Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E:y^2=x^3-4p^{2}x$. Further, let $N\left(p\right)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\ge 17$, then $N\left(p\right)\le 4$ or $1$ depending on whether $p\equiv 1(mod8)$ or $p\equiv -1(mod8)$.

We show a $p$-parity result in a $D_{2p^n}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W(E/K,1\oplus \eta \oplus \tau )={(-1)}^{〈1\oplus \eta \oplus \tau ,X_p(E/L)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}(L/K)=D_{2p^n}$, and $X_p(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_0$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture (using...

For $n\in {𝔽}_q\left[T\right]$ let $G$ be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve $X_0\left(𝔫\right)$. We determine all $𝔫$ and $G$ for which the quotient curve $G\setminus X_0\left(𝔫\right)$ is rational or elliptic.

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $K$ be an imaginary quadratic field, and let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$. Given a set $\Sigma$ of primes of $K$, containing the primes above $p$, and the primes of bad reduction for $E$, write $K_{\Sigma }$ for the maximal algebraic extension of $K$ which is unramified outside $\Sigma$. This paper is devoted to the study of the structure of the cohomology groups $H^i(K_{\Sigma }/K_{\infty },E_{p^{\infty }})$ for $i=1,2,$ and of the $p$-primary Selmer group Sel${}_{p^{\infty }}(E/K_{\infty })$, viewed as discrete modules over the Iwasawa algebra of $K_{\infty }/K.$

We determine the distribution over square-free integers $n$ of the pair $({dim}_{{𝔽}_2}{\mathrm{Sel}}^{\Phi }(E_n/\mathbb{Q}),{dim}_{{𝔽}_2}{\mathrm{Sel}}^{\widehat{\Phi }}(E_n^{\text{'}}/\mathbb{Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^{\text{'}}:y^2=x^3+4n^{2}x$ is the image of isogeny $\Phi :E_n\to E_n^{\text{'}}$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat{\Phi }$ is the isogeny dual to $\Phi$.

La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre $r_{\infty }$ du zéro en $s=1$ de la fonction $L$ d’une courbe elliptique $E$ définie sur $𝐐$ est égal au rang $r$ du groupe de ses points rationnels. On sait démontrer cette conjecture si $r_{\infty }=0$ ou $1$, mais on n’a aucun résultat reliant $r_{\infty }$ et $r$ si $r_{\infty }\ge 2$. Nous expliquerons comment Kato démontre que la fonction $L$$p$-adique attachée à $E$ a, en $s=1$, un...

For $E_b:y²=x³+b$, we establish Lang’s conjecture on a lower bound for the canonical height of nontorsion points along with upper and lower bounds for the difference between the canonical and logarithmic heights. These results are either best possible or within a small constant of the best possible lower bounds.

We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of bounded height....

A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to $1$ modulo $m$. We prove a similar result for polynomials $f\left(X\right)$ that are divisible in $(\mathbb{Z}/m\mathbb{Z})\left[X\right]$ by a polynomial of the form $1+X+\cdots +X^n$ for some $n\ge ϵdeg\left(f\right)$. We also formulate and prove an analogous statement for elliptic curves.