Consider the families of curves $C^{n,A}:y²=xⁿ+Ax$ and $C_{n,A}:y²=xⁿ+A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}\left(\mathbb{Q}\right)$ and $C_{3,A}\left(\mathbb{Q}\right)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}\left(\mathbb{Q}\right)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}\left(\mathbb{Q}\right)\left[2\right]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We...

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_E\left(p\right)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field ${}_p$. Let be the family of elliptic curves $E_{a,b}:y^2=x^3+ax+b$, where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1/\left|\right|{\sum }_{E\in }{\sum }_{p\le x}{e}_E^k\left(p\right)=C_{k}li\left(x^{k+1}\right)+O(\left(x^{k+1}\right)/{\left(logx\right)}^c$)$$as x → ∞, as long...

Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}:y²=xⁿ+a$ and $C_{n,a}:y²=x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}\left(\mathbb{Q}\right)$ and $J_{n,a}\left(\mathbb{Q}\right)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}\left(\mathbb{Q}\right)$. Namely, we show that $J_{8,a}{\left(\mathbb{Q}\right)}_{tors}=J_{8,a}\left(\mathbb{Q}\right)\left[2\right]$. In addition, we characterize the torsion parts of $J_{p,a}\left(\mathbb{Q}\right)$, where p is an odd...

Let $C_m:y^2=x^3-m^{2}x+p^2{q}^2$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p$, $q$ are distinct odd primes. We study the torsion part and the rank of $C_m\left(\mathbb{Q}\right)$. More specifically, we prove that the torsion subgroup of $C_m\left(\mathbb{Q}\right)$ is trivial and the $\mathbb{Q}$-rank of this family is at least 2, whenever $m\neg \equiv 0(mod3)$, $m\neg \equiv 0(mod4)$ and $m\equiv 2(mod64)$ with neither $p$ nor $q$ dividing $m$.

Let $D_m$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^{2}x+m^2$, where $m$ is an integer. In this paper we prove that the two points $P_{-1}=(-m,m)$ and $P_0=(0,m)$ on $D_m$ can be extended to a basis for $D_m\left(\mathbb{Q}\right)$ under certain conditions described explicitly.

Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=\mathbb{Q}$), we prove that this principle holds for all $\ell \equiv 1mod4$, and for $\ell \<7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb{Q}$ we show that, up to isomorphism, this is the only counterexample.

We give a family ${ℱ}_{h,\beta }$ of elliptic curves, depending on two nonzero rational parameters $\beta$ and $h$, such that the following statement holds: let $ℰ$ be an elliptic curve and let $ℰ\left[3\right]$ be its 3-torsion subgroup. This group verifies $\mathbb{Q}\left(ℰ\left[3\right]\right)=\mathbb{Q}\left({\zeta }_3\right)$ if and only if $ℰ$ belongs to ${ℱ}_{h,\beta }$ . Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer...

The rational completion $\overline{M}$ of an $R$-module $M$ can be characterized as a ${\tau }_M$-injective hull of $M$ with respect to a (hereditary) torsion functor ${\tau }_M$ depending on $M$. Properties of a torsion functor depending on an $R$-module $M$ are studied.

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$, for $r\>1$ and $p$ a prime number exceeding $2·{10}^{11}$. This includes the case of the curves $X_{\mathrm{split}}\left(p\right)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le {10}^{14}$, $p\ne 13$. The combination of those results completes the qualitative study of rational points on $X_0^{+}\left(p^r\right)$ undertook in our previous work, with the only exception of $p^r={13}^2$.