Elliptic curves over the perfect closure of a function field

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb F}_q)$ be an elliptic curve over ${\mathbb F}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer $n$ we denote by $ \# E({\mathbb F}_{q^n})$ the number of rational points on $E$ (including infinity) over the extension ${\mathbb F}_{q^n}$. Under a mild technical condition, we show that the sequence $\lbrace \# E({\mathbb F}_{q^n}) \rbrace_{n>0}$ contains at most $10^{200}$ perfect squares. If the mild condition is not satisfied, then $\#E({\mathbb F}_{q^n})$ is a perfect square for infinitely many $n$ including all th…

Perfect points on genus one curves and consequences for supersingular K3 surfaces

We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.

按高度计数具有7-同源的有理数域上的椭圆曲线

Counting elliptic curves over the rationals with a 7-isogeny

本文按高度计数了有理数域上具有7次循环同源的椭圆曲线数量，分别考虑了有理数域及其代数闭包上的同构类。

显式界定椭圆可除性序列中的完美幂

Explicitly bounding perfect powers in elliptic divisibility sequences

本文研究了 $j$-不变量为 $1728$ 的椭圆曲线生成的椭圆可除性序列，利用 $\mathbb{Q}$-曲线的模方法显式确定有限素数集，证明集合外的素数 $l$ 在序列中无 $l$ 次幂，并为 $\mathbb{Q}$-曲线计算框架提供示例。

Cyclotomic torsion points in elliptic schemes

An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin-Mumford conjecture stated by Masser and Zannier, we propose a family version of the above statement and prove it under a suitable integrality condition.

The perfect power problem for elliptic curves over function fields

We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p>3, let S denote a finite set of places of K and let E/K denote a non-isotrivial elliptic curve over K with j-invariant a p^s power but not p^(s+1) power in K. Fix a non-constant function f in K(E) with a pole of order N>0 at the zero element of E. We prove that there are only finitely many rational points P in E(K) such that for any valuation outside S for which f(P) is negative, that valuation of f(P) is divisible by so…

Perfect powers in products of terms of elliptic divisibility sequences

Diophantine problems involving recurrence sequences have a long history and is an actively studied topic within number theory. In this paper, we connect to the field by considering the equation \begin{align*} B_mB_{m+d}\dots B_{m+(k-1)d}=y^\ell \end{align*} in positive integers $m,d,k,y$ with $\gcd(m,d)=1$ and $k\geq 2$, where $\ell\geq 2$ is a fixed integer and $B=(B_n)_{n=1}^\infty$ is an elliptic divisibility sequence, an important class of non-linear recurrences. We prove that the above equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solu…

椭圆曲线分母乘积中的完美幂

Perfect powers in the product of denominators of elliptic curves

利用筛法论证估计有理点s元组中分母乘积为固定素数ℓ次完美幂的频率，并推广到有理函数情形。

四次方简化与椭圆障碍物在完美欧拉砖中的应用

Quartic reductions and elliptic obstructions for perfect Euler bricks

本文探讨了完美欧拉砖问题等价于是否存在互质整数a,b,m,n使两个表达式同时为完全平方数，并通过四次方简化和椭圆障碍物分析，验证了参数至10^3时无解。

环的完备化的混合特征类比及其几乎 Cohen-Macaulay 性质

A mixed characteristic analogue of the perfection of rings and its almost Cohen-Macaulay property

本文在混合特征完备局部整环上构造了一个类似正特征完美闭包的 perfectoid 环，并利用 André 平坦性引理和 Riemann 延拓定理证明了其几乎 Cohen-Macaulay 性质。

Elliptic Dedekind Domains Revisited

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and -- we feel -- simpler proofs of theorems of L. Claborn and C.R. Leedham-Green.

Canonical lifts of families of elliptic curves

We show that the canonical-lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti an…

无限族的原始海伦三角形，其中两边为完全平方数

An Infinite Family of Primitive Heron Triangles with Two Sides as Perfect Squares

该研究通过椭圆曲线理论证明存在无限多个原始海伦三角形，其中两边为完全平方数，并暗示可能存在三边均为完全平方数的原始海伦三角形。

Mordell-Weil groups and the rank of elliptic curves over large fields

Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $σ\in Gal(\bar{K}/K)$, the Mordell-Weil group $E(\bar{K}^σ)$ of $E$ over the fixed subfield of $\bar{K}$ under $σ$ has infinite rank.

Heegner points and Mordell-Weil groups of elliptic curves over large fields

Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $σ\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$ over the fixed subfield of $\bar{\bbq}$ under $σ$ has infinite rank. Our approach uses the modularity of $E/\bbq$ and a collection of algebraic points on $E$ -- the so-called {\em Heegner points} -- arising from the theory of complex multiplication.

椭圆可除序列中的完全幂

Perfect powers in elliptic divisibility sequences

本文研究椭圆可除序列中完全幂的有限性问题，通过构造全实域上的Frey曲线并结合模性与降阶论证，在三个附加假设下给出肯定回答，且在Langlands纲领标准猜想下可放宽假设得到相同结论。

具有亏格1伽罗瓦闭包的射影直线的例外覆盖：有限域上的算术形式

Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields

本文对几何伽罗瓦闭包亏格1的可分不可约驯服例外射影直线映射分类，给出有限域上算术固定域形式的编码准则，获秩2支撑公式等结果，明确特征与障碍的相关性质。

Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.

Radical entanglement for elliptic curves

Let $G$ be a commutative connected algebraic group over a number field $K$, let $A$ be a finitely generated and torsion-free subgroup of $G(K)$ of rank $r>0$ and, for $n>1$, let $K(n^{-1}A)$ be the smallest extension of $K$ inside an algebraic closure $\overline K$ over which all the points $P\in G(\overline K)$ such that $nP\in A$ are defined. We denote by $s$ the unique non-negative integer such that $G(\overline K)[n]\cong (\mathbb Z/n\mathbb Z)^s$ for all $n\geq 1$. We prove that, under certain conditions, the ratio between $n^{rs}$ and the degree $[K(n^{-1}A):K(G[n])]$ is bounded independ…

Non-isogenous elliptic curves and hyperelliptic jacobians II

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is …

Massey 积与椭圆曲线

Massey products and elliptic curves

本文研究完美域上绝对不可约光滑射影曲线中系数为 $\mathbb{Z}/\ell$ 的高阶 Massey 积消失问题，并完整刻画椭圆曲线上三重 Massey 积非消失的条件。

Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have…

椭圆曲线模型的一致性

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。

临界值的算术 II：临界椭圆曲线

The arithmetic of critical values II: critical elliptic curves

本文作为《临界值的算术》系列第二章，通过伽罗瓦理论直接证明双覆叠 $E_f$ 存在 3-同源，推导塞尔默伴随定理并给出相关应用与新雅可比簇例子。

Diophantine triples and construction of high-rank elliptic curves

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.

Perfect powers generated by the twisted Fermat cubic

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than 1. Moreover, if the first term is divisible by 6 and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an lth power for some l dividing the order of 2 in the first term.

Expansion in perfect groups

Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an integer q, denote by Ga_q the subgroup of Ga consisting of the elements that project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q with respect to the generating set S form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Ga is perfect.

Primitive Divisors of Certain Elliptic Divisibility Sequences

Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$. We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the elliptic divisibility sequence generated by $P$ always has a primitive divisor.

Heegner points and the rank of elliptic curves over large extensions of global fields

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our…

椭圆曲线的约化与同源

Reduction and isogenies of elliptic curves

本文研究具有素次数有理同源的椭圆曲线的约化Kodaira类型，在完备离散赋值环域的设定下展开分析，并得到有理数域上椭圆曲线的相关结论。

On elliptic curves induced by rational Diophantine quadruples

In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.

Torsion points of Drinfeld modules over large algebraic extensions of finitely generated function fields

Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.

Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves

For a non-square positive integer x, let k_x denote the distance between x^3 and the perfect square closest to x^3. A conjecture of Marshall Hall states that the ratios r_x = (x^(1/2))/k_x, are bounded above. (Elkies has shown that any such bound must exceed 46.6.) Let {x(n)} be the sequence of "Hall numbers": positive non-square integers for which r_x(n) exceeds 1. Extensive computer searches have identified approximately 50 Hall numbers. (It can be proved that infinitely many exist.) In this paper we study the minimum gap between consecutive Hall numbers. We prove that for all n, x(n + 1) - …

Rank gain of Jacobians over number field extensions with prescribed Galois groups

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$ gain rank over infinitely many $G$-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion which allows to deduce that "many" elliptic curves gain rank over infin…

On the topological cyclic homology of the algebraic closure of a local field

The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper with Ib Madsen, we used this comparison and an approximate evaluation of the structure of the pro-spectrum TR to evaluate the p-adic K-groups of a local field K of mixed characteristic (0,p) with perfect residue field. In this paper we completly determine the structure of the pro-spectrum TR for an algebrac closure of the local field K. This leads us to formulate a conjecture for the structure of t…

Rank zero elliptic curves induced by rational Diophantine triples

Rational Diophantine triples, i.e. rationals a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first su…

On iterations of rational functions over perfect fields

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…

Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves

We show that for k a perfect field of characteristic p, there exist endomorphisms of the completed algebraic closure of k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p a prime and C_p a completed algebraic closure of Q_p, there exist closed points of the Fargues-Fontaine curve associated to C_p whose residue fields are not (even abstractly) isomorphic to C_p as topological fields.

High rank elliptic curves induced by rational Diophantine triples

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank >= 7, which are both the current records for that k…

A note on rational and elliptic curves associated with the cuboid factor equations

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four equations with respect to six variables. The cuboid factor equations were derived from these four equations by symmetrization procedure. They constitute a system of eight polynomial equations, which has been solved parametrically. In the present paper its parametric solution is expressed through intersections or rational and elliptic curves arranged into parametric families.

关于丢番图三元组对应的椭圆曲线上整数点的杜杰拉（Dujella）猜想的反例

Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple

本文针对丢番图三元组对应椭圆曲线上整数点的杜杰拉猜想，构造无穷多组反例，证明存在x≠-1的整数点使得三个因子均非平方数，否定了该猜想的弱化问题。

Elliptic curves with complex multiplication and $Λ$-structures

This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda structures and that the structure morphism is a Lambda morphism. This implies that elliptic curves with complex multiplication can be canonically lifted to the Witt vectors of the base (these are big and global Witt vectors). We also show that elliptic curves with complex multiplication of Shimura typ…

动力序列：闭包性质和自动恒等式证明

Dynamical sequences: closure properties and automatic identity proving

本文研究了动力序列的闭包性质，展示了其涵盖多种经典序列，并提出算法证明动力序列恒等式。

An octic diophantine equation and related families of elliptic curves

We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.