Davide Lombardo
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Davide Lombardo has a total of 34 co-authors. Here are the top co-authors by number of shared publications:
- Elisa Lorenzo García
Institut de Mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland
4 shared publications - Matteo Verzobio
Institute of Science and Technology Austria
4 shared publications - John Voight
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755; \normalfont and Department of Mathematics and Statistics, University of Sydney, NSW 2006, Sydney, Australia
3 shared publications - Christophe Ritzenthaler
Institut de Biologie Moleculaire des Plantes du CNRS (UPR2357) and Universite de Strabourg
2 shared publications - Edgar Costa
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA
2 shared publications - Jackson Salvatore Morrow
Univ Montreal, Ctr Rech Math, Montreal, PQ H3T 1J4, Canada
2 shared publications
Latest Publications
- Open Access
- 12 August 2026
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 133 (2) , e70193
On 7‐adic Galois representations for elliptic curves over 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 ‐adic Galois representations attached to elliptic curves over . Currently, the classification is only complete for . The main difficulty for other primes arises from the need to understand elliptic curves whose mod‐ 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 . Here, we consider the case and show that the modular curve , of genus 69, has no non‐CM rational points. To achieve this, we establish a correspondence between the rational points on and the primitive integer solutions of the generalised Fermat equation , 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.
- 12 August 2026
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 133 (2) , e70193
Open AccessOn 7‐adic Galois representations for elliptic curves over 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 ‐adic Galois representations attached to elliptic curves over . Currently, the classification is only complete for . The main difficulty for other primes arises from the need to understand elliptic curves whose mod‐ 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 . Here, we consider the case and show that the modular curve , of genus 69, has no non‐CM rational points. To achieve this, we establish a correspondence between the rational points on and the primitive integer solutions of the generalised Fermat equation , 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.
- 29 August 2025
- Published by Springer Nature in Mathematische Annalen
- Vol. 393 (3-4) , 3713-3771
Classification of rational angles in plane lattices II
- 29 August 2025
- Published by Springer Nature in Mathematische Annalen
- Vol. 393 (3-4) , 3713-3771
Classification of rational angles in plane lattices II
- Open Access
- 6 February 2025
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- p. 1-49
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.- 6 February 2025
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- p. 1-49
Open AccessOn 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. - Open Access
- 3 August 2024
- Published by Springer Nature in Bollettino dell'Unione Matematica Italiana
- Vol. 18 (1) , 273-291
Galois representations in arithmetic geometry
We survey some recent recent results whose proofs depend in an essential way on the study of Galois representations. We discuss in particular the scarcity of rational points on ramified covers of abelian varieties, the problem of algorithmically computing endomorphism rings of abelian varieties over number fields, and a version of Kummer theory for commutative algebraic groups.
- 3 August 2024
- Published by Springer Nature in Bollettino dell'Unione Matematica Italiana
- Vol. 18 (1) , 273-291
Open AccessGalois representations in arithmetic geometry
We survey some recent recent results whose proofs depend in an essential way on the study of Galois representations. We discuss in particular the scarcity of rational points on ramified covers of abelian varieties, the problem of algorithmically computing endomorphism rings of abelian varieties over number fields, and a version of Kummer theory for commutative algebraic groups.
- Open Access
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.Open AccessClassification 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. - Open Access
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 $\ell$.Open AccessOn 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 $\ell$. - Open Access
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 $\mathbb{F}_q$. This allows us to prove, among other things, 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.Open AccessOn 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 $\mathbb{F}_q$. This allows us to prove, among other things, 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. - Open Access
- 26 January 2024
- Published by Springer Nature in Selecta Mathematica
- Vol. 30 (2) , 1-68
On the local-global principle for isogenies of abelian surfaces
Let $$\ell $$ be a prime number. We classify the subgroups G of $${\text {Sp}}_4({\mathbb {F}}_\ell )$$ and $${\text {GSp}}_4({\mathbb {F}}_\ell )$$ 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 a 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 $$\ell $$ .- 26 January 2024
- Published by Springer Nature in Selecta Mathematica
- Vol. 30 (2) , 1-68
Open AccessOn the local-global principle for isogenies of abelian surfaces
Let $$\ell $$ be a prime number. We classify the subgroups G of $${\text {Sp}}_4({\mathbb {F}}_\ell )$$ and $${\text {GSp}}_4({\mathbb {F}}_\ell )$$ 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 a 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 $$\ell $$ . - 1 January 2024
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 20 (3) , 1251-1284
The semi-infinite cohomology of Weyl modules with two singular points
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2024
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 20 (3) , 1251-1284
The semi-infinite cohomology of Weyl modules with two singular points
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- Open Access
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.Open AccessMonodromy 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.