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Let $Q(alpha)$ be the simplest cubic field, it is known that $Q(alpha)$ can be generated by adjoining a root of the irreducible equation $x^{3}-kx^{2}+(k-3)x+1=0$, where $k$ belongs to $Q$. In this paper we have established a relationship between $alpha$, $alpha$ and $k,k$ where $alpha$ is a root of the equation $x^{3}-kx^{2}+(k-3)x+1=0$ and $alpha$ is a root of the same equation with $k$ replaced by $k$ and $Q(alpha)=Q(alpha)$.
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We prove Manins conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.
Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a cubic number field. To do so, we determine the cubic points on the modular curves $X_1(N)$ for [N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.] As part of our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(mathbb{Q})$ is generated by $text{Gal}(bar{mathbb{Q}}/mathbb{Q})$-orbits of cusps of $X_1(N)_{bar{mathbb{Q}}}$ for $Nleq 55$, $N eq 54$.
We present a method for tabulating all cubic function fields over $mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $mathbb{F}_{q}^*$, up to a given bound $B$ on the degree of $D$. Our method is based on a generalization of Belabas method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires $O(B^4 q^B)$ field operations as $B rightarrow infty$. The algorithm, examples and numerical data for $q=5,7,11,13$ are included.
Manins conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manins conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manins conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.
For a cubic algebraic extension $K$ of $mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum $$ sumlimits_{n_{1}^2+n_{2}^2leq x}a_{K}(n_{1}^2+n_{2}^2). $$
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