In this article, we study the family of elliptic curves $E_{-2pq}: y^2=x^3-2pqx$, where $p$ and $q$ are distinct odd primes. Using a $2$-isogeny methods and some elementary techniques, we obtain explicit possibilities for the Mordell--Weil ranks, conditional on the Parity Conjecture. Moreover, in the rank-one case, we are also able to derive explicit conditions that are independent of the parity conjecture. Moreover, the main results depend only on the residue classes of $(p,q)$ modulo $8$ and the Legendre symbols $\legendre{p}{q}$.
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $E^d$ be its twist by the quadratic character $\chi_d$. We prove there are infinitely many twists $d$ which are sums of two squares such that $E^d$ has rank $1$. This result is achieved using moments of derivatives of modular $L$-functions, and particularly capture...
In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit...
In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree...
Let $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$ and $E^{(p)}$ be its conjugate. We give a bound on the minimal degree of an isogeny from $E$ to $E^{(p)}$ depending on $p$, and show that this bound is both asymptotically optimal as well as sharp in many cases. This bound is obtained by develo...
Yves Aubry, Roger Oyono, Christelle Vincent· 3 citations· ⚡1
We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which s...
We investigate the low degree rational cohomology groups of the moduli space of twisted Prym curves ${\overline{Pr}_{g,n}^{\hspace{0.05cm}(m_1, \ldots, m_n)}} $, where the integer twists $0\leq m_i\leq 1$ have even sum over $i$. We prove that these groups vanish in odd degree $\leq3$ and that the group in degree $2$ is...
C. Fontanari· 0 citations
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