Graham Everest, Helen King
Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In certain circumstances, the methods show only finitely many terms have length below a given bound.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider a particular case of an analog for elliptic curves to the Mersenne problem : finding explicitely all prime power terms in an elliptic divisibility sequence when descent via isogeny is possible. We explain how this question can be related to classical problems in diophantine geometry and we compute an explicit upper bound on the index of prime power terms in magnified elliptic divisibility sequences.
We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.
It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.
Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences called Lucas sequences. There has been much interest in cases where the terms of Lucas sequences are squares or cubes. In this work, using the Tate normal form having one parameter of elliptic curves with torsion points, the general terms and periods of all elliptic divisibility sequences with a zero term are given in terms of this parameter by means of Mazur's theorem, and it is shown that which term of h_{n} of an EDS can be a square or a cube by using the general terms of these sequences.
We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of $p$. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with $p > 2d+1$. We also deduce that the local-global principle for divisibility by powers of $7$ holds over quadratic f…
Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited.
For a pair $(E,P)$ of an elliptic curve $E/\mathbb{Q}$ and a nontorsion point $P\in E(\mathbb{Q})$, the sequence of \emph{elliptic Fermat numbers} is defined by taking quotients of terms in the corresponding elliptic divisibility sequence $(D_{n})_{n\in\mathbb{N}}$ with index powers of two, i.e. $D_{1}$, $D_{2}/D_{1}$, $D_{4}/D_{2}$, etc. Elliptic Fermat numbers share many properties with the classical Fermat numbers, $F_{k}=2^{2^k}+1$. In the present paper, we show that for magnified elliptic Fermat sequences, only finitely many terms are prime. We also define \emph{generalized elliptic Ferma…
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。
For every prime power p^n with p = 2 or 3 and n > 1 we give an example of an elliptic curve over Q containing a rational point which is locally divisible by p^n but is not divisible by p^n. For these same prime powers we construct examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.
The class number divisibility problem for number fields is one of the classical problems in algebraic number theory, which originated from Gauss' class number conjectures. The relation between the points on an elliptic curve and class number divisibility of a number field has been explored through the works of various mathematicians. Here, we explicitly construct an unramified abelian extension of a bi-quadratic field generated from points of a certain type of elliptic curve. Moreover, showing the $2$-Selmer rank of the said elliptic curve as $1$, we also construct an infinite family of bi-qua…
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.
Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (ζ_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that the existence of a counterexample to the local-global divisibility by $p^2$ in $\mathcal{E}$, assures the existence of a $k$-rational point of exact order $p$ in $\mathcal{E}$. Using the Merel Theorem, we then shrunk the known set of primes for which there could be a counterexample to the local-global divisibility by $p^2$.
Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal} ( k ( {\mathcal{E}}[p] ) / k )$-module ${\mathcal{E}}[p]$ is isomorphic to the $G$-module $( \mathbb{Z} / p \mathbb{Z} )^2$ and there exists $n \in \mathbb{N}$ such that the local-global divisibility by $p^n$ does not hold over ${\mathcal{E}} ( k )$.
We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with $|a|\le A$ and $|b| \le B$, where $A$ and $B$ are small relative to $x$. Specifically, we investigate behavior with respect to the Sato--Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer $m$.
Let $\mathbf{D}=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of $\mathscr{A}_{E,k}$. Also, we explain when $\mathscr{A}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of $\mathscr{A}_{E,k}$ using the Möbius function.
Let $p$ be a prime number and let $k$ be a number field. Let $E$ be an elliptic curve defined over $k$. We prove that if $p$ is odd, then the local-global divisibility by any power of $p$ holds for the torsion points of $E$. We also show with an example that the hypothesis over $p$ is necessary. We get a weak generalization of the result on elliptic curves to the larger family of ${\rm GL}_2$-type varieties over $k$. In the special case of the abelian surfaces $A/k$ with quaternionic multiplication over $k$ we obtain that for all prime $p$, except a finite number depending on $A$, the local-gl…
Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve
本文研究定义在数域上、经同源映射到椭圆曲线幂的阿贝尔簇中秩为1的点所诱导的整除序列，并揭示其与椭圆整除序列的联系。
Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728
本文研究 j=1728 有理椭圆曲线上的椭圆整除序列，证明 $B_{5m}$ 恒有素因子，并推广至 $B_{mp}$（$p\equiv1\pmod4$）。
In this paper we consider divisibility sequences obtained from square matrices. We work with of matrix divisibility sequences associated to a semigroup and arising from endomorphisms of an affine space. We prove that determinant divisibility sequences originated from powers of square matrices are generalized Lucas sequences.
Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This provides an analogue to the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that a…
Let $p$ be a prime number and let $ k $ be a number field, which does not contain the field $\mathbb{Q} (ζ_p + \bar{ζ_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\E$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorem of Merel, for $p$ larger than a constant depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. Nice and deep works give explicit small constants for…
We show that many existing divisibility sequences can be seen as sequences of determinants of matrix divisibility sequences, which arise naturally as Jacobian matrices associated to groups of maps on affine spaces.
We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1.
Some effectivity results for primitive divisors of elliptic divisibility sequences
本文证明了椭圆曲线上非挠点的分母序列在项数足够大时每项均存在本原除子，并给出了仅依赖于曲线模型的可有效计算的常数。
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…
We present a different proof of the characterization of non--degenerate recurrence sequences, which are also divisibility sequences, given by Van der Poorten, Bezevin, and Petho in their paper "A Full Characterisation of Divisibility Sequence". Our proof is based on an interesting determinant identity related to impulse sequences, arising from the evaluation of a generalized Vandermonde determinant. As a consequence of this new proof we can find a more precise form for the resultant sequence presented in their paper, in the general case of non--degenerate divisibility sequences having minimal …
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials that divide their compositions with every positive integer power. We completely characterize such polynomials in terms of their factorizations into cyclotomic polynomials using labeled Hasse diagrams, and construct new integer divisibility sequences based on them. We also show that, unlike the Fibonacci numbers, these non-classical sequences do not have the p…
Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper we give numerical evidence that suggests some conjectures on the 2-divisibility of certain sums of Heenger points on E of discriminant D dividing 4N. One of these conjectures suggests a possible link between the parity of the eigenvalue a_A(2) and the parity of the Shafarevich-Tate group of certain elliptic curves A of square conductor.
On the average congruence class bias for cyclicity and divisibility of the groups of $\mathbb{F}_p$-points of elliptic curves
本文改进了Banks和Shparlinski关于椭圆曲线模素数约化点群循环性与可除性的平均密度结果，将素数限制在算术级数中，并研究不同同余类间的统计偏差。
We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in sub-exponential time if and only if the elliptic curve discrete logarithm problem is solvable in sub-exponential time. We also relate the problem of EDS Association to the Tate pairing and the MOV, Frey-Rück and Shipsey EDS attacks on the elliptic curve discrete logarithm problem in the cases where these apply.
Divisibility Biases in the Orders of Elliptic Curve Reductions
本文利用Chebotarev密度定理，分析了椭圆曲线约化阶数被m整除的素数密度，证明对于Serre曲线该密度约为1/φ(m)且大于1/m，并验证了Banks-Shparlinski平均密度的正确性。
An elliptic divisibility sequence (EDS) is a sequence of integers W_0,W_1,W_2,... generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of W_n for unbounded nonsingular elliptic divisibility sequences. A typical case is Sign(W_n) = (-1)^[n*b] for an irrational real number b, where [x] denotes the greatest integer in x. As an application, we show that the associated sequence of absolute values |W_1|,|W_2|,|W_3|,... cannot be realized as the sequence counting fixed points of any (abstract) dynamical system.
Strong divisibility sequences and sieve methods
本文利用筛法及除数闭集工具，研究强整除序列中含大素因子整数与素数项的密度界，并以 Fibonacci 数和椭圆整除序列为例分析方法局限。
Let $P(m)$ denote the greatest prime factor of $m$. For integer $a>1$, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, $$P(a^n-1)\gg_{ε, a} n^{2-ε}$$ for any $ε>0$. We study analogues results for the corresponding divisibility sequence over the function field $\mathbb{F}_q(t)$ and for some divisibility sequences associated to elliptic curves over the rational field $\mathbb{Q}$.
We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,ζ_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $ζ_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(ζ_{p^r},\sqrt[m_1]{a})$, with $a\in K(ζ_{p^r})$ and $m_1|p^r$. In…
In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.
We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by $2^n$ formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields $\mathbb{F}_q$ with a prescribed (small) group $E(\mathbb{F}_q)$. In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.
Let D = (D_n)_{n\ge1} be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | D_n. In particular, given an index n in S(D), we explain how to construct elements nd in S(D), where d is either a prime divisor of D_n, or d is the product of the primes in an aliquot cycle for D. We also give bounds for the exceptional indices that are not constructed in this way.
On divisibility of Hecke eigenvalues of Ikeda lifts
本文估计了Ikeda提升的Hecke特征值被固定正整数整除的素数密度，并研究了相关椭圆Hecke本征形式的傅里叶系数在算术级数中的分布。
A recurrence relation for elliptic divisibility sequences
本文针对椭圆可除序列的两种非等价定义，证明在索引满足特定条件时，分母平方根序列也满足经典递推关系。
Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve
该研究针对椭圆曲线E:y²=x³-x，结合高斯CM筛框架与交叉项，证明了特定素数p下#E(F_p)/8的无平方因子性及素因子个数限制，为相关数论问题提供了可除性估计与分布结果。
On the Hasse principle for divisibility in elliptic curves
研究了数域上椭圆曲线中 $p^n$ 整除性的局部-全局原理，给出了伽罗瓦群生成元条件以判定原理成立并构造反例，推广了 $n=2$ 的已知结果。
We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new p…
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…
Tamagawa Numbers of Elliptic Curves with an $\ell$-isogeny
本文研究具有ℓ-同源映射的椭圆曲线，将其与带ℓ-挠点的椭圆曲线建立联系，并分析其整体Tamagawa数的ℓ-可除性。
We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these res…
For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is periodic. If K/Q_p is a local field, we prove (under certain hypotheses) that there is a power q=p^e so that for all m > 0, the limit as k goes to infinity of F_{mq^{k}}(P) exists in K and is algebraic over Q(E). We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.
We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.
We define the $\textit{Divisor Divisibility Sequence}$ associated to a Laurent polynomial $f\in\mathbb{Z}[X_1^{\pm1},\ldots,X_N^{\pm1}]$ to be the sequence $W_n(f)=\prod f(ζ_1,\ldots,ζ_N)$, where $ζ_1,\ldots,ζ_N$ range over all $n$'th roots of unity with $f(ζ_1,\ldots,ζ_N)\ne0$. More generally, we define $W_Λ(f)$ analogously for any finite subgroup $Λ\subset(\mathbb C^*)^N$. We investigate divisibility, factorization, and growth properties of $W_Λ(f)$ as a function of $Λ$. In particular, we give the complete factorization of $W_Λ(f)$ when $f$ has generic coefficients, and we prove an analytic …
We study some divisibility properties related to the factors of the discriminant of the characteristic polynomial of generalized Fibonacci sequences $(G_n)_{n\ge0}$ defined by $G_0=0$, $G_1=1$ and $G_n=pG_{n-1}+qG_{n-2}$ for $n\ge2$, where $p,q$ are given integers. As corollaries, we give some divisibility properties on some well known sequences.
Divisibility of orders of reductions of elliptic curves
本文研究椭圆曲线模素数约化群阶的整除与同余分布，给出可显式计算模 $m_E$ 同余类及素数密度的曲线族，并推广到 $\mathbb Q(t)$ 上的正密度情形。
The purpose of this memoir is to discuss two very interesting properties of integer sequences. One is the law of apparition and the other is the law of repetition. Both have been extensively studied by mathematicians such as Ward, Lucas, Lehmer, Hall, etc. However, due to the lack of a proper survey in this area, many results have been rediscovered many decades later. This along with the necessity of the usefulness of such theory calls for a survey on this topic.
Let $U = (U_n)_{n \geq 0}$ be a Lucas sequence and, for every prime number $p$, let $ρ_U(p)$ be the rank of appearance of $p$ in $U$, that is, the smallest positive integer $k$ such that $p$ divides $U_k$, whenever it exists. Furthermore, let $d$ be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes $p \leq x$ such that $d$ divides $ρ_U(p)$, as $x \to +\infty$.
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。
We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by $τ_{n+2}τ_{n-2}=ατ_{n+1}τ_{n-1}+βτ_n^2$ and initial values $τ_1=τ_2=τ_3=τ_4=1$, that the polynomial $τ_n(α,β)$ is a divisor of $τ_{n+k(2n-5)}(α,β)$ for all $n,k\in\mathbb Z$ and establish a similar result for the generalized Somos-5 sequence.
Let $R$ be a gcd-domain (for example let $R$ be a unique factorization domain), and let $(a_n)_{n\geqslant1}$ be a sequence of nonzero elements in $R$. We prove that $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for all $n,m\geqslant1$ if and only if $$a_n=\prod\limits_{d\mid n} c_d\quad\mbox{for} \ n\geqslant1, $$ where $c_1=a_1$ and $c_n=\mbox{lcm}(a_1,a_2,\dots,a_n)/\mbox{lcm}(a_1,a_2,\dots,a_{n-1})$ for $n\geqslant2$. All equalities with gcd and lcm are determined up to units of $R$.
On the simultaneous $3$-divisibility of class numbers of quadruples of real quadratic fields
本文研究了四元实二次域类数的3可除性，构造了无限多个此类四元组，并应用于生成具有3阶torsion子群的椭圆曲线。
Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some $n\geq 1$. The bound obtained depends only on $\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。