scholarly journals Corrigendum

1972 ◽  
Vol 71 (2) ◽  
pp. 431-431
Keyword(s):  
Special Reference ◽  
Elliptic Curves ◽  
London Math ◽  
Diophantine Equations

J. W. S. Cassels, A note on the division values of ℘(u).Monsieur Y. Hellegouarch has pointed out to me that the inequality (131) of my paper “A note on the division values of ℘ (u)” (Proc. Cambridge Phil. Soc.45 (1949), 169–172) does not follow from the argument hinted there and is almost certainly incorrect. All the argument shows is that the rather weaker estimate in (132) holds for m = pk as well as for m = 2pk.The erroneous claim is reproduced in my report “Diophantine equations with special reference to elliptic curves” (J. London Math. Soc.41 (1966), 193–291) in expression (17.14) on page 247.

scholarly journals On a class of quartic Diophantine equations

2021 ◽  
Vol 27 (1) ◽  
pp. 1-6
Author(s):  
F. Izadi ◽  
◽  
M. Baghalaghdam ◽  
S. Kosari ◽  
◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Nontrivial Solutions ◽  
Natural Numbers ◽  
Positive Rank

In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) { \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }, where a_i and n\geq3 are fixed arbitrary integers. We try to transform this quartic to a cubic elliptic curve of positive rank. We solve the equation for some values of a_i and n=3,4, and find infinitely many nontrivial solutions for each case in natural numbers, and show among other things, how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for n\geq 3, this paper will be restricted to the examples where n=3,4. Finally, we explain how to solve more general cases (n\geq 4) without giving concrete examples to case n\geq 5.

scholarly journals A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

2017 ◽  
Vol 60 (2) ◽  
pp. 411-434 ◽  
Author(s):  
MARUSIA REBOLLEDO ◽  
CHRISTIAN WUTHRICH
Keyword(s):  
Elliptic Curves ◽  
Half Plane ◽  
London Math ◽  
Number Fields ◽  
Arithmetic Geometry ◽  
Modular Curves ◽  
Torsion Points ◽  
Modular Curve ◽  
Upper Half Plane ◽  
Cartan Subgroups

AbstractModular curves likeX0(N) andX1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here byXnsp(p) andXnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures ofp-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc.(3)77(1) (1998), 1–38;J. Algebra231(1) (2000), 414–448).

special reference to elliptic curves

1996 ◽  
Vol 85 (3) ◽  
pp. 555-582 ◽  
Author(s):  
R. Parimala ◽  
R. Sujatha
Keyword(s):  
Special Reference ◽  
Elliptic Curves

scholarly journals Ternary Diophantine equations of signature (p, p, 3)

2004 ◽  
Vol 140 (6) ◽  
pp. 1399-1416 ◽  
Author(s):  
Michael A. Bennett ◽  
Vinayak Vatsal ◽  
Soroosh Yazdani
Keyword(s):  
Elliptic Curves ◽  
Diophantine Equations ◽  
Torsion Point ◽  
Good Reduction ◽  
Coprime Integers

In this paper, we develop machinery to solve ternary Diophantine equations of the shape Ax n + By n = C z3 for various choices of coefficients (A, B, C). As a byproduct of this, we show, if p is prime, that the equation x n + y n = pz3 has no solutions in coprime integers x and y with |xy| > 1 and prime n > p4p2. The techniques employed enable us to classify all elliptic curves over $\mathbb{Q}$ with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.

INTEGRAL POINTS ON CONGRUENT NUMBER CURVES

2013 ◽  
Vol 09 (06) ◽  
pp. 1619-1640 ◽  
Author(s):  
MICHAEL A. BENNETT
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Precise Description ◽  
Integral Points ◽  
Linear Forms ◽  
Integer Points ◽  
Congruent Number

We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 - N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of (non-trivial) computation.

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