Davide Lombardo’s research while affiliated with University of Pisa and other places

In this chapter we give two proofs of the fundamental density theorem of Chebotarev. The first directly generalises our proof of Dirichlet’s theorem. The second is more algebraic in nature and provides a nice illustration of the power of Frobenius elements in algebraic number theory.

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.

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 \'etale covers of such curves of genus 2 and the study of the torsion values of sections of certain doubly elliptic abelian schemes.

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.

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

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 $\mathbb{V}^{\lambda}$ corresponding to a dominant weight $\lambda$. This object plays an important role in the theory. In arXiv:2012.01858, we introduced a possible analogue $\mathbb{V}^{{\lambda},{\mu}}$ 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 $\mathbb{V}^{{\lambda},{\mu}}$ 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 $\tilde{\mathbb{V}}^{{\lambda},{\mu}}$ which, in the case of sl(2), enjoys all the expected properties of a Weyl module.

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields k of characteristic zero. For example, given a ramified cover $\pi : X \to A$ , where A is an abelian variety over k with a dense set of k -rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $\pi (X(k))$ is disjoint from C . Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.

We study local opers with two singularities for the case of the Lie algebra sl(2)$\mathfrak {sl}(2)$, and discuss their connection with a two-variables extension of the affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem describing the centre at the critical level, and an analogue of a result by Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules in terms of functions on the space of opers.