Carlo Pagano

Golod--Shafarevich for arithmetic surfaces

Timo Keller, Carlo Pagano

We construct Golod--Shafarevich towers of arithmetic surfaces over the integers. In particular, we obtain an arithmetic surface with a section and infinite geometric etale fundamen…

math.GR2026

High Rank and Multiplicity in Random and Perfect Profinite Groups

Carlo Pagano, Mark Shusterman

We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Saw…

$\Kab$ has the Bogomolov property for canonical heights

Andrea Ferraguti, Carlo Pagano

We show that for any number field $K$ and any rational map $f$ in $K (x)$ that is not conjugate to a power, (signed) Chebyshev or Lattès map, then $K^{ab}$ has the strong B…

Abelian dynamical Galois groups over global function fields

Andrea Ferraguti, Patrick Ingram, Carlo Pagano

We establish a function field analogue of a recent conjecture of Andrews--Petsche. Our main result characterizes abelian dynamical Galois groups to be precisely the isotrivial ones…

Chowla's non-vanishing conjecture over $F_{q} (T)$

Peter Koymans, Carlo Pagano, Mark Shusterman

Let $q \equiv 1 mod 8$ be a prime power. We prove that $L (\frac{1}{2}, χ) \neq = 0$ for 100% of the imaginary quadratic Dirichlet characters of $F_{q} (T)$ .

Hilbert's tenth problem for finitely generated rings

Peter Koymans, Carlo Pagano

This expository article covers the recent developments surrounding Hilbert's tenth problem for finitely generated rings. We start by recounting the history of Hilbert's tenth probl…

math.NT2025

Elliptic curves of rank one over number fields

Peter Koymans, Carlo Pagano

We prove that for every number field $K$ , there exist infinitely many elliptic curves $E$ over $K$ with rank exactly equal to 1.

Hilbert's tenth problem via additive combinatorics

Peter Koymans, Carlo Pagano

For all infinite rings $R$ that are finitely generated over $Z$ , we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic…

Diophantine stability and second order terms

Carlo Pagano, Efthymios Sofos

We establish a Galois-theoretic trichotomy governing Diophantine stability for genus $0$ curves. We use it to prove that the curve associated to the Hilbert symbol is Diophantine s…

Elliptic fibrations and $3 \cdot 2^{k}$

Peter Koymans, Carlo Pagano, Efthymios Sofos

We determine the order of magnitude for all exponential moments of the rank in a broad class of elliptic fibrations and for the $3 \cdot 2^{k}$ -torsion in the class group of quadrati…

Averages of multiplicative functions along equidistributed sequences

Stephanie Chan, Peter Koymans, Carlo Pagano +1

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

math.NT2024★ 1 cited

$6$ -torsion and integral points on quartic threefolds

Stephanie Chan, Peter Koymans, Carlo Pagano +1

We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine qu…

Malle's conjecture for fair counting functions

Peter Koymans, Carlo Pagano

We show that the naive adaptation of Malle's conjecture to fair counting functions is not true in general.

Frieze patterns over algebraic numbers

Michael Cuntz, Thorsten Holm, Carlo Pagano

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…

Abelian dynamical Galois groups for unicritical polynomials

Andrea Ferraguti, Carlo Pagano

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 Ga…

Two-step nilpotent extensions are not anabelian

Peter Koymans, Carlo Pagano

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 particu…

math.NT2022★ 2 cited

On Stevenhagen's conjecture

Peter Koymans, Carlo Pagano

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 use…

math.NT2021★ 1 cited

The size of arboreal images, I: exponential lower bounds for PCF and unicritical polynomials

Carlo Pagano

Let $f$ be a polynomial over a global field $K$ . For each $α$ in $K$ and $N$ in $Z_{\geq 0}$ denote by $K_{N} (f, α)$ the arboreal field $K (f^{- N} (α))$ and by $D_{N} (f, α)$ its d…

math.NT2021★ 1 cited

On Malle's conjecture for nilpotent groups, I

Peter Koymans, Carlo Pagano

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 t…

A sharp upper bound for the $2$ -torsion of class groups of multiquadratic fields

Peter Koymans, Carlo Pagano

Let $K$ be a multiquadratic extension of $Q$ and let $Cl^{+} (K)$ be its narrow class group. Recently, the authors \cite{KP} gave a bound for $∣ Cl^{+} (K) [2] ∣$ …

Effective convergence of coranks of random Rédei matrices

Peter Koymans, Carlo Pagano

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 thi…

math.NT2020★ 4 cited

Higher Rédei reciprocity and integral points on conics

Peter Koymans, Carlo Pagano

Fix an integer $l$ such that $∣ l ∣$ is a prime $3$ modulo $4$ . Let $d > 0$ be a squarefree integer and let $N_{d} (x, y)$ be the principal binary quadratic form of $\mathbb{Q}(\sqrt{d}…

Constraining images of quadratic arboreal representations

Andrea Ferraguti, Carlo Pagano

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-crit…