Related papers: The distribution of the number of points modulo an…

Enge and Schertz gave the method of using the double eta-quotient for the construction of elliptic curves over finite fields. In their method, it is necessary to count the number of rational points of elliptic curves corresponding to…

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $p \neq \ell$, the Frobenius automorphism associated to $p$ (unique up to conjugation) acts on the $\ell$-adic Tate module of…

Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer $g \geq 2$, we…

We determine in this paper the distribution of the number of points on the cyclic covers of $\mathbb{P}^1(\mathbb{F}_q)$ with affine models $C: Y^r = F(X)$, where $F(X) \in \mathbb{F}_q[X]$ and $r^{th}$-power free when $q$ is fixed and the…

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$…

Fix a positive integer $n$ and a finite field $\mathbb F_q$. We study the joint distribution of the rank of $E$, the $n$-Selmer group of $E$, and the $n$-torsion in the Tate-Shafarevich group of $E$ as $E$ varies over elliptic curves of…

We provide new upper bounds on N_q(g), the maximum number of rational points on a smooth absolutely irreducible genus-g curve over F_q, for many values of q and g. Among other results, we find that N_4(7) = 21 and N_8(5) = 29, and we show…

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a quadratic number field.

The distribution of the number of points on abelian covers of $\mathbb{P}1(\mathbb{F}_q)$ ranging over an irreducible moduli space has been answered in recent work by the author. Bucur, et al. determined the distribution over the whole…

In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there…

Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the associated torsion group. In a previous paper, the authors studied, for a given G, which possible groups G\leq H could appear such that H=E(K)_tors, for [K:Q]=2. In the…

The classification of elliptic curves E over the rationals Q is studied according to their torsion subgroups E_{tors}(Q) of rational points. Explicit criteria for the classification are given when E_{tors}(Q) are cyclic groups with even…

Consider an elliptic curve $E$ over a number field $K$. Suppose that $E$ has supersingular reduction at some prime $\mathfrak{p}$ of $K$ lying above the rational prime $p$. We completely classify the valuations of the $p^n$-torsion points…

Given an elliptic curve E over a finite field F_q of q elements, we say that an odd prime ell not dividing q is an Elkies prime for E if t_E^2 - 4q is a square modulo ell, where t_E = q+1 - #E(F_q) and #E(F_q) is the number of F_q-rational…

Following Lang and Trotter we describe a probabilistic model that predicts the distribution of primes $p$ with given Frobenius traces at $p$ for two fixed elliptic curves over $\mathbb{Q}$. In addition, we propose explicit Euler product…

Given a prime $p$, an elliptic curve $\E/\F_p$ over the finite field $\F_p$ of $p$ elements and a binary \lrs\ $\(u(n)\)_{n =1}^\infty$ of order~$r$, we study the distribution of the sequence of points $$ \sum_{j=0}^{r-1} u(n+j)P_j, \qquad…

Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…