Rational Points of Elliptic Curve y2=x3+k3

2018 ◽  
Vol 25 (01) ◽  
pp. 133-138
Author(s):  
Xia Wu ◽  
Yan Qin
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Class Number ◽  
Quadratic Field ◽  
Rational Numbers ◽  
Rational Points ◽  
Pell Equation ◽  
Rank Zero ◽  
Quadratic Twist

Let E be an elliptic curve defined over the field of rational numbers ℚ. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed(ℚ) is zero. Let E(k) denote the elliptic curve y2 = x3 + k. Then the quadratic twist E(1)d of E(1) by d is the elliptic curve [Formula: see text]. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d ≡ r (mod 24) such that rank [Formula: see text], using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E(k3)(ℚ) has rank zero.

scholarly journals An explicit Baker-type lower bound of exponential values

2015 ◽  
Vol 145 (6) ◽  
pp. 1153-1182 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen ◽  
Kalle Leppälä ◽  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Linear Form ◽  
Quadratic Field ◽  
Rational Numbers ◽  
Rational Points ◽  
Imaginary Quadratic Field ◽  
Ring Of Integers

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1, . . . , eαm,m⩾ 2, whereα0= 0,α1, . . . ,αmarem+ 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies onmare improved.

scholarly journals POINTS ON HYPERBOLAS AT RATIONAL DISTANCE

2012 ◽  
Vol 08 (04) ◽  
pp. 911-922
Author(s):  
EDRAY HERBER GOINS ◽  
KEVIN MUGO
Keyword(s):  
Elliptic Curve ◽  
Rational Numbers ◽  
Rational Points ◽  
Conic Section ◽  
Distance Set ◽  
Set Of Points ◽  
Distance Sets

Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article, we consider such a set of rational points restricted to a given hyperbola. To be precise, for rational numbers a, b, c, and d such that the quantity D = (ad - bc)/(2a2) is defined and non-zero, we consider rational distance sets on the conic section axy + bx + cy + d = 0. We show that, if the elliptic curve Y2 = X3 - D2X has infinitely many rational points, then there are infinitely many sets consisting of four rational points on the hyperbola such that their pairwise distances are rational numbers. We also show that any rational distance set of three such points can always be extended to a rational distance set of four such points.

scholarly journals Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

2004 ◽  
Vol 70 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Prime Ideal ◽  
Elliptic Curve ◽  
Class Number ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Good Reduction ◽  
Absolute Value ◽  
Frobenius Endomorphism ◽  
Trace Of Frobenius ◽  
Positive Integers

Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field

1981 ◽  
Vol 33 (1) ◽  
pp. 55-58 ◽  
Author(s):  
Hiroshi Takeuchi
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Real Quadratic Field ◽  
The Real ◽  
Root Of Unity ◽  
Image Position ◽  
Real Quadratic

Let p be an integer and let H(p) be the class-number of the fieldwhere ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).We designate by h(p) the class-number of the real quadratic field

scholarly journals Torsion points on elliptic curves defined over quadratic fields

1988 ◽  
Vol 109 ◽  
pp. 125-149 ◽  
Author(s):  
M. A. Kenku ◽  
F. Momose
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Prime Power ◽  
Quadratic Field ◽  
Rational Points ◽  
Negative Integer ◽  
Quadratic Fields ◽  
Prime Power Order ◽  
Torsion Points

Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13] [23] discussed the k-rational points on E of prime power order. For a prime number p, let n = n(k, p) be the least non negative integer such thatfor all elliptic curves E defined over a quadratic field k ([15]).

scholarly journals Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

2011 ◽  
Vol 14 ◽  
pp. 327-350 ◽  
Author(s):  
Robert L. Miller
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Rational Numbers ◽  
Rank Zero ◽  
Computer Assistance

AbstractWe 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 rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

scholarly journals A note on the Fourier coefficients of a Cohen–Eisenstein series

2016 ◽  
Vol 12 (05) ◽  
pp. 1149-1161
Author(s):  
Srilakshmi Krishnamoorthy
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Eisenstein Series ◽  
Class Number ◽  
Fourier Coefficients ◽  
Rank Zero ◽  
Tate Group ◽  
Class Number Formula ◽  
Free Level ◽  
Number Formula

We prove a formula for the coefficients of a weight [Formula: see text] Cohen–Eisenstein series of square-free level [Formula: see text]. This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let [Formula: see text] be an odd prime number. For any elliptic curve [Formula: see text] defined over [Formula: see text] of rank zero and square-free conductor [Formula: see text], if [Formula: see text], under certain conditions on the Shafarevich–Tate group [Formula: see text], we show that [Formula: see text] divides [Formula: see text] if and only if [Formula: see text] divides the class number [Formula: see text] of [Formula: see text]

Periods of Modular Forms and Imaginary Quadratic Base Change

2010 ◽  
Vol 53 (3) ◽  
pp. 571-576
Author(s):  
Mak Trifković
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Class Number ◽  
Quadratic Field ◽  
Base Change ◽  
Imaginary Quadratic Field ◽  
Period Ratio ◽  
Upper Half Plane ◽  
Periods Of Modular Forms

AbstractLet f be a classical newform of weight 2 on the upper half-plane , E the corresponding strong Weil curve, K a class number one imaginary quadratic field, and F the base change of f to K. Under a mild hypothesis on the pair (f, K), we prove that the period ratio is in ℚ. Here ΩF is the unique minimal positive period of F, and ΩE the area of E(ℂ). The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.

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