相关论文: Multiplicative and linear dependence in finite fie…

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x + b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ by \begin{equation*}…

In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in…

The primary objective of this paper is the study of different instances of the elliptic Stark conjectures of Darmon, Lauder and Rotger, in a situation where the elliptic curve attached to the modular form $f$ has split multiplicative…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $\Delta(X)$.…

We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only…

Let O_K be a discrete valuation ring with field of fractions K and perfect residue field. Let E be an elliptic curve over K, let L/K be a finite Galois extension and let O_L be the integral closure of O_K in L. Denote by X' the minimal…

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskr\k{e}cki--Stoll uses the…

Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to…

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic)…

We show that if p is a prime, then all elliptic curves defined over the cyclotomic Z_p extension of Q are modular.

Let $E / \mathbb{Q}$ and $A / \mathbb{Q}$ be elliptic curves. We can construct modular points derived from $A$ via the modular parametrisation of $E$. With certain assumptions we can show that these points are of infinite order and are not…

Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the…

Let $E$ be an elliptic curve defined over ${\mathbb Q}$. For a prime $p$ of good reduction for $E$, denote by $e_p$ the exponent of the reduction of $E$ modulo $p$. Under GRH, we prove that there is a constant $C_E\in (0, 1)$ such that $$…

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of…

Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $\Gamma$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\in\Gamma$ if and only if $P\in…

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length…

For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points…