An elementary proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves

关于Mazur和Tate的精细化“Birch--Swinnerton-Dyer型”猜想

On the refined `Birch--Swinnerton-Dyer type' conjectures of Mazur and Tate

该研究证明了Mazur和Tate对Birch-Swinnerton-Dyer猜想的精细化猜想的大部分内容，通过等变Tamagawa数猜想的零秩分量方法，还得到了比原预测更精细的结果。

具有超奇异约化的椭圆曲线的反分圆Mazur-Tate猜想研究

On the anticyclotomic Mazur-Tate conjecture for elliptic curves with supersingular reduction

该短注针对具有超奇异约化的椭圆曲线，研究其Selmer群Fitting理想相关的Mazur-Tate“弱”主猜想的反分圆类似问题。

Mazur-Tate elements of non-ordinary modular forms

We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case.

Congruences between Ramanujan's tau function and elliptic curves, and Mazur--Tate elements at additive primes

We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.

加法素数处椭圆曲线的Mazur–Tate元的岩泽不变量

On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes

该研究探讨有理数域上椭圆曲线在加法约化素数处Mazur–Tate元的λ-不变量，结合Sage计算数据给出加法潜在超奇异情形的实例与猜想，并将结果推广到相关尖点Hecke特征形式的对应不变量。

关于惰性素数处反圆塔的Mazur--Tate精细猜想

On the Mazur--Tate refined conjecture for the anticyclotomic tower at inert primes

本文针对$p\geq5$为惰性素数且椭圆曲线在该处超奇异约化的情形，利用Burungale--Büyükboduk--Lei的结果，证明了虚二次域反圆塔上的弱Mazur--Tate精细猜想。

具有大于2的塞尔权重的非平凡模形式的马祖尔 - 泰特元素

Mazur-Tate elements of non-ordinary modular forms with Serre weight larger than two

研究固定奇素数$p$下，权重$k$且层数与$p$互素的非平凡本征尖形式$f$（$p>k - 1$），通过计算渐近公式及结合定理，得到塞尔权重相关的马祖尔 - 泰特元素的$\lambda$-不变量描述，推广了相关结果。

与拉马努金τ函数相关的Mazur–Tate元的Lambda不变量以及与艾森斯坦级数的同余关系

Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series

该研究针对权12模形式$\Delta$，证明其与特定艾森斯坦级数模$p$（$p\in\{3,5,7\}$）同余，并推导了相关Mazur–Tate元的岩泽不变量公式，验证了此前的数值数据。

On analogues of Mazur-Tate type conjectures in the Rankin-Selberg setting

We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic cu…

An archimedian analog of Iwasawa theory

We will show a conjecture which reduces Mazur-Tate-Teitelbaum conjecture to the known cases. In order to explain its background we will develop an archimedian analog of Iwasawa theory. Moreover consequences of the conjecture which are related to Birch and Swinnerton-Dyer conjecture will be discussed.

关于Mazur-Tate元素的λ-不变量最大性的研究

On the maximality of the $λ$-invariants of Mazur--Tate elements

本文研究了Mazur-Tate元素的λ-不变量最大性，证明其在满足Greenberg猜想的情况下要么稳定到p-进L-函数的不变量，要么在某些条件下达到最大值，并探讨了与Eisenstein级数边界符号相关的同源关系。

关于马祖尔 - 泰特元素的木田公式的一个类似物

An analogue of Kida's formula for Mazur-Tate elements

研究\(\mathbb{Q}\)上椭圆曲线马祖尔 - 泰特元素的岩泽不变量，证明其\(\mu\) - 不变量消失在扩张中保持，\(\lambda\) - 不变量有过渡公式，统一了相关解析岩泽不变量的木田公式，还得到特定\(p\) - 进\(L\) - 函数的木田型公式。

Trivial zeros of p-adic L-functions at near central points

Using the $\scr L$-invariant constructed in our previous paper we prove a Mazur-Tate-Teitelbaum style formula for derivatives of p-adic L-functions of elliptic modular forms at near central points. In the second version of the paper the case of potentially crystalline reduction is also covered.

Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of …

Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian Variety over a Number Field

The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non-trivial elements of the Shafarevich-Tate group of an elliptic curve by means of the Mordell-Weil group of an ambient curve. In this paper, we generalize Cremona and Mazur's work and give precise conditions under which such a construction can be made for the Shafarevich-Tate gro…

模形式的显式互反律与岩泽理论

Explicit reciprocity laws and Iwasawa theory for modular forms

本研究证明模形式本征形式的Mazur-Tate元素位于分圆Zp扩张对偶塞尔梅克群的菲廷理想中，通过p进局部朗兰兹对应构造局部上同调类与代数类似物，结合Kato欧拉系统和显式互反律完成证明。

A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the …

Even Galois Representations and the Fontaine--Mazur conjecture II

We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of $\Gal(\Qbar/\Q)$ which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous work of the author. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric (= de Rham) deformations.

椭圆曲线的Tate-Shafarevich群中的可见2-挠元

Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve

本文证明椭圆曲线的Tate-Shafarevich群中任意两个2-挠元可在同一阿贝尔曲面中可见，推广了先前仅知单个2-挠元的结果，并解释了Cremona和Mazur论文中的一些原始观察。

Even Galois Representations and the Fontaine-Mazur Conjecture

We prove some cases of the Fontaine-Mazur conjecture for even Galois representations. In particular, we prove, under mild hypotheses, that there are no irreducible two-dimensional ordinary even Galois representations of $\Gal(\Qbar/\Q)$ with distinct Hodge-Tate weights. If $K/\Q$ is an imaginary quadratic field, we also prove (again, under certain hypotheses) that $\Gal(\Qbar/K)$ does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight. Finally, we prove that any weakly compatible family of two dimensional irreducible Galois representations of $\Gal(\Qb…

关于Shafarevich-Tate群的可见性范畴

On the Visibility category of the Shafarevich--Tate group

本文研究了椭圆曲线的Shafarevich-Tate群中非平凡元素的可见性范畴，探讨了最小对象的性质，并通过两种构造方法分析了最小可视化阿贝尔簇的生成。

A remark on potentially semi-stable representations of Hodge-Tate type (0,1)

In this note we complement a part of a theorem of Fontaine-Mazur. We show that if $(V,ρ)$ is an irreducible finite dimensional representation of the Galois group $Gal({\bar K}/K)$ of a finite extension of $K\Q_p$, of Hodge-Tate type $(0,1)$ then it is potentially semi-stable if and only if it is potentially crystalline. This was proved by Fontaine-Mazur for dimension two and $p\geq 5$ by their classfication theorem.

Recognizing Galois representations of K3 surfaces

Under the assumption of the Hodge, Tate and Fontaine-Mazur conjectures we give a criterion for a compatible system of l-adic representations to be isomorphic to the second cohomology of a K3 surface.

Irreducibility of geometric Galois representations and the Tate conjecture for a family of elliptic surfaces

Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.

BSD型精化猜想的数值研究

Numerical study of refined conjectures of the BSD type

本文通过SageMath数值实验检验Mazur和Tate提出的三个BSD型猜想，发现其原始陈述存在偏差，但经轻微修正后猜想成立。

有理数域上具有秩1和非平凡2-部分Tate-Shafarevich群的椭圆曲线的$\mathbb{Z}_2$扩张

Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the $\mathbb{Z}_2$-extension of $\mathbb{Q}$

本文研究了二次扭曲椭圆曲线在$\mathbb{Z}_2$扩张上的秩和Tate-Shafarevich群性质，通过Heegner点同余性质和Coates-Wiles定理的等变版本构造了具有秩1和无限Tate-Shafarevich群的椭圆曲线。

Modularity of some potentially Barsotti-Tate Galois representations

We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the Fontaine-Mazur conjectures, specifically, the modularity of certain potentially Barsotti-Tate Galois representations. The proof follows the template of Wiles, Taylor-Wiles, and Breuil-Conrad-Diamond-Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the integers of a local field. This makes crucial use of the filtered ϕ_1-modules of C. Breuil.

Visibility of ideal classes

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich-Tate groups of elliptic curves. The analogue for an abelian number field $K$ is capitulation of ideal classes of $K$ in the minimal cyclotomic field containing $K$. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of nontrivial Shafarevich-Tate elements might be much more common for elliptic curves of positive rank than f…

Experimental Evidence on a Refined Conjecture of the BSD type

Let $E/\mathbb{Q}$ be an elliptic curve of level $N$ and rank equal to $1$. Let $p$ be a prime of ordinary reduction. We experimentally study conjecture $4$ of B. Mazur and J. Tate in his article "Refined Conjectures of the Birch and Swinnerton-Dyer Type". We report the computational evidence.

On the universal p-adic sigma and Weierstrass zeta functions

For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.

Points of order 13 on elliptic curves

We pick up the study of 13-torsion in elliptic curves where Mazur and Tate left off 45 years ago. We consider various questions concerning elliptic curves defined over the maximal totally real subfield of the 13th cyclotomic field (where J_1(13) acquires everywhere good reduction), and over quadratic extensions.

$p$-adic $L$-函数的二阶导数与秩二CM椭圆曲线的Shafarevich--Tate群

Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves

本文为秩二CM椭圆曲线证明了分圆型判据：在显式排除的素数外，p-adic L函数二阶Taylor系数为单位当且仅当分圆调节子为单位且Sha消失，并解决了Coates等人遗留的p=577情形。

Equivariant Iwasawa theory for elliptic curves

We discuss abelian equivariant Iwasawa theory for elliptic curves over $\mathbb{Q}$ at good supersingular primes and non-anomalous good ordinary primes. Using Kobayashi's method, we construct equivariant Coleman maps, which send the Beilinson-Kato element to the equivariant $p$-adic $L$-functions. Then we propose equivariant main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application, we prove a case of a conjecture of Mazur-Tate.

Visualizing elements of Sha[3] in genus 2 jacobians

Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get …

On derivatives of Kato's Euler system for elliptic curves

In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves $E$ over $\mathbb{Q}$. This conjecture, which we refer to as the `Generalized Perrin-Riou Conjecture', predicts a precise congruence relation between a `Darmon-type derivative' of the zeta element of $E$ over an arbitrary real abelian field and the critical value of an appropriate higher derivative of the $L$-function of $E$ over $\mathbb{Q}$. We prove that the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou's conjecture on the logarithm of Kato's …

On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence towards our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=E_K is the base-change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases.…

椭圆曲线的算术统计与丢番图稳定性

Arithmetic statistics and diophantine stability for elliptic curves

本文研究有理数域椭圆曲线在素数幂循环扩张中的Mordell-Weil群与Tate-Shafarevich群稳定性，提出“Sha”-稳定性概念，并利用算术统计与岩泽理论证明非CM椭圆曲线在几乎所有素数处稳定。

On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction

In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main conjecture" à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the "strong main conjecture" suggested by the second named author under the validity of the $\pm$-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "$\pm$-objects",…

Coleman-Gross height pairings and the $p$-adic sigma function

We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Colema…

任意同源映射的除法多项式

Division polynomials for arbitrary isogenies

本文扩展了除法多项式定义至任意椭圆曲线同源映射，包括核不和身份元相加的映射，推导了递推关系、经典椭圆函数关系、源曲率与目标曲率间的关系链规则及高维推广。

与 $y^2=x^3-23$ 的模 $23$ 同余都是平凡的

Congruences modulo $23$ to $y^2=x^3-23$ are trivial

本文运用 Mazur 策略、Jacobian 的 Tate 模及 BSD 新实例确定扭曲模曲线有理点，证明与 $y^2=x^3-23$ 的模 $23$ 同余均平凡。

模曲线的伽罗瓦扭曲的 L-函数与有理点

L-functions and rational points for Galois twists of modular curves

本文应用 Mazur 策略研究模曲线伽罗瓦扭曲 $Y_E(p)$ 的有理点，确定其 Jacobian 的 Tate 模与 $L$-函数因子的函数方程符号，并证明部分因子满足 Bloch-Kato 猜想及形式浸入定理。

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。