Related papers: A $p$-Converse theorem for Real Quadratic Fields

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal…

We calculate K-theory of a crossed product $C^*$-algebra of the noncommutative torus with real multiplication by elliptic curve $\mathscr{E}(K)$ over a number field $K$. This result is used to evaluate the rank and the Shafarevich-Tate…

Consider a real algebraic curve with set of real points $R\neq\emptyset$ and complexification $P\supset R$. Let $f$ be an algebraic function on $P$ with devisor of critical points $D\subset P$. We prove that $f$ is real after a…

We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent…

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary…

Given a field $F$, an \'etale extension $L/F$ and an Azumaya algebra $A/L$, one knows that there are extensions $E/F$ such that $A \otimes_F E$ is a split algebra over $L \otimes_F E$. In this paper we bound the degree of a minimal…

We introduce a $p$-adic $L$-function $\mathscr L_{A/L}$ associated to an ordinary elliptic curve $A$ over a global function field $K$ of characteristic $p$ together with a $\mathbb{Z}_{p}^{d}$-extension $L/K$, $d=0$ allowed, unramified…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of…

Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of characteristic $\neq 2$, let $s$ be the unique non-negative integer such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$, and let $k$ denote the dimension of the anisotropic part…

Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian…

Let $\ell$ and $p \geq 3$ be different primes. Let $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ be elliptic curves with isomorphic $p$-torsion. Assume that $E$ has potentially multiplicative reduction. We classify when all…

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a…

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

Let $S$ be a closed Riemann surface of genus $g \geq 2$ and $\varphi$ be a conformal automorphism of $S$, of prime order $p$ such that $S/\langle \varphi \rangle$ has genus zero. Let ${\mathbb K} \leq {\mathbb C}$ be a field of definition…

This is a translation of a research announcement by Anas G. Nasybullin from 1976, in which he states formulas for the p-primary part of the Tate-Shafarevich group of an elliptic curve in cyclotomic $\Z_p$-extensions of number fields.

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity…

Let $K$ be a field with a discrete valuation, and let $p$ and $\ell$ be (possibly equal) primes which are not necessarily different from the residue characteristic. Given a superelliptic curve $C : y^p = f(x)$ which has split degenerate…

Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied…

If F is a global function field of characteristic p>3, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of…