A. Agboola
We study Rubin's variant of the $p$-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable $p$-adic $L$-function that lie outside the range of $p$-adic interpolation.
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We give new examples of weight three cusp forms on noncongruence subgroups of SL(2, Z) whose Scholl representation is modular and which satisfy three term Atkin-Swinnerton-Dyer relations.
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.
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 Birch and Swinnerton-Dyer's cubic surfaces
本文推广了 Birch 和 Swinnerton-Dyer 1975 年的工作，利用现代计算机代数软件证明了更大范围的三次曲面存在 Brauer--Manin 障碍以阻碍 Hasse 原理，从而对无限多个三次曲面验证了 Colliot-Thélène--Sansuc 猜想。
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…
We give a brief description of the Birch-Swinnerton-Dyer conjecture and present related conjectures. We describe the relation between the nilpotent orbits of SL(2,R) and CM points.
We prove that a majority (in fact, $>66\%$) of all elliptic curves over $\mathbb Q$, when ordered by height, satisfy the Birch and Swinnerton-Dyer rank conjecture.
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…
Let E be an elliptic curve over Q with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of $p$-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when $E(K)$ is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin's original conjecture no longer applies, and the relevant special value of the appropriate $p$-adic L-function is equ…
Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces
本文给出Cassels--Swinnerton-Dyer猜想在三次曲面情形（含光滑与奇异）的完整证明，通过两则独立证明补全缺失的4次点情形，并利用Ma的提升论证推广至任意域。
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…
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.
In this paper we exhibit a noncongruence subgroup $\G$ whose space of weight 3 cusp forms $S_3(\G)$ admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to $S_3(\G)$ by A. Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.
We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we have proved the formula for 16714 of the 16725 such curves of conductor less than 5000.
In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over $\mathbb{Q}$. We prove the existence of explicit infinite families of quadratic twists with analytic ranks $0$ and $1$ for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank $1$. In addition, we show that these families of quadratic twists satisfy the $2$-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of …
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…
Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense $p$-adic analogues of Hecke recursion satisfied by classic Hecke eigenforms. They actually appeared in different context and sometimes can be obtained using the theory of formal groups. In this survey paper, we introduce the Atkin and Swinnerton-Dyer congruences, and discuss some recent progress on this topic.
We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjectures. We do calculations in some cases and raise questions about their relations, if any, with the predictions of various average rank models that have been considered.
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…
Given an elliptic curve E over Q with complex multiplication having good reduction at 2, we investigate the 2-adic valuation of the algebraic part of the L-value at 1 for a family of quadratic twists. In particular, we prove a lower bound for this valuation in terms of the Tamagawa number in a form predicted by the conjecture of Birch and Swinnerton-Dyer.
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。
In this paper, we imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematica (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of $4, 5, 6, ..., 2n+2$ variables of degree $2n+1$ for infinitely many integers $n$. It is contrastive to Swinnerton-Dyer's original construction that we do not need any concrete calculation in the proof of local solubility.
Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of Jetchev-Skinner-Wan, we use Iwasawa theory to establish the $p$-part of the Birch and Swinnerton-Dyer formula for $E$. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction.
In this paper we show the Birch and Swinnerton-Dyer conjecture for a certain elliptic curve over $\mathbb{Q}(\sqrt[4]{5})$ is equivalent to the same conjecture for a certain pair of hyperelliptic curves of genus 2 over $\mathbb{Q}$. We numerically verify the conjecture for these hyperelliptic curves. Moreover, we explain the methods used to find this example, which turned out to be a bit more subtle than expected.
In the present paper, we prove, for a large class of elliptic curves defined over $\mathbb{Q}$, the existence of an explicit infinite family of quadratic twists with analytic rank $0$. In addition, we establish the $2$-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch--Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over $\mathbb{Q}$ without complex multiplication.
We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves $E/\mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove under mild hypotheses one of the inequalities predicted by the rank part of our conjectures, as well as the predicted leading coefficient formula up to a $p$-adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in 1996 for certain Heegner distributions, and as application …
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…
Based upon new global class field concepts leading to Langlands two-dimensional global correspondences,a modular representation of cusp forms is proposed in terms of global elliptic (bisemi)modules which are (truncated) Fourier series over R.As application,the conjectures of Shimura-Taniyama-Weil,Birch-Swinnerton-Dyer and Riemann are analyzed.
Introduction to the Birch and Swinnerton-Dyer Conjecture
本文简要介绍Birch和Swinnerton-Dyer猜想，通过回顾丢番图方程、定义椭圆曲线的代数秩与解析秩，提出二者相等的核心猜想。
We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence'' of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.
Let E be an elliptic curved defined over $\Q$ and let $K/\Q$ be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for $h^1(E\times_{\Q} K)(1)$ viewed as a motive over $\Q$ with coefficients in $\Q[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of $\Q$.
We give a brief description of the Birch and Swinnerton-Dyer Conjecture which is one of the seven Clay problems.
Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{Ω_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.
The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non-reduced components in the special fibre of their Néron model. In this present paper we explain how the real period can be computed, and how the verification has been extended to many more hyperelliptic curves, some of genus 3, 4 and 5, without using modular methods.
In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long, Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied special families of cuspforms for noncongruence arithmetic subgroups. It was found that the Fourier coefficients of these modular forms at infinity satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the $p$-adic analogue of the three-term recursions satisfied by the coefficients of classical Hecke eigenforms. In this paper, we first consider Atkin and Swinnerton-Dyer type congruences which generalize the three-ter…
We study infinite families of quadratic and cubic twists of the elliptic curve $E = X_0(27)$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the value of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.
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.
For every odd prime $p$, we exhibit families of irreducible Artin representations $τ$ with the property that for every elliptic curve $E$ the order of the zero of the twisted $L$-function $L(E,τ,s)$ at $s\!=\!1$ must be a multiple~of~$p$. Analogously, the multiplicity of $τ$ in the Selmer group of $E$ must also be divisible by $p$. We give further examples where $τ$ can moreover be twisted by any character that factors through the $p$-cyclotomic extension, and examples where the $L$-functions are those of twists of certain Hilbert modular forms by Dirichlet charaters. These results are conject…
We give a survey on the relationship between non-commutative Iwasawa theory and the Equivariant Tamagawa Number Conjecture following the work of (in alphabetical order) Burns/Flach, Coates/Sujatha, Fukaya/Kato, Huber/Kings and others.
Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be …
Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant $Δ_E$. For primes $p$ of good reduction, let $N_p$ be the number of points modulo $p$ and write $N_p=p+1-a_p$. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies $$\lim_{x\to\infty}\frac{1}{\log x}\sum_{\substack{p\leq x\\ p\nmid Δ_{E}}}\frac{a_p\log p}{p}=-r+\frac{1}{2},$$ where $r$ is the order of the zero of the $L$-function $L_{E}(s)$ of $E$ at $s=1$, which is predicted to be the Mordell-Weil rank of $E(\mathbb{Q})$. We show that if the above limit exits, then the limit equals $-r+1/2$. We also relate th…
The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture
本研究确定了有理数域上椭圆曲线二次扭族的2-Selmer秩分布，证明其二次扭的2^∞-Selmer上秩0和1各占50%，进而推出Birch和Swinnerton-Dyer猜想蕴含Goldfeld猜想，部分情形的分布与Poonen-Rains模型不同。
The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication
本文在 L(E/F,1) 非零且扭转扩张为阿贝尔的假设下，完整证明了具有复乘的椭圆曲线及 CM 阿贝尔簇的 Birch-Swinnerton-Dyer 猜想及其 Gross 等变细化。
In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the $p$-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic $L$-function over $\mathbb Q$ whose local factors agree with those of the $l$-adic Scholl representations …
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。
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) and L^\flat(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functi…
Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions?
本文探讨数域K/Q的Birch和Swinnerton-Dyer猜想，假设V_K具有上同调性质并验证Serre关于ζ_K(s)在s=1/2处消失阶的猜想，证明存在自然的 functor K→V_K 并且它们同构，其共同自同构群是2- torsion 和 Abel 的。
Let $E$ be an optimal elliptic curve defined over $\mathbb{Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $E$. We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to $E$ and describe two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical…
Atkin and Swinnerton-Dyer congruences for meromorphic modular forms
本文将Atkin和Swinnerton-Dyer同余推广到亚纯模形式，证明p-进递推关系包含Scholl同余与万有椭圆曲线极点纤维的贡献，并在CM极点情形约化为二项关系，部分证明磁性猜想。
A visual perspective on the Birch and Swinnerton-Dyer conjecture through a family of approximations of $L$-functions
本文通过一族 $L$-函数近似，给出了椭圆曲线解析秩的视觉解释，并精确刻画了近似误差，为 Birch 和 Swinnerton-Dyer 猜想提供了新的视角。
Derived Bockstein regulators and anticyclotomic $p$-adic Birch and Swinnerton-Dyer conjectures
本文借助 Nekovář 的思想定义导出 Bockstein 调节子并建立下降形式体系，据此由相关主猜想推出两类 p-adic BSD 猜想，并将一般 motive 的 Euler 系统导数理论推广至导出情形。
Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 a…
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。
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…
The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves
本文研究了二次 twists 椭圆曲线的 Birch--Swinnerton-Dyer 精确公式，推导了 $2$-adic 估值下界，并证明了存在显式无限族的二次 twists 具有解析秩 0。
The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre showed the first zero occurs by O(1/loglog(N_E)), where N_E is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller …
A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields
本文在Iwasawa主猜想及其他假设下，证明了全实域的全虚二次扩域上，模半稳定椭圆曲线$E$的Birch和Swinnerton-Dyer公式的$p$部分变体，给出了相关$p$进单位范围内的等式关系。
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.…
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…
Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $α$ and $m\in\{1,2,3\}$, we have \begin{align*} \sum_{k=0}^{p^αn-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\equiv\bigg(\frac{m(m-4)}{p}\bigg)\sum_{k=0}^{p^{α-1}n-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\pmod{p^{2α}}, \end{align*} where $(x)_k=x(x+1)\cdots(x+k-1)$ and $\big(\frac{\cdot}{\cdot}\big)$ denotes the Legendre symbol. Also, when $m=4$, \begin{align*} \sum_{k=0}^{p^αn-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\equiv p\sum_{k=0}^{p^{α-1}n-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\pmod{p^{2α}}. \end{align*…
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