Davide Lombardo
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View article: The intrinsic connectivity between the default mode and dorsal attention networks is an independent fMRI biomarker of Alzheimer's disease pathology burden
The intrinsic connectivity between the default mode and dorsal attention networks is an independent fMRI biomarker of Alzheimer's disease pathology burden Open
The mechanism of neurocognitive failure in Alzheimer's disease remains obscure. While the mainstream hypothesis in the field posits that brain tau pathology is the only process that drives cognitive decline in AD, other complementary mecha…
View article: On 7-adic Galois representations for elliptic curves over $\mathbb{Q}$
On 7-adic Galois representations for elliptic curves over $\mathbb{Q}$ Open
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 $\mathbb{Q}$. Curr…
View article: On the <i>L</i>-polynomials of curves over finite fields
On the <i>L</i>-polynomials of curves over finite fields Open
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 cha…
View article: Examples of effectivity for integral points on certain curves of genus 2
Examples of effectivity for integral points on certain curves of genus 2 Open
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 det…
View article: Galois representations in arithmetic geometry
Galois representations in arithmetic geometry Open
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 algorithm…
View article: Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians Open
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 …
View article: Classification of rational angles in plane lattices II
Classification of rational angles in plane lattices II Open
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 latt…
View article: The semi-infinite cohomology of Weyl modules with two singular points
The semi-infinite cohomology of Weyl modules with two singular points Open
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 . This object plays an important role in the theory. In a…
View article: Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$
Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$ Open
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}…
View article: Monodromy groups of Jacobians with definite quaternionic multiplication
Monodromy groups of Jacobians with definite quaternionic multiplication Open
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 …
View article: The semi-infinite cohomology of Weyl modules with two singular points
The semi-infinite cohomology of Weyl modules with two singular points Open
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 $λ$. Th…
View article: COM volume 158 issue 11 Cover and Back matter
COM volume 158 issue 11 Cover and Back matter Open
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View article: On the distribution of rational points on ramified covers of abelian varieties
On the distribution of rational points on ramified covers of abelian varieties Open
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 va…
View article: A family of quintic Thue equations via Skolem's $p$-adic method
A family of quintic Thue equations via Skolem's $p$-adic method Open
In this semi-expository article we solve the diophantine equation $m^5+(4 \cdot 5^4 b^4)mn^4 - n^5=1$ for all integers $b \neq 0$. This gives an example of a family of quintic Thue equations that can be solved completely by using nothing m…
View article: Torsion bounds for a fixed abelian variety and varying number field
Torsion bounds for a fixed abelian variety and varying number field Open
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 …
View article: On the local-global principle for isogenies of abelian surfaces
On the local-global principle for isogenies of abelian surfaces Open
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 $\ma…
View article: Effective Kummer Theory for Elliptic Curves
Effective Kummer Theory for Elliptic Curves Open
Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha $ be the set of $N$-division points of $\alpha $ in $E(\overline {K})$. We prove strong effective and u…
View article: Non-isogenous abelian varieties sharing the same division fields
Non-isogenous abelian varieties sharing the same division fields Open
Two abelian varieties $A_1, A_2$ over a number field $K$ are called strongly iso-Kummerian if the torsion fields $K(A_1[d])$ and $K(A_2[d])$ coincide for all $d \geq 1$. For all $g \geq 4$ we construct geometrically simple, strongly iso-Ku…
View article: Decomposing Jacobians Via Galois covers
Decomposing Jacobians Via Galois covers Open
Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closur…
View article: Some uniform bounds for elliptic curves over $\mathbb Q$
Some uniform bounds for elliptic curves over $\mathbb Q$ Open
We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois co…
View article: Decomposing Jacobians Via Galois covers
Decomposing Jacobians Via Galois covers Open
Let $phi : X \to Y$ be a (possibly ramified) cover, with $X$ and $Y$ of strictly positive genus. We develop tools to identify the Prym variety of $phi$, up to isogeny, as the Jacobian of a quotient curve $C$ of the Galois closure of a comp…
View article: Some uniform bounds for elliptic curves over Q
Some uniform bounds for elliptic curves over Q Open
We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider, in particular, the subgroup of scalars in the image of Galois, the first Galois co -homolo…
View article: Local opers with two singularities: the case of $\mathfrak{sl}(2)$
Local opers with two singularities: the case of $\mathfrak{sl}(2)$ Open
We study local opers with two singularities for the case of the Lie algebra 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 cen…
View article: Hilbert's irreducibility theorem for abelian varieties
Hilbert's irreducibility theorem for abelian varieties Open
We prove a version of Hilbert's irreducibility theorem for abelian varieties over finitely generated fields of characteristic zero, thereby generalizing earlier results of Zannier for powers of non-CM elliptic curves. Our work proves a con…
View article: On the distribution of rational points on ramified covers of abelian varieties
On the distribution of rational points on ramified covers of abelian varieties Open
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 $π: X \to A$, where $A$ is an abelian variet…