arXiv subjects

Jessica Alessandrì

Publications and source records attributed to Jessica Alessandrì.

5 recordsLinked to original sources

arXiv2026

Torsion points on $\rm{GL}_2$-type abelian varieties

It is well known that the rational torsion of an abelian variety defined over a number field injects into the reduction modulo any sufficiently large prime, so the order of the torsion group divides the greatest common divisor of the sizes of points on the reduction at each prime. Drawing inspiration from Katz's Inventiones paper (1981), we investigate the converse to this for abelian varieties of $\rm GL_2$-type and exhibit a conjectural list of possible torsion orders for modular abelian varieties over $\mathbb Q$ of dimension up to $5$.

math.NT↗

arXiv2025

Fermat near misses and the integral Hilbert Property

We consider the Diophantine equation $x^4 + y^4 - w^2 = n$ for $n \in \mathbb{Z}$, which is related to near misses for the quartic case of Fermat's Last Theorem. For certain $n$ we show that the set of solutions is infinite, or more generally not thin. Our approach is via the geometry of del Pezzo surfaces of degree $2$, and we prove a more general result on non-thinness of integral points on double conic bundle surfaces.

math.NT↗

arXiv2025

On the Hasse principle for divisibility in elliptic curves

Let $p$ be a prime number and $n$ a positive integer. Let $E$ be an elliptic curve defined over a number field $k$. It is known that the local-global divisibility by $p$ holds in $E/k$, but for powers of $p^n$ counterexamples may appear. The validity or the failing of the Hasse principle depends on the elliptic curve $E$ and the field $k$ and, consequently, on the group $\mathrm{Gal}(k(E[p^n])/k)$. For which kind of these groups does the principle hold? For which of them can we find a counterexample? The answer to these questions was known for $n=1,2$, but for $n\geq 3$ they were still open. We show some conditions on the generators of $\mathrm{Gal}(k(E[p^n])/k)$ implying an affirmative answer to the local-global divisibility by $p^n$ in $E$ over $k$, for every $n\geq 2$. We also prove that these conditions are necessary by producing counterexamples in the case when they do not hold. These last results generalize to every power $p^n$, a result obtained by Ranieri for $n=2$.

math.NT↗

arXiv2023

Local-global divisibility on algebraic tori

We give a complete answer to the local-global divisibility problem for algebraic tori. In particular, we prove that given an odd prime $p$, if $T$ is an algebraic torus of dimension $r< p-1$ defined over a number field $k$, then the local-global divisibility by any power $p^n$ holds for $T(k)$. We also show that this bound on the dimension is best possible, by providing a counterexample of every dimension $r \geq p-1$. Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the $p^n$-torsion point of $T$, the local-global divisibility still holds for tori of dimension less than $3(p-1)$.

math.NT↗

arXiv2022

On 7-division fields of CM elliptic curves

Let $\mathcal{E}$ be a CM elliptic curve defined over a number field $K$, with Weiestrass form $y^3=x^3+bx$ or $y^2=x^3+c$. For every positive integer $m$, we denote by ${\mathcal{E}}[m]$ the $m$-torsion subgroup of ${\mathcal{E}}$ and by $K_m:=K({\mathcal{E}}[m])$ the $m$-th division field, i.e. the extension of $K$ generated by the coordinates of the points in ${\mathcal{E}}[m]$. We classify all fields $K_7$. In particular we give explicit generators for $K_7/K$ and produce all Galois groups ${\textrm{Gal}}(K_7/K)$. We also show some applications to the Local-Global Divisibility Problem and to modular curves.

math.NT↗