Musa Demirci

Rational Points on Elliptic Curves

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} to F_{p^{r}}.

y^{2}=x^{3}+a^{3}

Let H(λq) be the Hecke group associated to λq=2cosπq for q≥3 integer. In this paper, we determine the constant term of the minimal polynomial of λq denoted by Pq∗(x).MSC:12E05, 20H05.

in

In this note, we find all the solutions of the Diophantine equation x2 + 2a · 3b · 11c = yn, in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.

{\bf F}_p

In this note, we find all the solutions of the Diophantine equation x2 + 2a · 3b · 11c = yn, in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.

where

In this note, we find all the solutions of the Diophantine equation x2 + 2a · 3b · 11c = yn, in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.

p\equiv 1\pmod6

Let H(λq) be the Hecke group associated to λq=2cosπq for q≥3 integer. In this paper, we determine the constant term of the minimal polynomial of λq denoted by Pq∗(x).MSC:12E05, 20H05.

is Prime

Let H(λq) be the Hecke group associated to λq=2cosπq for q≥3 integer. In this paper, we determine the constant term of the minimal polynomial of λq denoted by Pq∗(x).MSC:12E05, 20H05.

The constant term of the minimal polynomial of cos ( 2 π / n ) Open image in new window over ℚ

The number λq = 2 cos π/q, q∈N, q≥3,, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of this number by means of the better known Chebycheff polynomials for odd q and give some of their properties.

On the diophantine equation x 2 + 2 a · 3 b · 11 c = y n

The number λq = 2 cos π/q, q∈N, q≥3, appears in the study of Hecke groups which are Fuchsian groups of the first kind, and in the study of regular polyhedra. Here we obtained the minimal polynomial of this number by means of the better known Chebycheff polynomials and the set of roots on the extension Q(λq). We follow some kind of inductive method on the number q. The minimal polynomial is obtained for even q.

On the diophantine equation x2 + 2a · 3b · 11c = yn

In [4], Greenberg showed that n≤6t3 so that μ = nt≤6t4 for a normal subgroup N of level n and index μ having t parabolic classes in the modular group Γ. Accola, [1], improved these to n≤6t2 always and n≤t2 if Γ/N is not abelian. In this work we generalise these results to Hecke groups. We get results between three parameters of a normal subgroup, i.e. the index μ, the level n and the parabolic class number t. We deal with the case q = 4, and then obtain the generalisation to other q. Two main problems here are the calculation of the number of normal subgroups and the determination of the bounds on the level n for a given t.