Some new families of positive-rank elliptic curves arising from Pythagorean triples

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

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A NOTE ON HIGHER TWISTS OF ELLIPTIC CURVES

Glasgow Mathematical Journal ◽

10.1017/s0017089510000066 ◽

2010 ◽

Vol 52 (2) ◽

pp. 371-381 ◽

Cited By ~ 2

Author(s):

MACIEJ ULAS

Keyword(s):

Elliptic Curves ◽

Higher Twists ◽

Positive Rank

AbstractWe show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v] such that the cubic twists of the curves E1, E2 by D(u, v) have positive rank over ℚ(u, v). We also prove that for any quadruple of pairwise distinct elliptic curves Ei, i = 1, 2, 3, 4, with j-invariant j = 0, there exists a polynomial D ∈ ℤ[u] such that the sextic twists of Ei, i = 1, 2, 3, 4, by D(u) have positive rank. A similar result is proved for quadruplets of elliptic curves with j-invariant j = 1, 728.

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Moments of the Critical Values of Families of Elliptic Curves, with Applications

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-049-5 ◽

2010 ◽

Vol 62 (5) ◽

pp. 1155-1181 ◽

Cited By ~ 1

Author(s):

Matthew P. Young

Keyword(s):

Elliptic Curves ◽

Large Family ◽

Critical Values ◽

Central Values ◽

The Family ◽

Order Of Magnitude ◽

Rank 2 ◽

Family Based ◽

First Moment ◽

Positive Rank

AbstractWe make conjectures on the moments of the central values of the family of all elliptic curves and on themoments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.Furthermore, as arithmetical applications, we make a conjecture on the distribution of ap's amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).

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On a class of quartic Diophantine equations

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.1.1-6 ◽

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.

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