arXiv subjects
Publications and source records attributed to Carlo Pagano.
On a projective variety defined over a global field, any Brauer--Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.
We show that the naive adaptation of Malle's conjecture to fair counting functions is not true in general.
Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jorgensen and the first two authors. In this paper we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field $\mathbb{Q}(\sqrt{d})$ where $d<0$. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non-zero frieze patterns for any given height. We then show that apart from the cases $d\in \{-1,-2,-3,-7,-11\}$ all non-zero frieze patterns over the rings of integers $\mathcal{O}_d$ for $d<0$ have only integral entries and hence are known as (twisted) Conway-Coxeter frieze patterns.
This paper introduces a systematic approach towards the inverse problem for arboreal Galois representations of finite index attached to quadratic polynomials. Let $F$ be a field of characteristic $\neq 2$, $f\in F[x]$ be monic and quadratic and $ρ_f$ be the arboreal Galois representation associated to $f$, taking values in the group $Ω_{\infty}$ of automorphisms of the infinite binary tree. We give a complete description of the maximal closed subgroups of each closed subgroup of index at most two of $Ω_{\infty}$ in terms of linear relations modulo squares among certain universal functions evaluated in elements of the critical orbit of $f$. We use such description in order to derive necessary and sufficient criteria for the image of $ρ_f$ to be a given subgroup of index two of $Ω_\infty$. These depend exclusively on the arithmetic of the critical orbit of $f$. Afterwards, we prove that if $ϕ=x^2+t\in\mathbb Q(t)[x]$, then there exist exactly five distinct subgroups of index two of $Ω_{\infty}$ that can appear as images of $ρ_{ϕ_{t_0}}$ for infinitely many $t_0\in\mathbb Q$, where $ϕ_{t_0}$ is the specialized polynomial. We show that two of them appear infinitely often, and if Vojta's conjecture over $\mathbb Q$ holds true, then so do the remaining ones. Finally, we give an explicit description of the derived series of each subgroup of index two. Using this, we introduce a sequence of combinatorial invariants for subgroups of index two of $Ω_\infty$. With a delicate use of these invariants we are able to establish that such subgroups are pairwise non-isomorphic as topological groups, a result of independent interest. This implies, in particular, that the five aforementioned groups are pairwise distinct topological groups, and therefore yield five genuinely different instances of the infinite inverse Galois problem over $\mathbb Q$.
Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,α)$, where $f$ is a polynomial over a number field $K$ and $α\in K$, such that the dynamical Galois group of the pair $(f,α)$ is abelian. In this paper we focus on the case of unicritical polynomials $f$, and more general dynamical systems attached to sequences of unicritical polynomials. After having reduced the conjecture to the post-critically finite case, we establish it for all polynomials with periodic critical orbit, over any number field. We next establish the conjecture in full for all monic unicritical polynomials over any quadratic number field. Finally we show that for any given degree $d$ there exists a finite, explicit set of unicritical polynomials that depends only on $d$, such that if $f=ux^d+1$ is a unicritical polynomial over a number field $K$ that lies outside such exceptional set, then there are at most finitely many basepoints $α$ such that the dynamical Galois group of $(f,α)$ is abelian. To obtain these results, we exploit in multiple ways the group theory of the generic dynamical Galois group to force diophantine relations in dynamical quantities attached to $f$. These relations force in all cases, outside of the ones conjectured by Andrews--Petsche, a contradiction either with lower bounds on the heights in abelian extensions, in the style of Amoroso--Zannier, or with the computation of rational points on explicit curves, carried out with techniques from Balakrishnan--Tuitman and Siksek.
We prove the existence of two non-isomorphic number fields $K$ and $L$ such that the maximal two-step nilpotent quotients of their absolute Galois groups are isomorphic. In particular, one may take $K$ and $L$ to be any of the imaginary quadratic number fields of discriminant -11, -19, -43, -67, -163. Furthermore, we give an explicit combinatorial description of these Galois groups in terms of a generalization of the Rado graph. A critical ingredient in our proofs is the back-and-forth method from model theory.
We generalize a classical reciprocity law due to Rédei using our recently developed description of the $2$-torsion of class groups of multiquadratic fields. This result is then used to prove a variety of new reflection principles for class groups, one of which involves a symbol similar to the spin symbol as defined in work of Friedlander, Iwaniec, Mazur and Rubin. We combine these reflection principles with Smith's techniques to prove Stevenhagen's conjecture on the solubility of the negative Pell equation.
Let $f$ be a polynomial over a global field $K$. For each $α$ in $K$ and $N$ in $\mathbb{Z}_{\geq 0}$ denote by $K_N(f,α)$ the arboreal field $K(f^{-N}(α))$ and by $D_N(f,α)$ its degree over $K$. It is conjectured that $D_N(f,α)$ should grow as a double exponential function of $N$, unless $f$ is post-critically finite (PCF), in which case there are examples like $D_N(x^2,α) \leq 4^{N}$. There is evidence conditionally on Vojta's conjecture. However, before the present work, no unconditional non-trivial lower bound was known for post-critically infinite $f$. In the case $f$ is PCF, no non-trivial lower bound was known, not even under Vojta's conjecture. In this paper we give two simple methods that turn the finiteness of the critical orbit into an exploitable feature, also in the post-critically infinite case. First, assuming GRH for number fields, we establish for all PCF polynomials $f$ of degree at least $2$ and all $α$ outside of the critical orbits of $f$, the existence of a positive constant $c(f,α)$ such that $$D_N(f,α) \geq \text{exp}(c(f,α)N), $$ for all $N$, which is sharp up to, possibly, improve the constant $c(f,α)$. Second, we show unconditionally that if $f$ is post-critically infinite over any number field $K$ and unicritical, then, for each $α$ in $K$, there exists a positive constant $c(f,α)$ such that $$D_N(f,α) \geq \text{exp}(c(f,α)N), $$ for all $N$. The main input here is to work modulo a suitably chosen prime and use a construction available for PCF unicritical polynomials with periodic critical orbit.
We develop an abstract framework for studying the strong form of Malle's conjecture for nilpotent groups $G$ in their regular representation. This framework is then used to prove the strong form of Malle's conjecture for any nilpotent group $G$ such that all elements of order $p$ are central, where $p$ is the smallest prime divisor of $\# G$. We also give an upper bound for any nilpotent group $G$ tight up to logarithmic factors, and tight up to a constant factor in case all elements of order $p$ pairwise commute. Finally, we give a new heuristical argument supporting Malle's conjecture in the case of nilpotent groups in their regular representation.
Let $K$ be a multiquadratic extension of $\mathbb{Q}$ and let $\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors \cite{KP} gave a bound for $|\text{Cl}^{+}(K)[2]|$ only in terms of the degree of $K$ and the number of ramifying primes. In the present work we show that this bound is sharp in a wide number of cases. Furthermore, we extend this to ray class groups.
In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss's genus theory. In this paper we extend Gauss's work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith \cite{smith2} in his recent breakthrough on Goldfeld's conjecture and the Cohen--Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides an explicit description for the group $\text{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives a sharp upper bound for the size of $\text{Cl}^{+}(K)[2]$. For every positive integer $n$, we find infinitely many multi-quadratic number fields $K$ such that $[K:\mathbb{Q}]$ equals $2^n$ and $\text{Gal}(L/\mathbb{Q})$ is a universal expansion group. Such fields $K$ are obtained using Smith's notion of additive systems and their basic Ramsey-theoretic behavior.
In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-critically finite, and we combine our results with local class field theory to classify quadratic pairs over $\mathbb Q$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence towards Jones' finite index conjecture.
We give effective estimates for the $l^1$-distance between the corank distribution of $r \times r$ Rédei matrices and the measure predicted by the Cohen--Lenstra heuristics. To this end we pinpoint a class of stochastic processes, which we call $c$-transitioning. These stochastic processes are well approximated by Markov processes, and we give an effective ergodic theorem for such processes. With this tool we make effective a theorem of Gerth \cite{Gerth} that initiated the study of the Cohen--Lenstra heuristics for $p = 2$. Gerth's work triggered a series of developments that has recently culminated in the breakthrough of Smith \cite{Smith}. The present work will be used in upcoming work of the authors on further applications of Smith's ideas to the arithmetic of quadratic fields. To this end we extend our main result to several other families of matrix spaces that occur in the study of integral points on the equation $x^2 - dy^2 = l$ as $d$ varies.
We show that for $100\%$ of the odd, squarefree integers $n > 0$ the $4$-rank of $\text{Cl}(\mathbb{Q}(i, \sqrt{n}))$ is equal to $ω_3(n) - 1$, where $ω_3$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.
Using a recent breakthrough of Smith, we improve the results of Fouvry and Klüners on the solubility of the negative Pell equation. Let $\mathcal{D}$ denote the set of fundamental discriminants having no prime factors congruent to $3$ modulo $4$. Stevenhagen conjectured that the density of $D$ in $\mathcal{D}$ such that the negative Pell equation $x^2-Dy^2=-1$ is solvable with $x,y\in\mathbb{Z}$ is $58.1\%$, to the nearest tenth of a percent. By studying the distribution of the $8$-rank of narrow class groups $\mathrm{CL}^+(D)$ of $\mathbb{Q}(\sqrt{D})$, we prove that the infimum of this density is at least $53.8\%$.
Using a recent breakthrough of Alexander Smith, we prove that $l^\infty$-class groups of cyclic degree $l$ fields have the distribution conjectured by Gerth under GRH.
We show how to use the combinatorial notion of jump sets to parametrize the possible structures of the group of principal units of local fields, viewed as filtered modules. We establish a natural bijection between the set of jump sets and the orbit space of a $p$-adic group of filtered automorphisms acting on a free filtered module. This, together with a Markov process on Eisenstein polynomials, culminates into a mass-formula for unit filtrations. As a bonus the proof leads in many cases to explicit invariants of Eisenstein polynomials, yielding a link between the filtered structure of the unit group and ramification theory. Finally, with the basic theory of filtered modules developed here, we recover, with a more conceptual proof, a classification, due to Miki, of the possible sets of upper jumps of a wild character: these are all jump sets, with a set of exceptions explicitly prescribed by the jump set of the local field and the size of its residue field.