arXiv subjects
Publications and source records attributed to Davide Lombardo.
We carry out an arithmetical study of analytic functions $f: [0,1] \to [0,1]$ that by restriction induce a bijection $\mathbb{Q} \cap [0,1] \to \mathbb{Q} \cap [0,1]$. The existence of such functions shows that, unless $f(x)$ has some additional property of an algebraic nature, very little can be said about the distribution of rational points on its graph. Some more refined questions involving heights are also explored.
Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$ such that, for every prime $\ell>\ell_0(A,K)$, the image of the absolute Galois group of $K$ in $\operatorname{Aut}(T_\ell(A))$ is as large as it is allowed to be by endomorphisms and polarizations.
We answer a question raised by Hindry and Ratazzi concerning the intersection between cyclotomic extensions of a number field $K$ and extensions of $K$ generated by torsion points of an abelian variety over $K$. We prove that a weak version of the property $(μ)$ they introduced holds for any abelian variety, while the same is not true for a slightly stronger version of the property they also considered.
Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute Galois group of $K$ naturally attached to $E_1 \times \cdots \times E_n$. The main result is an explicit bound for the index of this image in $\left\{ (x_1,\ldots,x_n) \in \operatorname{GL}_2(\hat{\mathbb{Z}})^n \bigm\vert \det x_i = \det x_j \;\; \forall i,j \right\}$.
We prove a uniform version of a finiteness conjecture due to Rasmussen and Tamagawa in the case of CM abelian varieties. This extends recent results concerning CM elliptic curves to CM abelian varieties of arbitrary dimension.
We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).
For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of $E$ at the various primes. The bound is effective and only depends on $[K:\mathbb{Q}]$ and on the stable Faltings height of $E$. We also prove a result relating the structure of subgroups of $\operatorname{GL}_2(\mathbb{Z}_\ell)$ to certain Lie algebras naturally attached to them.
Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we also describe the full Galois group of $K(A_{\text{tors}})/K$. This makes explicit previous results of Serre and Ribet, and strengthens a theorem of Banaszak, Gajda and Krasoń. Our bounds are especially sharp in case the CM type of $A$ is nondegenerate.
We study open subgroups $G$ of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$ in terms of some associated Lie algebras without assuming that $G$ is a pro-$\ell$ group, thereby extending a theorem of Pink. The result has applications to the study of families of Galois representations.