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 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...

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 $p^r={13}^2$.

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...

Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by $y^2=f\left(x\right)$. In this paper we prove that the coefficient at $x^{\frac{1}{2}p(p-1)}$ in the $p$–th division polynomial of $E$ equals the coefficient at $x^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}(p-1)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...

Let $𝒜$ be a commutative algebraic group defined over a number field $k$. We consider the following question:A complete answer for the case of the multiplicative group ${𝔾}_m$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $𝒜$ is either an elliptic curve or a torus of small dimension with respect to $r$. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime. ...

ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p,q;Y}(\Omega ,ℬ)$..........................................................73. Some properties of the spaces $H_{p,q;Y}(\Omega ,ℬ)$...................................................104. Some examples of the spaces $H_{p,q;Y}(\Omega ,ℬ)$....................................................155....

Let $k$ be an algebraically closed field of characteristic $p\>0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...