Related papers: Quantum dynamics of elliptic curves

Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx…

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb…

We construct a sequence of $n-1$ cyclic exact sequences that can be used to compute the $K$-theory of the $C^\star$-algebra crossed product $A \ltimes {\mathbb Z}_n$.

Let $p$ be a prime and let $K$ be a finite extension of $\mathbb{Q}_p$. Let $E/K$ be an elliptic curve with additive reduction. In this paper, we study the topological group structure of the set of points of good reduction of $E(K)$. In…

We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$, the…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…

Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic,…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…

The aim of this paper is to study certain family of elliptic curves $\{\mathscr{X}_H\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathscr{X}/F$ associated to a cyclic cubic extension $K/F$.…

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve…

It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.

We classify the possible torsion structures of rational elliptic curves over quintic number fields. In addition, let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G,…

For a directed graph $E$, we compute the $K$-theory of the $C^*$-algebra $C^*(E)$ from the Cuntz-Krieger generators and relations. First we compute the $K$-theory of the crossed product $C^*(E)\times_\gamma\IT$, and then using duality and…

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C*-algebras defining our quantum…

We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of…

For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced crossed products $A^{\otimes G}\rtimes_rG$ of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for…

Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_2$-extension over $\mathbb{Q}$. For each integer $n\geq1$, let $\mathbb{Q}_n$ denote the unique subfield in $\mathbb{Q}_\infty$ such that $[\mathbb{Q}_\infty:\mathbb{Q}]=2^n$. Denote by…

Let $\Omega$ be a tiling space and let $G$ be the maximal group of rotations which fixes $\Omega$. Then the cohomology of $\Omega$ and $\Omega/G$ are both invariants which give useful geometric information about the tilings in $\Omega$. The…

We compute the convolution product on the equivariant K-groups of the cyclic quiver variety. We get a q-analogue of double-loop algebras, closely related to the toroidal quantum groups previously studied by the authors. We also give a…

By focusing on the family $E:y^2=x^3+a$, we present strategies for determining the structure of the torsion subgroup of the Mordell-Weil group of an elliptic curve, $E(K)$, over quadratic field $K$. Generalizations of the Nagell-Lutz…