Related papers: Class groups of imaginary quadratic points on $X_1…

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than…

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note, we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in…

For each prime number $\ell$ and for each imaginary quadratic order of class number one or two, we determine all the possible $\ell$-adic Galois representations that occur for any elliptic curve with complex multiplication by such an order…

Let $A$ be an abelian surface over $\mathbb{Q}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur's theorem for elliptic curves, we show that the torsion subgroup of $A(\mathbb{Q})$ is…

Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that $E(K)_{\operatorname{tors}}\supseteq \mathscr{T}\cong…

For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height''…

A class number formula is proved for extended ring class fields $L_{\mathcal{O},9}$ over imaginary quadratic fields $K_d = \mathbb{Q}(\sqrt{-d})$, in which the prime $p = 3$ splits, by determining the fields generated by the periodic points…

We investigate the number ${\Cal F}(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of ${\Cal F}(h)$. We also establish a modest non-trivial upper bound for ${\Cal F}(h)$ and…

For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky,…

We give another solution to the class number one problem by showing that imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$ correspond to integral points on a genus two curve $\mscrK_3$. In fact one can find all rational…

We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of…

For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the…

It is shown that the class number for negative discriminant $D$ can be expressed in terms of the base $B$ expansions of reduced fractions $\frac{x}{|D|}$, where $B$ is an integer prime to $D$. This result is then formulated to obtain…

In this paper we obtain a complete list of imaginary $n$-quadratic fields with class groups of exponent $3$ and $5$ under ERH for every positive integer $n$ where an $n$-quadratic field is a number field of degree $2^n$ represented as the…

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma = +1or $-1.Here the Mordell-Weil group and the…

Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for…

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…

We start with $n$-torsions in the Jacobian of an $m$-gonal curve and produce $n$-torsions in the class group of certain number field $K$.

Let $f$ be a positive definite integral quadratic form in $d$ variables. In the present paper, we establish a direct link between the genus representation number of $f$ and the order of higher even $K$-groups of the ring of integers of real…

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and…