A Shafarevich-Faltings Theorem For Rational Functions

On the vanishing of almost all primary components of the Shafarevich-Tate group of elliptic curves over the rationals

The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve $E$ over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question. We present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of Galois cohomology. Known results on the finiteness of the Shafarevich-Tate group are mentioned, including results of Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin. We then prove the vanishing…

Examples of non-simple abelian surfaces over the rationals with non-square order Tate-Shafarevich group

Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate-Shafarevich group of A must, if finite, be a square or twice a square. The situation for A not principally polarised remains unclear. For each k in {1,2,3,5,6,7,10,13} we construct a non-simple non-principally polarised abelian surface over the rationals whose Tate-Shafarevich group has order k times a square. To obtain this result, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture.

Faltings heights of big CM cycles and derivatives of L-functions

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Mass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtain…

扭曲的 $\mathbb Q$-同源类中的 Faltings 椭圆曲线

Faltings elliptic curves in twisted $\mathbb Q$-isogeny classes

本文研究二次扭曲后 $\mathbb Q$-同源类图中的最小 Faltings 高度曲线，确定其对应顶点，并给出顶点被选中的概率及其与模函数 $p$-进赋值的关系。

A dynamical Shafarevich theorem for twists of rational morphisms

Let $K$ denote a number field and a finite set $S$ of places of $K$ and $ϕ:\PP^n\rightarrow\PP^n$ be rational morphism defined over $K$. The main result of this paper proves that there are only finitely many twists of $ϕ$ defined over $K$ which have good reduction at all places outside $S$. This answers a question of Silverman in the affirmative.

阿贝尔簇中超曲面的Shafarevich猜想

The Shafarevich conjecture for hypersurfaces in abelian varieties

本文证明：在维数至少为4的固定阿贝尔簇中，表示给定丰富类且在固定素数集外具有良好约化的光滑超曲面，在平移意义下仅有有限多个，从而验证了该情形的Shafarevich猜想。

Finiteness of the Tate-Shafarevich Groups for Elliptic Curves over the Field of Rational Numbers

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\underline{III}(E)$ be a certain group of equivalence classes of homogeneous spaces of $E$ called its Tate-Shafarevich group. We show in this paper that this group has finite cardinality and discuss its role in the Birch and Swinnerton-Dyer Conjecture. In particular, our result implies the Parity Conjecture, or the Birch and Swinnerton-Dyer Conjecture modulo 2. It also removes the finiteness condition of $\underline{III}(E)$ from previous results in the literature of this subject and makes possible for some computation problems concerning th…

Logarithmic good reduction of abelian varieties

Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the existence of a projective, regular model of $A$, generalizing a result of Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.

Elliptic curves with large Tate-Shafarevich groups over $\mathbb{F}_q(t)$

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite sequences of explicit elliptic curves $E$, for which we prove that their Tate-Shafarevich group $\mathrm{III}(E)$ is finite and satisfies $|\mathrm{III}(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes the exponential differential height of $E$. The elliptic curves in these sequences are pairwise neither isogenous nor geometrically isomorphic. We furthe…

更高的Fitting理想与反圆环谢夫列夫-塔特群的结构

Higher Fitting ideals and the structure of anticyclotomic Shafarevich-Tate groups

本文研究了反圆环Z_p扩展上椭圆曲线塞尔默和谢夫列夫-塔特群的结构，通过双分欧拉系统描述了更高Fitting理想，并提供了相关的新结果。

Rational points and ramified covers of products of two elliptic curves

Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified covers of abelian varieties, and Faltings's finiteness theorem for rational points on higher genus curves.

Effective Chabauty for symmetric powers of curves

Faltings' theorem states that curves of genus $g \geq 2$ have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than $g$, can be used to give a good effective bound on the number of rational points of curves of genus $g > 1$. We draw ideas from nonarchimedean geometry to show that we can also give an effect…

On Tate-Shafarevich groups of some elliptic curves

This is an updated version of ANT-0166. Generalizing results of Stroeker and Top we show that the 2-ranks of the Tate-Shafarevich groups of the elliptic curves $y^2 = (x+k)(x^2+k^2)$ can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves.

Large Shafarevich-Tate groups over quadratic number fields

Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for every $r>0$, there are infinitely many quadratic twists $E^d/\mathbf{Q}$ of $E/\mathbf{Q}$ such that $\mathrm{dim}_{\mathbf{F}_2}(\mathrm{ST}(E^d/K)[2]) > r$

Structure of Tate-Shafarevich groups of elliptic curves over global function fields

The structure of the Tate-Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups from the monograph [1] and hence they are direct sums of finite cyclic groups where the orders of these cyclic components are invariants of the Tate-Shafarevich group. This decomposition of the Tate-Shafarevich groups into direct sums of finite cyclic groups depends on the behaviour of Drinfeld-Heegner points on these elliptic curves. These are points analogous to Heegner points on elliptic curves over the rational numbers.

算术曲面的Golod--Shafarevich塔

Golod--Shafarevich for arithmetic surfaces

该研究构造了整数环上算术曲面的Golod--Shafarevich塔，得到带截面且几何étale基本群无限的曲面，并取一般纤维获得有理数域上曲线，其有理点完全分裂且亏格趋于无穷，对数导子线性增长，回答了多个开放问题。

Density of rational points on isotrivial rational elliptic surfaces

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We also prove that these surfaces satisfy a variant of weak-weak approximation. Our results are conditional on the finiteness of Tate-Shafarevich groups for elliptic curves over the field of rational numbers.

Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups

We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|\sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $\sza$. For instance, $410536^2$ is the true order of $\sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.

双椭圆曲面的约化

Reduction of bielliptic surfaces

本文研究双椭圆曲面，证明具有有理点时 Shafarevich 猜想成立，无有理点时一般不成立，并建立剩余特征非 2 或 3 时 Néron 模型的潜在存在性。

Local-global questions for tori over $p$-adic function fields

We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. As an application, we show that the failure of the local-global principle for rational points on …

Partial zeta functions of algebraic varieties over finite fields

By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well known rationality theorem. In general, the partial zeta function is probably not rational. But a theorem of Faltings says that the partial zeta function is always nearly rational.

Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich groups for symmetric powers of elliptic curves

For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\mathrm{Cl}_K$ of $K$ as a $\mathrm{Gal}(K/\mathbb{Q})$-module. More precisely, for any $j$ with $1\leqslant j \leqslant p-2$, we give a condition that $\mathrm{Cl}_K\otimes \mathbb{F}_p$ has the symmetric power $\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\mathrm{Gal}(K/\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\mathrm{Sym}^j V_p E$. Here $V_p E$ denotes the rational $p$-adic Tate module of …

关于Tate-Shafarevich群的算术：基于Kolyvagin猜想

On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture

本文通过Heegner点研究半稳定椭圆曲线Tate-Shafarevich群p-primary部分的有限性，给出Kolyvagin猜想的简洁证明，并部分恢复Çiperiani--Wiles定理，方法独立于分圆Iwasawa主猜想。

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F is a global function field of characteristic p>3, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a…

具有完整 $\sqrt{3}$-层结构的亏格 2 曲线与 Tate-Shafarevich 群

Genus two curves with full $\sqrt{3}$-level structure and Tate-Shafarevich groups

本文显式参数化具有完整 $\sqrt{3}$-层结构的亏格 2 曲线参数曲面，并据此构造 Abel 曲面，证明其几乎所有二次扭曲的 3-Selmer/Tate-Shafarevich 非平凡，同时给出平均秩与有理点数统计。

Rational points on elliptic curves

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that for a fixed power greater than 1, there are only finitely many rational points. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points, these points are bounded in number in an explicit fashion, and that they are effectively computable.

Second p descents on elliptic curves

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate group such that pD = C and obtains explicit models for these D as curves in projective space. This leads to a practical algorithm for performing 9-descents on elliptic curves over the rationals.

Diophantine problems and $p$-adic period mappings

We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of $p$-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of $p$-adic Hodge theory, and explicit topological computations of monodromy. By the same methods w…

具有大Tate-Shafarevich群的$\mathbb{F}_{q}(t)$上的阿贝尔曲面

Abelian surfaces over $\mathbb{F}_{q}(t)$ with large Tate-Shafarevich groups

本文构造了有理函数域上的阿贝尔曲面序列，其Tate-Shafarevich群有限且大，通过证明BSD猜想并分析Néron模型与L-函数，利用特征和的角分布估计中心值，得到群阶的渐近公式。

Bivariate polynomial injections and elliptic curves

For every number field $k$, we construct an affine algebraic surface $X$ over $k$ with a Zariski dense set of $k$-rational points, and a regular function $f$ on $X$ inducing an injective map $X(k)\to k$ on $k$-rational points. In fact, given any elliptic curve $E$ of positive rank over $k$, we can take $X=V\times V$ with $V$ a suitable affine open set of $E$. The method of proof combines value distribution theory for complex holomorphic maps with results of Faltings on rational points in sub-varieties of abelian varieties.

对角三次曲面上的有理点

Rational Points on Diagonal Cubic Surfaces

本文在假设椭圆曲线的Tate-Shafarevich群有限的前提下，证明了特定对角三次曲面存在有理点。

有理数域上椭圆曲线所附带的adelic伽罗瓦表示的有效界

Effective bounds for adelic Galois representations attached to elliptic curves over the rationals

本文研究了无复乘的有理数域上椭圆曲线所附带的adelic伽罗瓦表示的像指数，给出了一个明确的上界，并改进了Zywina和Lombardo之前的成果。

An Errata for: Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

In [2], the author claims that the fields $\mathbb{Q}(D_4^\infty)$ defined in the paper and the compositum of all $D_4$ extensions of $\mathbb{Q}$ coincide. The proof of this claim depends on a misreading of a celebrated result by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification of torsion structures of $E$ defined over $\mathbb{Q}$ when base changed to the compositum of all $D_4$ extensions of $\mathbb{Q}$ main results of [2]. All the main results in [2] are still correct except that we are no longer able to prove that these two fields are equal.

On a conjecture of Gross and Zagier

Gross and Zagier conjectured that if the analytic rank of a rational elliptic curve is 1, then the order of the rational torsion subgroup of the elliptic curve divides the product of Tamagawa number, Manin constant, and the square root of the order of Tate--Shafarevich group over an imaginary quadratic field. In this paper, we show that this conjecture is true.

Elliptic curves with long arithmetic progressions have large rank

For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a non-trivial arithmetic progression implies that the Mordell-Weil rank is large, and similarly for $y$-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond's quantitative extension of results of Faltings.

$\mathcal{D}$-elliptic sheaves and odd Jacobians

We examine the existence of rational divisors on modular curves of $\mathcal{D}$-elliptic sheaves and on Atkin-Lehner quotients of these curves over local fields. Using a criterion of Poonen and Stoll, we show that in infinitely many cases the Tate-Shafarevich groups of the Jacobians of these Atkin-Lehner quotients have non-square orders.

阿贝尔簇上有理挠点与小点位的一致性

Uniformity in rational torsion and small points on abelian varieties

本文提出一种源于沃伊塔证明思路的方法，针对整体域上的阿贝尔簇，明确证明了一致有界性猜想与朗-西尔弗曼猜想，还给出特征p>0函数域上椭圆曲线的相关明确界限，并在提出的弱于Hindry版本的高维斯皮罗猜想下对数域证明了这两个猜想。

On quadratic rational maps with prescribed good reduction

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…

On The Birch and Swinnerton-Dyer Conjecture for CM Elliptic Curves over $\BQ$

For CM elliptic curve over rational field with analytic rank one, for any potential good ordinary prime p, not dividing the number of roots of unity in the complex multiplication field, we show the p-part of its Shafarevich-Tate group has order predicted by the Birch and Swinnerton-Dyer conjecture.

Critically separable rational maps in families

Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich's theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro's co…

通过轨道树形表示的动态Néron-Ogg-Shafarevich准则

A Dynamical Néron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations

本文通过轨道树形表示研究了非阿基米德局部域上理性映射的严格良好降解准则，揭示了残余度下降与纤维判别式在控制残余纤维的微分性中的作用，并通过具体例子验证了该准则。

A heuristic for boundedness of ranks of elliptic curves

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.

On the soft $p$-converse to a theorem of Gross-Zagier and Kolyvagin

We give a proof of a soft version of the $p$-converse to a theorem of Gross--Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at $p >3$ under the irreducibility assumption on the residual representation. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain that the Mordell-Weil rank is one and the Tate-Shafarevich group is finite if and only if the analytic rank is one for every elliptic curve over the rationals.

Central values of $L$-functions of cubic twists

We are interested in finding for which positive integers $D$ we have rational solutions for the equation $x^3+y^3=D.$ The aim of this paper is to compute the value of the $L$-function $L(E_D, 1)$ for the elliptic curves $E_D: x^3+y^3=D$. For the case of $p$ prime $p\equiv 1\mod 9$, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate $L(E_D, 1)$ to the square of a trace of a modular function at a CM point. This offers a criterion for when the integer $D$ is the sum of two rational cubes. Furthermore, when $L(E_D, 1)$ is nonzero we get a formul…

具有有理双切线的平面四次曲线上的双覆盖下降

Two-cover descent on plane quartics with rational bitangents

本文针对 Q 上 28 条双切线全为有理的平面四次曲线实现双覆盖下降，结合模空间构造算例，判定有理点存在性并分析局部障碍、双-Selmer 秩与 Tate–Shafarevich 群。

Serre 开像定理的改进界

Improved bounds for Serre's open image theorem

本文通过分析偏差群的商并利用 Rouse 和 Zureick-Brown 对 2-adic 表示像的刻画，在广义黎曼假设下进一步改进了 Mayle-Wang 关于 Serre 开像定理中常数 $C_E$ 的显式界，并得到改进的有效同源定理。

椭圆曲线生成的挠域的理想类群统计

Class group statistics for torsion fields generated by elliptic curves

研究椭圆曲线生成的挠域的理想类群模 p 的 Galois 稳定商的平均非消失性，结果依赖于 Delaunay 和 Poonen-Rains 的预测。

关于一族椭圆曲线的扭和其 $ L $-函数

On Twists of A Family of Elliptic Curves and Their $ L-$Function

本文研究了一族特定椭圆曲线的二次扭及其 $ L $-函数的算术性质，通过经典工作和条件限制，确定了规范指数、根数等关键量，并探讨了 Mordell-Weil 群与 Selmer 群之间的关系，为验证 BSD 猜想提供证据。