arXiv subjects

Francesco Campagna

Publications and source records attributed to Francesco Campagna.

10 recordsLinked to original sources

arXiv2026

Arithmetic unlikely intersections in powers of the multiplicative group

Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its ``size" should not be too big compared to the ``complexity" of the subgroup scheme. We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-M\'erai-Ostafe-Sha. We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.

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arXiv2026

Semistable abelian varieties over $\mathbb{Q}$ with bad reduction at 19 only: an overview of the Fontaine--Schoof strategy

In this paper we provide an overview of a strategy pioneered by Fontaine and heavily refined by Schoof to classify abelian varieties with prescribed bad reduction. Throughout the overview, we prove various non-trivial background results turning it into an introduction for readers unacquainted with this topic. With the overview completed, we provide explicit examples of the strategy in action. At first we give introductory examples, classifying semistable abelian varieties over $\mathbb{Q}$ with bad reduction at exactly one of 3 or 5 up to isogeny over $\mathbb{Q}$. We then move onto a harder example, proving the analogous result for 19, which is new.

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arXiv2024

Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the André-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.

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arXiv2022

Cyclic reduction densities for elliptic curves

For an elliptic curve $E$ defined over a number field $K$, the heuristic density of the set of primes of $K$ for which $E$ has cyclic reduction is given by an inclusion-exclusion sum $δ_{E/K}$ involving the degrees of the $m$-division fields $K_m$ of $E$ over $K$. This density can be proved to be correct under assumption of GRH. For $E$ without complex multiplication (CM), we show that $δ_{E/K}$ is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of $E$ and a universal infinite Artin-type product. For $E$ admitting CM over $K$ by a quadratic order ${\mathcal{O}}$, we show that $δ_{E/K}$ admits a similar `factorization' in which the Artin type product also depends on ${\mathcal{O}}$. For $E$ admitting CM over $\bar K$ by an order ${\mathcal{O}}\not\subset K$, which occurs for $K={\bf Q}$, the entanglement of division fields over $K$ is non-finite. In this case we write $δ_{E/K}$ as the sum of two contributions coming from the primes of $K$ that are split and inert in ${\mathcal{O}}$. The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities.

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arXiv2022

Ordinary isogeny graphs over $\mathbb{F}_p$: the inverse volcano problem

We give a detailed presentation of $\ell$-isogeny graphs associated with ordinary elliptic curves defined over $\mathbb{F}_p$. We then focus on the following inverse problem: given an abstract volcano $V$, do there always exist primes $\ell, p \in \mathbb{N}$ such that the ordinary $\ell$-isogeny graph over $\mathbb{F}_p$ contains $V$ as a connected component? We provide an affirmative answer to this question.

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arXiv2022

Effective bounds on differences of singular moduli that are S-units

Given a singular modulus $j_0$ and a set of rational primes $S$, we study the problem of effectively determining the set of singular moduli $j$ such that $j-j_0$ is an $S$-unit. For every $j_0 \neq 0$, we provide an effective way of finding this set for infinitely many choices of $S$. The same is true if $j_0=0$ and we assume the Generalized Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a "uniformity conjecture" for singular $S$-units.

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arXiv2022

Around the support problem for Hilbert class polynomials

Let $H_D(T)$ denote the Hilbert class polynomial of the imaginary quadratic order of discriminant $D$. We study the rate of growth of the greatest common divisor of $H_D(a)$ and $H_D(b)$ as $|D| \to \infty$ for $a$ and $b$ belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many $D$ every prime ideal dividing $H_D(a)$ also divides $H_D(b)$, what can we say about $a$ and $b$? If we replace $H_D(T)$ by $T^n-1$ and the Dedekind domain is a ring of $S$-integers in some number field, then these are classical questions that have been investigated by Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and Corrales-Rodrigáñez-Schoof.

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arXiv2022

How big is the image of the Galois representations attached to CM elliptic curves?

Using an analogue of Serre's open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve $E$ defined over a number field $F$ a natural number $\mathcal{I}(E/F)$ which describes how big the image of the Galois representation associated to $E$ is. We show how one can compute $\mathcal{I}(E/F)$, using a closed formula that we obtain from the classical theory of complex multiplication.

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arXiv2021

Entanglement in the family of division fields of elliptic curves with complex multiplication

For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F = K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglement in the family of division fields of $E$.

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arXiv2020

On singular moduli that are S-units

Recently Yu. Bilu, P. Habegger and L. Kühne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.

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