Milnor $K$-group attached to a torus and Birch-Tate conjecture

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 …

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.

Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one

Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that u…

A Weil-etale version of the Birch and Swinnerton-Dyer formula over function fields

We give a reformulation of the Birch and Swinnerton-Dyer conjecture over global function fields in terms of Weil-etale cohomology of the curve with coefficients in the Neron model, and show that it holds under the assumption of finiteness of the Tate-Shafarevich group.

A divisibility related to the Birch and Swinnerton-Dyer conjecture

Let $E/\mathbb{Q}$ be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of $E/\mathbb{Q}$ divides the product of the order of the Shafarevich--Tate group of $E/\mathbb{Q}$, the (global) Tamagawa number of $E/\mathbb{Q}$, and the Tamagawa number of $E/\mathbb{Q}$ at infinity. This consequence of the Birch and Swinnerton-Dyer conjecture was noticed by Agashe and Stein in 2005. In this paper, we prove this divisibility statement unconditionally in many cases, in…

Second Isogeny Descents and the Birch and Swinnerton-Dyer Conjectural Formula

Let h be a p-isogeny of elliptic curves. We describe how to perform h-descents on the nontrivial elements in the Shafarevich-Tate group which are killed by the dual isogeny h'. This makes computation of p-Selmer groups of elliptic curves admitting a p-isogeny over Q feasible for p = 5,7 in cases where an isogeny descent is insufficient and a full p-descent would be infeasible. As an application we complete the verification of the full Birch and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank zero or one and conductor less than 5000.

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.

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…

Explicit Sato-Tate type distribution for a family of $K3$ surfaces

In the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy with Birch's result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces $A_λ(p)$ of a certain family of $K3$ surfaces $X_λ$ with generic Picard rank $19$ is the $O(3)$ distribution. This distribution, which we denote by $\frac{1}{4π}f(t),$ is quite different from the semicircular distribution. It is su…

On a BSD-type formula for L-values of Artin twists of elliptic curves

This is an investigation into the possible existence and consequences of a Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate-Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular we exhibit settings where the different p-primary components of the Tate-Shafarevich group do not behave independently of one another. We also gi…

Birch and Swinnerton-Dyer conjecture in the complex multiplication case and the congruent number problem

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic curves with complex multiplications. The key method of the proof is to reduce the Galois action of infinite order on the Tate module of an elliptic curve to that of finite order by using the $p$-adic Hodge theory. As a corollary, we can determine whether a given natural number is a congruent number (congruent number problem). This problem is one of the olde…

关于正特征情形下 Birch 和 Swinnerton-Dyer 猜想的一个精细化

On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic

本文提出正特征函数域上阿贝尔簇的 Birch 和 Swinnerton-Dyer 猜想精细化版本，融合 $L$-级数首项同余与 Selmer 复形 Galois 结构限制，并在假定 Tate-Shafarevich 群有限的条件下对一类重要例子给出完整证明。

A classical family of elliptic curves having rank one and the $2$-primary part of their Tate-Shafarevich group non-trivial

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.

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…

关于满足Birch-Swinnerton-Dyer猜想的椭圆曲线识别

On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

本文基于Burungale-Skinner-Tian-Wan的研究，提出一个显式算法，应用于LMFDB数据库，识别出所有导数不超过500,000的椭圆曲线，这些曲线有无限多个二次扭曲满足强Birch-Swinnerton-Dyer猜想，并提供了Radziwiłł和Soundararajan猜想的数值证据。

On pairs of p-adic analogues of the conjectures of Birch and Swinnerton-Dyer

For a weight two modular form and a good prime $p$, we construct a vector of Iwasawa functions $(L_p^\sharp,L_p^\flat)$. In the elliptic curve case, we use this vector to put the $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer for ordinary [MTT] and supersingular [BPR] primes on one footing. Looking at $L_p^\sharp$ and $L_p^\flat$ individually leads to a stronger conjecture containing an extra zero phenomenon. We also give an explicit upper bound for the analytic rank in the cyclotomic direction and an asymptotic formula for the $p$-part of the analytic size of the Šafarevič…

Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero

Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank zero, i.e., such that the $L$-function $L_E(s)$ of $E$ does not vanish at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the product of the order of the Shafarevich-Tate group of $E$ and the orders of the arithmetic component groups of $E…

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.

On refined conjectures of Birch and Swinnerton-Dyer type for Hasse-Weil-Artin L-series

We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult archimedean considerations related to refinements of the conjecture of De…

On an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic $p$. We prove the prime-to-$p$ part conditionally on the finiteness of the $p$-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-$p$ part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-$p$ part for (1) relative elliptic curve…

A visible factor for analytic rank one

Let $E$ be an optimal elliptic curve of conductor $N$, such that the $L$-function of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ also vanishes to order one at $s=1$. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in $E(K)$ of the subgroup generated by the Heegner point is equal to the product of the Manin constant of $E$, the Tamagawa numbers of $E$, and the square root of the order of the Shafarevich-Tate…

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.…

Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-…

Conjectural estimates on the Mordell-Weil and Tate-Shafarevich groups of an abelian variety

We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Néron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators…

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…

Park City lectures on elliptic curves over function fields

These are the notes from a course of five lectures at the 2009 Park City Math Institute. The focus is on elliptic curves over function fields over finite fields. In the first three lectures, we explain the main classical results (mainly due to Tate) on the Birch and Swinnerton-Dyer conjecture in this context and its connection to the Tate conjecture about divisors on surfaces. This is preceded by a "Lecture 0" on background material. In the remaining two lectures, we discuss more recent developments on elliptic curves of large rank and constructions of explicit points in high rank situations.

Elliptic curves with large rank over function fields

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.

Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tama…

关于中心 $L$-值与 Tate-Shafarevich 群的 $3$-部分增长

On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group

本文研究椭圆曲线 $X^3+Y^3=λZ^3$ 中心 $L$-值代数部分的 $3$-进赋值下界，将其推广到 $3$ 整除 $λ$ 的情形，并结合 Tate-Shafarevich 群的 $3$-部分增长验证下界与 Birch 和 Swinnerton-Dyer 猜想一致。

椭圆曲线 Shafarevich-Tate 群的机器学习方法

Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves

研究通过前馈神经网络分类器和回归模型，基于BSD猜想不变量预测椭圆曲线Shafarevich-Tate群的阶，准确率超过0.9，并成功应用于秩为29的椭圆曲线。

The 2-parity conjecture for elliptic curves with isomorphic 2-torsion

The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version of this conjecture called the parity conjecture and prove the following. Let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness of the Shafarevich-Tate groups of $E_1$ and $E_2$, we show that the Birch and Swinnerton-Dyer conjecture correctly predicts the parity of the rank of $E_1\times E_2$. Using this result, we complet…

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。

阿贝尔曲面的奇偶猜想

Parity conjecture for abelian surfaces

在Tate--Shafarevich群有限假设下，证明Birch--Swinnerton-Dyer猜想正确预测半稳定主极化阿贝尔曲面秩的奇偶性，并对Jacobian情形附加2-adic好普通约化条件。

关于模阿贝尔曲面的 Birch–Swinnerton-Dyer 猜想

On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces

本文利用亏格2 Siegel模形式spin表示的Euler系统，证明模阿贝尔曲面在中心L值非零等条件下的BSD有限性结论及Iwasawa主猜想的部分包含关系。

A sheaf-theoretic reformulation of the Tate conjecture

Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l di…

非全纯变种上的一些Beilinson-Bloch猜想

The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields

本文证明了非全纯变种上Beilinson-Bloch猜想，并推导出中间Jacobians上的Birch-Swinnerton-Dyer猜想，同时推广了有限域上的Tate猜想。

Kato欧拉系统在模形式中的精细应用

Refined applications of Kato's Euler systems for modular forms

本文探讨了Kato欧拉系统在高权模形式中的精细应用，证明了Mazur-Tate猜想并给出岩泽主猜想的新解释，同时涉及对BSD猜想的应用。

孪生素数椭圆曲线 $y^2=x(x-2)(x-p)$ 的解析秩与代数秩

On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$

研究孪生素数椭圆曲线E_p的解析秩与代数秩，利用2-Selmer计算和Cassels-Tate配对，在低余秩BSD假设下证明二者相等、Tate-Shafarevich群有限并给出精确BSD公式。

无限多对具有相同BSD不变量的非同构椭圆曲线

Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants

研究具有相同\(L\)函数和BSD不变量相关算术数据的非同构椭圆曲线（BSD双胞胎），展示一族\(j\)不变量两两不同的BSD双胞胎，并证明即使施加素数处柯达依符号和平凡判别式相等条件，仍存在无限多BSD双胞胎。

Fermat椭圆曲线扭曲的中心$L$-值的Tamagawa数可除性

Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve

研究Fermat椭圆曲线扭曲的中心L值代数部分的3-adic赋值，揭示Tate-Shafarevich群3-部分与惰性素因子个数的关系，并证明在分裂素数乘积情形下BSD猜想预测的阶为完全平方。

曲线 Jacobian 簇的秩奇偶性

Parity of ranks of Jacobians of curves

本文研究带自同构群作用的曲线 Jacobian 簇 Selmer 群，利用局部不变量刻画 Mordell–Weil 秩奇偶性，并据此给出椭圆曲线奇偶性猜想的新证明。