arXiv subjects

Matteo Verzobio

Publications and source records attributed to Matteo Verzobio.

15 recordsLinked to original sources

arXiv2026

Sums of three powerful numbers

Let $p,q,r\geq 2$ and consider the Campana orbifold \[ \left( \mathbb{P}^1, \left(1-\tfrac1p\right)[0] +\left(1-\tfrac1q\right)[1] +\left(1-\tfrac1r\right)[\infty] \right). \] Primitive positive Campana points on this orbifold correspond to solutions of $a+b=c$ in which $a$, $b$, and $c$ are respectively $p$-full, $q$-full, and $r$-full. We establish upper bounds for the number of such points of bounded height in a broad range of exponents, with a power-saving over the trivial bound. The main analytic input is an estimate for primitive integral points in lopsided boxes on generalized Fermat surfaces $a_1x^p+a_2y^q+a_3z^r=0$, which is uniform in the coefficients.

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arXiv2025

Selmer groups of families of elliptic curves with an $\ell$-isogeny

For certain families of elliptic curves admitting a rational isogeny of prime degree $\ell$, we establish a central limit theorem for the Tamagawa ratio and derive bounds on its average value. By using the Tamagawa ratio to bound the size of the $\ell$-isogeny Selmer group from below, we show that for $\ell \in\{ 2, 3, 5, 7, 13\}$, there exist elliptic curves with arbitrarily large $\ell$-Selmer groups.

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arXiv2025

Counting rational points on smooth hypersurfaces with high degree

Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^n$ with $n\geq 4$ and degree $d\geq 50$, then the set of rational points on $X$ of height bounded by $B$ have cardinality $O_{n,d,\varepsilon}(B^{n-2+\varepsilon})$. If $X$ is smooth and has degree $d\geq 6$, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.

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arXiv2025

The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic numbers. We determine these densities by calculating the corresponding $p$-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of $\ell$-torsion for $\ell>3$ prime.

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arXiv2024

Counting integer points on affine surfaces with a side condition

We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.

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arXiv2024

Strong divisibility sequences and sieve methods

We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence which only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper there is an appendix by Sandro Bettin on divisor closed sets, that we use to study the density of prime terms that appear in strong divisibility sequences.

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arXiv2023

Some effectivity results for primitive divisors of elliptic divisibility sequences

Let $P$ be a non-torsion point on an elliptic curve defined over a number field $K$ and consider the sequence $\{B_n\}_{n\in \mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive divisor for $n$ greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.

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arXiv2023

Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve

Let $A$ be an abelian variety defined over a number field $K$, $E/K$ be an elliptic curve, and $\phi:A\to E^m$ be an isogeny defined over $K$. Let $P\in A(K)$ be such that $\phi(P)=(Q_1,\dots, Q_m)$ with $\text{Rank}_\mathbb{Z}(\langle Q_1,\dots, Q_m\rangle)=1$. We will study a divisibility sequence related to the point $P$ and show its relation with elliptic divisibility sequences.

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arXiv2022

On the local-global principle for isogenies of abelian surfaces

Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface $A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.

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arXiv2022

Common valuations of division polynomials

In this note we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi_n$ and $\phi_n$ associated with a sequence $\{nP\}_{n\in\mathbb{N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula is identical with the result of Yabuta-Voutier for the case of finite extension of $\mathbb{Q}_{p}$ and generalizes to the case of non-standard Kodaira types for non-perfect residue fields.

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arXiv2021

Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728

Take a rational elliptic curve defined by the equation $y^2=x^3+ax$ in minimal form and consider the sequence $B_n$ of the denominators of the abscissas of the iterate of a non-torsion point; we show that $B_{5m}$ has a primitive divisor for every $m$. Then, we show how to generalize this method to the terms in the form $B_{mp}$ with $p$ a prime congruent to $1$ modulo $4$.

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arXiv2021

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let $P$ and $Q$ be two points on an elliptic curve defined over a number field $K$. For $α\in \text{End}(E)$, define $B_α$ to be the $\mathcal{O}_K$-integral ideal generated by the denominator of $x(α(P)+Q)$. Let $\mathcal{O}$ be a subring of $\text{End}(E)$, that is a Dedekind domain. We will study the sequence $\{B_α\}_{α\in \mathcal{O}}$. We will show that, for all but finitely many $α\in \mathcal{O}$, the ideal $B_α$ has a primitive divisor when $P$ is a non-torsion point and there exist two endomorphisms $g\neq 0$ and $f$ so that $f(P)=g(Q)$. This is a generalization of previous results on elliptic divisibility sequences.

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arXiv2021

A recurrence relation for elliptic divisibility sequences

In literature, there are two different definitions of elliptic divisibility sequences. The first one says that a sequence of integers $\{h_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it verifies the recurrence relation $h_{m+n}h_{m-n}h_{r}^2=h_{m+r}h_{m-r}h_{n}^2-h_{n+r}h_{n-r}h_{m}^2$ for every natural number $m\geq n\geq r$. The second definition says that a sequence of integers $\{\beta_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it is the sequence of the square roots (chosen with an appropriate sign) of the denominators of the abscissas of the iterates of a point on a rational elliptic curve. It is well-known that the two sequences are not equivalent. Hence, given a sequence of the denominators $\{\beta_n\}_{n\geq 0}$, in general does not hold $\beta_{m+n}\beta_{m-n}\beta_{r}^2=\beta_{m+r}\beta_{m-r}\beta_{n}^2-\beta_{n+r}\beta_{n-r}\beta_{m}^2$ for $m\geq n\geq r$. We will prove that the recurrence relation above holds for $\{\beta_n\}_{n\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$.

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arXiv2019

Primitive divisors of sequences associated to elliptic curves

Let $\{nP+Q\}_{n\geq0}$ be a sequence of points on an elliptic curve defined over a number field $K$. In this paper, we study the denominators of the $x$-coordinates of this sequence. We prove that, if $Q$ is a torsion point of prime order, then for $n$ large enough there always exists a primitive divisor. Later on, we show the link between the study of the primitive divisors and the Lang-Trotter conjecture.

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arXiv2018

On the $L$-polynomials of curves over finite fields

We discuss, in a non-Archimedean setting, the distribution of the coefficients of $L$-polynomials of curves of genus $g$ over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension $g+1$. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

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