. Explicit imaginary and real bi-quadratic $p$-rational fields are also given for each prime $p$. Using a recent method developed by Greenberg, we deduce the existence of Galois extensions of $\mathbf{Q}$ with Galois group isomorphic to an open subgroup of $GL_n(\mathbf{Z_p})$, for $n =4$ and $n =5$ and at least for all the primes $p <192.699.943$.
Given a family of quadratic twists of a fixed elliptic curve defined over $\mathbb{Q}$, we investigate the average rank in the subfamily of twists having a nontorsion rational point of almost minimal height. We show in particular that the average analytic rank is greater than one.
The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. We prove this conjecture for the exterior square case when the number field is a real multi-quadratic Galois extension of any degree of the rationals.
We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.
We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.
Let $ξ, ζ$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $τ$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q ξ\|, \|q ζ\| \} > q^{-1 + τ}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.
In a recent paper of Blomer, Bourgain, Radziwiłł and Rudnick, the authors proved the existence of small gaps between eigenvalues of the Laplacian in a rectangular billiard with sides $π$ and $π/\sqrtα$, i.e. numbers of the form $αm^2+ n^2$, whenever $α$ is a quadratic irrationality of certain types. In this note, we extend their results to all positive quadratic irrationalities $α$.
We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that the index is bounded, in most cases, independently of the polynomial.
Rational integers as sums of units -- the quadratic case
本文研究了二次数域中自然数作为k个单位和的渐进行为，解决了Jarden和Narkiewicz在2007年提出的问题。
Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.
We formulate a conjecture about the distribution of the canonical height of the lowest non-torsion rational point on a quadratic twist of a given elliptic curve, as the twist varies. This conjecture seems to be very deep and we can only prove partial results in this direction.
Permutation Rational Functions over Quadratic Extensions of Finite Fields
本文研究有限域二次扩张上的置换有理函数，利用与二次型相关的特征和确定特殊 $q$-二次多项式的零点数，并据此判定给定有理函数是否为置换。
Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms
该研究拓展Heath-Brown的有理三次型成果，证明14变量以上虚二次数域三次型非平凡表示零，推导得s≥627时有理三次型对有非平凡解，还证明33变量以上有理三次超曲面含有理直线、对应三次型有殆素数解。
In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadrat…
Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are integral over every prime in $P$. This answers one of the principal questions posed by Conway and Sloane in their book {\em Sphere packings, lattices and groups}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol 290, Springer-Verlag, New York, 1999; page 402.
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…
For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX^2+c$, starting at $0$, always recurs after $O(q/\log\log q)$ steps. For $X^2+1$ the same is true for any starting value. We suggest that the traditional "Birthday Paradox" model is inappropriate for iterates of $X^3+c$, when $q$ is 2 mod 3.
We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we prove that the Mordell-Weil rank of each such elliptic surface is at most $6$ over $\mathbb Q$. In fact, we show that the Mordell-Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over $\mathbb Q$.
Dynatomic Galois groups for a family of quadratic rational maps
本文研究具有周期 2 临界点的二次有理映射族，并对其第三和第四个动力原子 Galois 群给出显式结构描述。
Rational and isolated quadratic points on hyperelliptic curves of rank 0 and small genus
本文提出一种利用Kummer簇上Mordell-Weil群像来计算秩0超椭圆曲线有理点和孤立二次点的方法，并用Magma实现，在大量曲线上验证。
It is shown that $c=-29/16$ is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map $f_c(x) = x^2+c$ has a rational periodic point of period $3$. Several arithmetic conditions on the set of all such rational numbers $c$ and the rational orbits of $f_c(x)$ are proved. A graph on the numerators of the rational $3$-periodic points of maps $f_c$ is considered which reflects connections between solutions of norm equations from the cubic field of discriminant $-23$.
On the torsion growth in quadratic number fields for elliptic curves defined over the rationals
本文研究了在二次数域中椭圆曲线 torsion 子群的增长问题，通过分析导数与素数之间的关系，揭示了有理数域上椭圆曲线的 torsion 增长特性。
Let $\mathbb K$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$ be a rational function. This paper studies the number $Δ_{α, R}(n)$ of distinct solutions of $R^{(n)}(x)=α$ over the algebraic closure $\overline{\mathbb K}$ of $\mathbb K$, where $α\in \overline{\mathbb K}$ and $R^{(n)}$ is the $n$-fold composition of $R$ with itself. With the exception of some pairs $(α, R)$, we prove that $Δ_{α, R}(n)=c_{α, R}\cdot d^n+O_{α, R}(1)$ for some $0<c_{α, R}\le 1<d$. The number $d$ is readily obtained from $R$ and we provide estimates on $c_{α, R}$. Moreover we prove that the…
We compute the genus of a rational quadratic form in terms of the K-theory of a C*-algebra attached to the adelic orthogonal group of the form. As a corollary, one gets a higher composition law for the rational quadratic forms. As an illustration, we consider the Gauss composition of the binary quadratic forms.
Given a map f:Z-->Z and an initial argument alpha, we can iterate the map to get a finite set of iterates modulo a prime p. In particular, for a quadratic map f(z)=z^2 +c, c constant, work by Pollard suggests that this set should have length on the order of p^(1/2). We give a heuristic argument that suggests that the statistical properties of this set might be very similar to the Birthday Problem random variable X_n, for an n=p day year, and offer considerable experimental evidence that the limiting distribution of these set lengths, divided by p^(1/2), for p\leq x as x goes to infinity, conve…
Rational lines on cubic hypersurfaces II
本文研究了三次超曲面上的有理直线问题，通过引入二次型线性空间和虚二次数域上的三次型结果，将三次超曲面的有理直线存在下界从33降低至29。
For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational points. In other words, the main result asserts that there is a finite set $P(k)$ of prime numbers depending on $k$ such that: if there is a prime divisor of the discriminant of $B$ which is not in $P(k)$, then $M^B$ has no $k$-rational points. Moreover, we can take $P(k)$ to satisfy the following: There is an effectively computable constant $C(k)$ depending…
Rational solutions of pairs of diagonal equations, one cubic and one quadratic
本文通过获得指数和 $αx^3+βx^2$ 的 32/3 阶矩的最优估计，在局部可解性假设下证明变量数超过 10 时一对三次和二次对角方程存在非平凡整数解。
For a quadratic endomorphism of the affine line defined over the rationals we consider the problem of bounding the number of rational points that eventually land at a given constant after iteration, called pre-images of the constant. In the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems," it was shown that the number of rational pre-images is bounded as one varies the morphism in a certain one-dimensional family. Explicit values of the constant for pre-images of zero and -1 defined over the rational numbers were addressed in subsequent articles. This article addresses …
We provide a complete classification of possible graphs of rational preperiodic points of quadratic rational functions defined over the rationals with a rational periodic critical point of period 3, under two assumptions: that these functions have no periodic points of period at least 5 and the conjectured enumeration of points on a certain genus 6 affine plane curve. We show that there are exactly six such possible graphs, and that rational functions satisfying the conditions above have at most eleven rational preperiodic points.
In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
Density formulas for $p$-adically bounded primes for hypergeometric series with rational and quadratic irrational parameters
本文研究超几何级数的 $p$-进有界素数密度，将有理参数情形推广至 $_nF_{n-1}$ 并给出精确公式，同时为二次无理参数情形建立下界及条件性精确公式。
It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve that classifies periodic points of period~4 is birational to $X_1(16)$, whose rational points had been previously computed. We prove there are none with $N=5$. Here the relevant curve has genus~$14$, but it has a genus~$2$ quotient, whose rational points we compute by performing a~$2$-descent on its Jacobian and applying a refinement of the method of Chabau…
Torsion of Rational Elliptic Curves over the $\mathbb{Z}_p$-Extensions of Quadratic Fields
本文证明对二次域的 $\mathbb{Z}_p$-扩张（$p>5$），有理椭圆曲线的挠子群不增大，并据此分类了可实现的挠子群。
Let $Z$ be a quadratic hypersurface of $\mathbb{P}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform rational approximation is the inverse $1/ρ$ of an explicit Pisot number $ρ<2$ depending only on $n$ if the Witt index (over $\mathbb{Q}$) of the quadratic form $q$ defining $Z$ is at most $1$, and that it is equal to $1$ otherwise. Furthermore there are points of $Z$ which realize this maximum. They constitute a countably infinite set in the first case, and an…
We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell--Weil group.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $ϕ:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space ${\mathcal M}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduct…
Fix a rational basepoint b and a rational number c. For the quadratic dynamical system f_c(x) = x^2+c, it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational parameter c. In this short note we investigate the dependence of the bound on the basepoint b, assuming a strong form of the Mordell Conjecture.
The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.
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 $S=\{x^2+c_1, x^2+c_2,\dots, x^2+c_s\}$ be a set of quadratic polynomials with rational coefficients, and let $P$ be a rational basepoint. We classify the pairs $(S,P)$ for which $P$ has finite orbit for $S$, assuming that the maximum period length for each individual polynomial is at most three (conjectured by Poonen). In particular, under these hypotheses we prove that if $s\geq4$, then there are no points $P$ with finite orbit for $S$. Moreover, we use this perspective to formulate an analog of the Morton-Silverman Conjecture for sets of rational functions.
Given a pair of elliptic curves $E_1$ and $E_2$ over the rational field $\mathbb Q$ whose $j$-invariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals $d$ such that the $d$-quadratic twists of $E_1$ and $E_2$ are both of positive rank. We construct infinite families of pairs of elliptic curves $E_1$ and $E_2$ over $\mathbb Q$ such that for each pair there exist infinitely many square-free rationals $d$ for which the $d$-quadratic twists of $E_1$ and $E_2$ are both of rank at least 2.
We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of quadratic polynomials $x^2+c$, using techniques in the theory of rational point…
In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadra…
Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups
本文研究二次有理映射在有理数域上非交换自同构群下的有理数周期点问题，证明了不存在周期N≥4的有理数周期点，并给出了周期1、2、3的参数化结果，同时证明了预周期点数量不超过6，最终分类出5种预周期点图谱。
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division division algebras and zero divisors in $M_2(D)$, the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.
In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.
We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit nonabelian Chabauty.
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a quadratic number field.
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results abo…
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.
In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic f…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。