Formal groups and some arithmetic properties of elliptic curves

1979 ◽  
pp. 630-658 ◽  
Author(s):  
Noriko Yui
Keyword(s):  
Elliptic Curves ◽  
Formal Groups ◽  
Arithmetic Properties

scholarly journals FORMAL GROUPS AND INVARIANT DIFFERENTIALS OF ELLIPTIC CURVES

2015 ◽  
Vol 92 (1) ◽  
pp. 44-51
Author(s):  
MOHAMMAD SADEK
Keyword(s):  
Power Series ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Series Expansion ◽  
Power Series Expansion ◽  
Formal Groups ◽  
Frobenius Endomorphism ◽  
Congruence Relations ◽  
Invariant Differential ◽  
Weierstrass Equations

In this paper, we find a power series expansion of the invariant differential ${\it\omega}_{E}$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we derive several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.

scholarly journals Arithmetic properties of coefficients of L-functions of elliptic curves

2018 ◽  
Vol 187 (2) ◽  
pp. 247-273
Author(s):  
Ahmet M. Güloğlu ◽  
Florian Luca ◽  
Aynur Yalçiner
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Properties

FORMAL GROUPS OF JACOBIAN VARIETIES OF HYPERELLIPTIC CURVES

2010 ◽  
Vol 06 (07) ◽  
pp. 1701-1716
Author(s):  
FUMIO SAIRAIJI
Keyword(s):  
Elliptic Curves ◽  
Characteristic Zero ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Formal Groups ◽  
Jacobian Varieties ◽  
Explicit Constructions ◽  
Modulo P

Let k be a field of characteristic zero. In this paper, we discuss two explicit constructions of the formal groups Ĵ of the Jacobian varieties J of hyperelliptic curves C over k. Our results are generalizations of the classical constructions of formal groups of elliptic curves. As an application of our results, we may decide the type of bad reduction of J modulo p when C is a hyperelliptic curve over ℚ.

scholarly journals On cubic Thue equations and the indices of algebraic integers in cubic fields

2019 ◽  
Vol 13 (3) ◽  
pp. 774-786
Author(s):  
Abdelmejid Bayad ◽  
Mohammed Seddik
Keyword(s):  
Elliptic Curves ◽  
Algebraic Integers ◽  
Integer Points ◽  
Arithmetic Properties ◽  
Cubic Form ◽  
Cubic Fields ◽  
Thue Equation ◽  
The Common

Let F(x; y) = ax3 + bx2y + cxy2 + dy3 ? Z[x,y] be an irreducible cubic form. In this paper, we investigate arithmetic properties of the common indices of algebraic integers in cubic fields. For each integer k such that v2(k)??0 (mod 3) and 2v2(-2b3 - 27a2d + 9abc) = 3v2(b2 - 3ac), we prove that the cubic Thue equation F(x,y) = k has no solution (x,y) ? Z2. As application, we construct parametrized families of twisted elliptic curves E : ax3 + bx2 + cx + d = ey2 without integer points (x,y).

A REFINEMENT OF RAMANUJAN'S CONGRUENCES MODULO POWERS OF 7 AND 11

2012 ◽  
Vol 08 (04) ◽  
pp. 865-879
Author(s):  
MARIE JAMESON
Keyword(s):  
Partition Function ◽  
Elliptic Curves ◽  
Arithmetic Properties

Ramanujan's famous congruences for the partition function modulo powers of 5, 7, and 11 have inspired much further research. For example, in 2002 Lovejoy and Ono found subprogressions of 5jn + β5(j) for which Ramanujan's congruence mod 5j could be strengthened to a statement modulo 5j+1. Here we provide the analogous results modulo powers of 7 and 11. We require the arithmetic properties of two special elliptic curves.

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