Showing papers on "Prime number published in 2013"
TL;DR: How the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 · 1018 and how the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood are described.
Abstract: This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 · 1018. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 · 1026. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood (with an error that appears to be O( √ t log log t), where t is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in 1982 by Heath-Brown. The Goldbach conjecture [13] is a famous mathematical problem whose proof, or disproof, has so far resisted the passage of time [20, Problem C1]. (According to [1], Waring and, possibly, Descartes also formulated similar conjectures.) It states, in its modern even form, that every even number larger than four is the sum of two odd prime numbers, i.e., that n = p + q. Here, and in what follows, n will always be an even integer larger than four, and p and q will always be odd prime numbers. The additive decomposition n = p + q is called a Goldbach partition of n. The one with the smallest p will be called the minimal Goldbach partition of n; the corresponding p will be denoted by p(n) and the corresponding q by q(n). It is known that up to a given number x at most O(x) even integers do not have a Goldbach partition [30], and that every large enough even number is the sum of a prime and the product of at most two primes [24]. Furthermore, according to [48], every odd number greater that one is the sum of at most five primes. As described in Table 1, over a time span of more than a century the even Goldbach conjecture was confirmed to be true up to ever-increasing upper limits. Section 1 describes the methods that were used by the first author, with computational help from the second and third authors, and others, to set the limit of verification of the Goldbach conjecture at 4 · 10. Section 2 presents a small subset of the empirical data that was gathered during the verification, namely, counts and first occurrences of primes in minimal Goldbach partitions, and counts and first occurrences of prime gaps, and compares it with the predictions made by Received by the editor May 21, 2012 and, in revised form, December 6, 2012. 2010 Mathematics Subject Classification. Primary 11A41, 11P32, 11N35; Secondary 11N05, 11Y55.
178 citations
25 Jun 2013
TL;DR: The results strengthen the previously observed inefficiency of composite-order bilinear groups and advocate the use of prime-order group whenever possible in protocol design.
Abstract: We provide software implementation timings for pairings over composite-order and prime-order elliptic curves. Composite orders must be large enough to be infeasible to factor. In the literature, protocols use orders which are product of 2 up to 5 large prime numbers. Our contribution is three-fold. First, we extend the results of Lenstra concerning the RSA modulus sizes to multi-prime modulus, for various security levels. We then implement a Tate pairing over a composite order supersingular curve and an optimal ate pairing over a prime-order Barreto-Naehrig curve, both at the 128-bit security level. Thirdly we use our implementation timings to deduce the total cost of the homomorphic encryption scheme of Boneh, Goh and Nissim and its translation by Freeman in the prime-order setting. We also compare the efficiency of the unbounded Hierarchical Identity Based Encryption protocol of Lewko and Waters and its translation by Lewko in the prime order setting. Our results strengthen the previously observed inefficiency of composite-order bilinear groups and advocate the use of prime-order group whenever possible in protocol design.
103 citations
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TL;DR: In this paper, the authors provide software implementation timings for pairings over composite-order and prime-order elliptic curves, both at the 128-bit security level, and compare the efficiency of the homomorphic encryption scheme of Boneh, Goh and Nissim and its translation by Freeman in the prime order setting.
Abstract: We provide software implementation timings for pairings over composite-order and prime-order elliptic curves. Composite orders must be large enough to be infeasible to factor. They are modulus of 2 up to 5 large prime numbers in the literature. There exists size recommendations for two-prime RSA modulus and we extend the results of Lenstra concerning the RSA modulus sizes to multi-prime modulus, for various security levels. We then implement a Tate pairing over a composite order supersingular curve and an optimal ate pairing over a prime-order Barreto-Naehrig curve, both at the 128-bit security level. We use our implementation timings to deduce the total cost of the homomorphic encryption scheme of Boneh, Goh and Nissim and its translation by Freeman in the prime-order setting. We also compare the efficiency of the unbounded Hierarchical Identity Based Encryption protocol of Lewko and Waters and its translation by Lewko in the prime order setting. Our results strengthen the previously observed inefficiency of composite-order bilinear groups and advocate the use of prime-order group whenever possible in protocol design.
88 citations
TL;DR: In this paper, it was shown that the Sylow p -subgroups of a finite group G/Z are abelian if p does not divide the integers for all lying over the group.
Abstract: Let Z be a normal subgroup of a finite group G , let ??Irr(Z) be an irreducible complex character of Z , and let p be a prime number. If p does not divide the integers ?(1)/?(1) for all ??Irr(G) lying over ? , then we prove that the Sylow p -subgroups of G/Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture
63 citations
TL;DR: In this article, the authors established function field versions of two classical conjectures on prime numbers, namely, the number of primes in intervals (x,x+x +x^epsilon] is about xπ(x)/log x, for d^(1+delta)
Abstract: In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p
56 citations
TL;DR: The enumeration of distinct skew cyclic codes over R are given and it is shown that these codes are equivalent to either cyclic code or quasi-cyclic codes.
Abstract: 【In this paper, we study a special class of linear codes, called skew cyclic codes, over the ring $R=F_p+vF_p$ , where $p$ is a prime number and $v^2=v$ . We investigate the structural properties of skew polynomial ring $R[x,{\theta} ] $ and the set $R[x,{\theta} ] /(x^n-1)$ . Our results show that these codes are equivalent to either cyclic codes or quasi-cyclic codes. Based on this fact, we give the enumeration of distinct skew cyclic codes over R.】
52 citations
1 Jan 2013
TL;DR: It is shown that it is easy to find a proof of primality for a prime p if the complete factorization of p - 1 is known, and every prime p can be deterministically supplied with aProof of its primality in O((logp)3) arithmetic steps with integers at most p.
Abstract: In this paper we present several algorithms that can find proofs of primality in deterministic polynomial time for some primes. In particular we show this for any prime p for which the complete prime factorization of p − 1 is given. We can also show this when a completely factored divisor of p − 1 is given that exceeds p l/4+e. And we can show this if p − 1 has a factor F exceeding p e with the property that every prime factor of F is at most (log p)2/e. Finally, we present a deterministic polynomial time algorithm that will prove prime more than x 1-e primes up to x, The key tool we use is the idea of a smooth number, that is, a number with only small prime factors. We show an inequality for their distribution that perhaps has independent interest.
42 citations
TL;DR: In this paper, the generic member of a family of K3 surfaces admitting a non-symplectic automorphism of finite order admits also a symplectic automomorphism of the same order.
Abstract: In this paper we investigate when the generic member of a family of K3 surfaces admitting a non–symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain conditions, a K3 surface admitting a non–symplectic automorphism of prime odd order, p, also admits a non–symplectic automorphism of order 2p. This generalizes a previous result by J. Dillies for p = 3.
33 citations
TL;DR: In this paper, it was shown that the value set of the function S ( n ) is exactly the set of all prime numbers, and that S( n) is the least prime greater than 2 n − 2.
29 citations
TL;DR: In this paper, the potential modularity of E over F modulo 2 has been shown to have a meromorphic continuation to C and satisfies the expected functional equation, and the integer ords = 1 L(E/F, s) ∈ Z is well defined.
Abstract: Note that potential modularity of E [Wi, Thm. A.1] implies that the L-function L(E/F, s) has a meromorphic continuation to C and satisfies the expected functional equation ([T2, proof of Cor. 2.2]; [N2, 12.11.6]). As a result, the integer ords=1 L(E/F, s) ∈ Z is well-defined. Various special cases of Theorem A (for F 6= Q) were proved in [N2], [Ki] and [N6]. If the p-primary part of X(E/F ) is finite for some prime number p, then sp(E/F ) = rkZE(F ) and the statement of Theorem A is the conjecture of Birch and Swinnerton-Dyer for E over F modulo 2. Theorem B. Let g = ∑∞ n=1 anq n ∈ S2r(Γ0(N)) (r ≥ 1) be a normalised (a1 = 1) newform, let L = Q(a1, a2, . . .) be the (totally real) number field generated by its coefficients. For any prime p of L above a rational prime p 6= 2, denote by Vp(g) the two-dimensional representation of GQ = Gal(Q/Q) over Lp attached to g: det (1−X Frgeom(l) | Vp(g)) = 1− alX + l2r−1X2 (l pN).
28 citations
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TL;DR: In this paper, the authors interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties.
Abstract: We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist three-dimensional quartic hypersurface domains and modules of finite length with irrational Hilbert-Kunz multiplicity. From this we deduce that also the Hilbert-Kunz multiplicity of a local noetherian domain might be an irrational number.
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TL;DR: In this paper, a broad generalisation of Dwork's formal congruences theorem for generalized hypergeometric series with rational parameters is presented, which holds for any prime number p and not only for almost all primes.
Abstract: Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the "Eisenstein constant" of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It essentially contains all the similar univariate integrality results in the litterature.
1 Jan 2013
TL;DR: Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which Pn 1 pn ¤ B.
Abstract: Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had previously seemed intractable. In this article we will discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and sketch a proof of his theorem. Zhang even proved the result with B 70 000 000. A co-operative team, polymath8, collaborating only on-line, has been able to lower the value of B to 4680, and it seems plausible that these techniques can be pushed somewhat further, though the limit of these methods seem, for now, to be B 12.
TL;DR: In this paper, it was shown that if G = 〈g〉 is a cyclic group with order of a product of two prime powers and gcd(|G|, 6) = 1, then every minimal zero-sum sequence S of the form S = (g)(n2g), n3g),n4g) has index 1.
Abstract: Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n1g)⋅…⋅(nlg) where g ∈ G and n1, …, nl ∈ [1, ord(g)], and the index ind(S) of S is defined to be the minimum of (n1+⋯+nl)/ord(g) over all possible g ∈ G such that 〈g〉 = G. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|, 6) = 1 has index 1. In this paper, we show that if G = 〈g〉 is a cyclic group with order of a product of two prime powers and gcd(|G|, 6) = 1, then every minimal zero-sum sequence S of the form S = (g)(n2g)(n3g)(n4g) has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of G is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.
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TL;DR: In this paper, it was shown that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a suitable range, generalizing the case N=2 treated by E. Fouvry, S. Ganguly, E. Kowalski and P. Michel.
Abstract: We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a suitable range, generalizing the case N=2 treated by E. Fouvry, S. Ganguly, E. Kowalski and P. Michel. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.
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TL;DR: In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao as mentioned in this paper showed that for any coprime integer $a$ and $D$ with bounded gaps in the congruence class $a \bmod D, there exist infinitely long strings of consecutive primes whose successive gaps form an increasing (resp. decreasing) sequence.
Abstract: In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x) = \{gx + h_j\}_{j=1}^k$ of linear forms in $\mathbb{Z}[x]$, the set $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \in \mathbb{N}$. In this note, we deduce that $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ consecutive primes for infinitely many $n \in \mathbb{N}$. We answer an old question of Erd\H os and Turan by producing strings of $m + 1$ consecutive primes whose successive gaps $\delta_1,\ldots,\delta_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $\delta_{j-1} \mid \delta_j$ for $2 \le j \le m$. For any coprime integers $a$ and $D$ we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class $a \bmod D$.
TL;DR: In this article, all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p, where p is a prime number, were shown to be connected normal with order 2δp, δ = 0, 1, 2 and p prime.
Abstract: We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p, where p is a prime number. As a consequence we prove if |G| = 2δp, δ = 0, 1,2 and p prime, then Λ = Cay(G, S) is a connected normal \(\tfrac{1} {2}\) arc-transitive Cayley graph only if G = F4p, where S is an inverse closed generating subset of G which does not contain the identity element of G and F4p is a group with presentation \(F_{4p} = \left\langle {\left. {a,b} \right|a^p = b^4 = 1,b^{ - 1} ab = a^\lambda } \right\rangle\), where λ2 ≠ −1 (mod p).
TL;DR: In this paper, the existence of certain hypothetical sets of zeros of Dirichlet L-functions lying off the critical line has been shown to imply that a set of real x of asymptotic density 1.
Abstract: We show, for any q > 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet L-functions lying off the critical line implies that �(x;q,a) < �(x;q,b) for a set of real x of asymptotic density 1.
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21 Aug 2013TL;DR: In this paper, a run-time branch trace value is determined to be divisible, without a remainder, by a first prime number of the plurality of prime numbers of the program.
Abstract: Exemplary methods, apparatuses, and systems assign a plurality of branch instructions within a computer program to a plurality of prime numbers. Each branch instruction is assigned a unique prime number within the plurality of prime numbers. A run-time branch trace value is determined to be divisible, without a remainder, by a first prime number of the plurality of prime numbers. The run-time branch trace value was generated during execution of the computer program. An output is generated indicating that a first branch instruction assigned to the first prime number was executed.
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TL;DR: This short note answers the related question: “Is the set of all quotients of Gaussian primes dense in the complex plane?”
Abstract: It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"
TL;DR: In this paper, the irreducibility conditions for polynomials of the form f(X) + pkg(X), with f and g relatively prime poynomials with integer coefficients, were shown.
Abstract: We provide irreducibility conditions for polynomials of the form f(X) + pkg(X), with f and g relatively prime polynomials with integer coefficients, deg f < deg g, p a prime number and k a positive integer. In particular, we prove that if k is prime to deg g - deg f and pk exceeds a certain bound depending on the coefficients of f and g, then f(X) + pkg(X) is irreducible over ℚ.
TL;DR: In this paper, the authors introduced the q-analogue of p-adic log gamma functions with weight alpha and gave a relationship be- tween weighted padic q-log gamma functions and q-extension of Genocchi and Euler numbers.
Abstract: In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship be- tween weighted p-adic q-log gamma functions and q-extension of Genocchi and Euler numbers with weight alpha. Assume that p is a fixed odd prime number. Throughout this paper Z, Zp, Qp and Cp will denote the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Also we denote N � = N ( {0} and exp(x) = e x . Let vp : Cp ! Q ( {1} (Q is the field of rational numbers) denote the p-adic valuation of Cp normalized so that vp (p) = 1. The absolute value on Cp will be denoted as |·|, and |x| p = p v p(x)
TL;DR: A partial proof is given by showing that a class of primitive sequences of order 2n'+1 over Z/(M) is distinct modulo 2, where n' is a positive integer.
Abstract: Let M be a square-free odd integer and Z/(M) the integer residue ring modulo M . This paper studies the distinctness of primitive sequences over Z/(M) modulo 2. Recently, for the case of M=pq, a product of two distinct prime numbers p and q, the problem has been almost completely solved. As for the case that M is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2n'+1 over Z/(M) is distinct modulo 2, where n' is a positive integer. Besides as an independent interest, this paper also involves two distribution properties of primitive sequences over Z/(M), which are related closely to our main results.
TL;DR: In this article, it was shown that the spectrum of the Riemann zeta function can be interpreted as a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory.
Abstract: The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1/2. Hilbert and Polya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Using the construction of the so-called super-zeta functions or secondary zeta functions built over the Riemann nontrivial zeros and the regularity property of one of this function at the origin, we show that it is possible to extend the Hilbert–Polya conjecture to systems with countably infinite number of degrees of freedom. The sequence of the nontrivial zeros of the Riemann zeta function can be interpreted as the spectrum of a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory. However, if one considers the same situation with numerical sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers, the associated functional integral cannot be constructed. Finally, we discuss possible relations between the asymptotic behavior of a sequence and the analytic domain of the associated zeta function.
TL;DR: In this article, the authors prove that the rational torsion subgroup of the modular Jacobian variety coincides with the 0-cuspidal class group up to 2-torsion.
Abstract: Let $N \geq 5$ be a prime number. Conrad, Edixhoven and Stein have conjectured that the rational torsion subgroup of the modular Jacobian variety $J_1(N)$ coincides with the 0-cuspidal class group. We prove this conjecture up to 2-torsion. To do this, we study certain ideals of the Hecke algebras, called the Eisenstein ideals, related to modular forms of weight 2 with respect to $\varGamma_1(N)$ that vanish at the 0-cusps.
1 Jan 2013
TL;DR: This work uses pruned enumeration algorithms to find lattice vectors close to a specific target vector for the prime number lattice and generates multiplicative prime number relations modulo N that factorize a given integer N.
Abstract: We use pruned enumeration algorithms to find lattice vectors close to a specific target vector for the prime number lattice. These algorithms generate multiplicative prime number relations modulo N that factorize a given integer N. The algorithm New Enum performs the stages of exhaustive enumeration of close lattice vectors in order of decreasing success rate. For example an integer N ≈ 1014 can be factored by about 90 prime number relations modulo N for the 90 smallest primes. Our randomized algorithm generated for example 139 such relations in 15 minutes. This algorithm can be further optimized. The optimization for larger integers N is still open.
TL;DR: In this paper, the authors give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q,d) = gcd (q, d), q, d − 1) = 1.
Abstract: In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d − 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p (d − 1)n then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φn+2(1 − d) where Φn+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.
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TL;DR: In this paper, the authors study the structure of a quadratic field as a Z_p-module and give numerical results, together with interpretations via Cohen-Lenstra's heuristics.
Abstract: We fix a prime number $p$ and $\K$ a number field, we denote by $M$ the maximal abelian $p$-extension of $\Ko$ unramified outside $p$. The aim of this paper is to study the $\Z_p$-module $\gal(M/\Ko)$ and to give a method to effectively compute its structure as a $\Z_p$-module. Then we give numerical results, for real quadratic fields, together with interpretations via Cohen-Lenstra's heuristics.
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TL;DR: In this article, Tian's induction method was generalized to study the Birch-Swinnerton-Dyer conjecture for the quadratic twist of certain elliptic curves defined over $\mathbb Q".
Abstract: In this paper, we show that Tian's induction method can be generalised to study the Birch-Swinnerton-Dyer conjecture for the quadratic twists, both with global root number $+1$ and with global root number $-1$, of certain elliptic curves $E$ defined over $\mathbb Q$. In particular, for the curve $E = X_0(49)$ we prove the following results. Let $q_1, \ldots, q_r$ be distinct primes which are congruent to $1$ modulo $4$ and inert in the field $F = \mathbb Q(\sqrt{-7})$, and let $E^{(R)}$ be the twist of $E$ by the quadratic extension $\mathbb Q(\sqrt{R})/\mathbb Q$, where $R=q_1\ldots q_r$. Then we show that the complex L-series of $E^{(R)}$ does not vanish at $s=1$, and the full Birch-Swinnerton-Dyer conjecture is true for $E^{(R)}$. Let $l_0$ be a prime number which is congruent to $3$ modulo $4$, and is such that $7$ splits in the field $K = \mathbb Q(\sqrt{-l_0})$. If we assume in addition that all of the primes $q_1, \ldots, q_r$ are inert in $K$ as well as in $F$, then we prove that the complex $L$-series of the twist of $E$ by $\mathbb Q(\sqrt{-l_0R})/\mathbb Q$ always has a simple zero at $s=1$. Similar results are obtained for certain other elliptic curves defined over $\mathbb Q$.
TL;DR: In this paper, the S-ramified Iwasawa module is defined as the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, they give the explicit formulae of the ℤp-ranks of the S -ramifiedIwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear
Abstract: For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic ℤp-extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, we give the explicit formulae of the ℤp-ranks of the S-ramified Iwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.