arXiv subjects

Davide Lombardo

Publications and source records attributed to Davide Lombardo.

At least 19 recordsLinked to original sources

arXiv2026

Equationless quadratic Chabauty for non-split Cartan modular curves

The aim of this article is to describe an equationless method for determining the rational points on the non-split Cartan curve $X_{\rm{ns}}^+(N)$ of prime level $N \geqslant 13$. Instead of using a projective model for the modular curve, our method uses the moduli interpretation of the curve, namely we work directly with elliptic curves and Cartan level structures. To accomplish this, we use the geometric version of the quadratic Chabauty method. We show that this can be combined with algorithms for divisor arithmetic developed by Makdisi and Mascot so as to apply to modular curves. As an illustration, we rederive the set of rational points on the curve $X_{\rm{ns}}^+(13)$.

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arXiv2026

Finding the complement of an elliptic curve inside a Jacobian

This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $\pi:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $\pi':X\to E'$ independent of $\pi$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here.

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arXiv2026

Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the local Galois structure of the $p$-power torsion of $E$. We pay special attention to the case where $E$ has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered $(\varphi,\operatorname{Gal}(K/\mathbb{Q}_p))$-module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves $E/\mathbb{Q}$: when the mod $p$ representation of $E$ has non-split Cartan image and $E$ doesn't have CM, the $p$-adic image must be the full preimage of the normaliser of a mod $p^n$ non-split Cartan for some $n \geq 1$. As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the $j$-invariant.

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arXiv2025

Cohomology of varieties over the maximal Kummer extension of a number field

Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the Galois group of the maximal Kummer extension of $K$ in odd degrees. In particular, every abelian variety has finite torsion over the maximal Kummer extension. This improves results by R\"ossler and the second author as well as Murotani and Ozeki. We also show that finiteness of torsion of a given abelian variety over non-abelian solvable extensions of $K$ is not controlled by the Galois group of the extension.

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arXiv2025

Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$

Serre's uniformity question asks whether there exists a bound $N>0$ such that, for every non-CM elliptic curve $E$ over $\mathbb{Q}$ and every prime $p>N$, the residual Galois representation $ρ_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{Aut}(E[p])$ is surjective. The work of many authors has shown that, for $p>37$, this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup $C_{ns}^+(p)$. Zywina has further proved that, whenever $ρ_{E,p}$ is not surjective for $p>37$, its image is either $C_{ns}^+(p)$ or a certain subgroup $G(p)$ of $C_{ns}^+(p)$ of index $3$. Recently, Le Fourn and Lemos showed that the index-$3$ case cannot arise for $p>1.4 \cdot 10^7$. We strengthen this result by proving that the image of $ρ_{E, p}$ is not conjugate to $G(p)$ for any prime larger than $5$.

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arXiv2025

Connected monodromy fields of Jacobians with complex multiplication

We describe an algorithm to compute the minimal field of definition of the Tate classes on powers of a Jacobian $J$ with potential complex multiplication. This field arises as a natural invariant of the Galois representations attached to $J$. We also give closed formulas expressing the periods of anti-holomorphic differential forms on $J$ in terms of the periods of the holomorphic ones.

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arXiv2025

On 7-adic Galois representations for elliptic curves over $\mathbb{Q}$

In recent years, significant progress has been made on Mazur's Program B, with many authors beginning a systematic classification of all possible images of $p$-adic Galois representations attached to elliptic curves over $\mathbb{Q}$. Currently, the classification is only complete for $p \in \{2,3,13,17\}$. The main difficulty for other primes arises from the need to understand elliptic curves whose mod-$p^n$ Galois representations are contained in the normaliser of a non-split Cartan subgroup. Equivalently, this amounts to determining the rational points on the modular curves $X_{ns}^+(p^n)$. Here, we consider the case $p=7$ and show that the modular curve $X_{ns}^+(49)$, of genus 69, has no non-CM rational points. To achieve this, we establish a correspondence between the rational points on $X_{ns}^+(49)$ and the primitive integer solutions of the generalised Fermat equation $a^2 + 28b^3 = 27 c^7$, the resolution of which can be reduced to determining the rational points of several genus-three curves. Furthermore, we reduce the complete classification of $7$-adic images to the determination of the rational points of a single plane quartic.

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arXiv2025

Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. In this paper, we compute the Sato-Tate group of $J_m$. Currently, there is no general algorithm that computes this invariant. We also describe the Sato-Tate group of an abelian variety, generalizing existing results that apply only to non-degenerate varieties, and prove an extension of a well-known formula of Gross-Koblitz that relates values of the classical and $p$-adic gamma functions at rational arguments.

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arXiv2025

Explicit open image theorems for abelian varieties with trivial endomorphism ring

Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$. Assuming that the image $G_{\ell^\infty}$ of the natural Galois representation attached to the Tate module $T_\ell(A)$ is $\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all sufficiently large primes $\ell$, we provide a semi-effective bound $\ell_0(A/K)$ such that $G_{\ell^\infty}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all primes $\ell > \ell_0(A/K)$. The bound is given in terms of the Faltings height of $A$ and of the cardinality of the residue field at a suitably generic place of $K$. We also describe an algorithmic approach to obtain better bounds for abelian threefolds over $\mathbb{Q}$.

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arXiv2025

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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arXiv2024

Examples of effectivity for integral points on certain curves of genus 2

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively determined. Our method is based on the construction of degree-3 étale covers of such curves of genus 2 and the study of the torsion values of sections of certain doubly elliptic abelian schemes.

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arXiv2024

Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. We compute several interesting arithmetic invariants of $J_m$: its decomposition up to isogeny into simple abelian varieties, the minimal field $\mathbb{Q}(\operatorname{End}(J_m))$ over which its endomorphisms are defined, and its connected monodromy field $\mathbb{Q}(\varepsilon_{J_m})$. Currently, there is no general algorithm that computes the last invariant. For large enough values of $m$, the abelian varieties $J_m$ provide non-trivial examples of high-dimensional phenomena, such as degeneracy and the non-triviality of the extension $\mathbb{Q}(\varepsilon_{J_m})/\mathbb{Q}(\operatorname{End}(J_m))$.

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arXiv2024

Classification of rational angles in plane lattices II

This paper is a continuation of an earlier one, and completes a classification of the configurations of points in a plane lattice that determine angles that are rational multiples of $π$. We give a complete and explicit description of lattices according to which of these configurations can be found among their points.

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arXiv2024

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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arXiv2023

The semi-infinite cohomology of Weyl modules with two singular points

In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the Weyl module $\mathbb{V}^λ$ corresponding to a dominant weight $λ$. This object plays an important role in the theory. In arXiv:2012.01858, we introduced a possible analogue $\mathbb{V}^{λ,μ}$ of the Weyl module in the setting of opers with two singular points, and in the case of sl(2) we proved that it has the "correct" endomorphism ring. In this paper, we compute the semi-infinite cohomology of $\mathbb{V}^{λ,μ}$ and we show that it does not share some of the properties of the semi-infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this reason, we introduce a new module $\tilde{\mathbb{V}}^{λ,μ}$ which, in the case of sl(2), enjoys all the expected properties of a Weyl module.

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arXiv2023

Monodromy groups of Jacobians with definite quaternionic multiplication

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. We show that for all even $g \geq 4$, there exist infinitely many geometrically nonisogenous abelian varieties $A$ over $\mathbb{Q}$ of dimension $g$ where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of $A$. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication.

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arXiv2023

Torsion bounds for a fixed abelian variety and varying number field

Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We study the smallest real number $β_A$ such that for any finite extension $L/K$ and $\varepsilon>0$, we have $|A(L)_{\operatorname{tors}}| \leq C \cdot [L:K]^{β_A+\varepsilon}$, where the constant $C$ depends only on $A$ and $\varepsilon$ (and not $L$). Assuming the Mumford--Tate conjecture for $A$, we will show that $β_A$ agrees with the conjectured value of Hindry and Ratazzi. We also give a similar bound for the maximal order of a torsion point in $A(L)$.

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arXiv2023

Cohomology of groups acting on vector spaces over finite fields

Let ${\mathbf{F}}_q$ be the finite field with $q=p^m$ elements and $G$ be a subgroup of ${\rm{GL}}_n({\mathbf{F}}_q)$. A famous theorem of Nori published in 1987 states that there exists a (non-effective) constant $c(n)$, depending only on $n$, such that if $p>c(n)$ and $G$ acts semisimply on ${\mathbf{F}}_p^n$, then $H^1(G,{\mathbf{F}}_p^n)=0$. We solve the long-standing problem, also considered by Serre of giving an effective proof of Nori's Theorem. Our approach yields the optimal constant $c(n)=n+2$. We also prove a more general version of Nori's theorem, namely, that for all powers $q$ of $p$, if $G$ acts semisimply on ${\mathbf{F}}_q^n$ and $p>n+2$, then $H^1(G,{\mathbf{F}}_q^n)$ is trivial. We apply these results to refine a criterion, proved by Çiperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields.

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