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Carlo Pagano

Publications and source records attributed to Carlo Pagano.

At least 19 recordsLinked to original sources
arXiv2026

High Rank and Multiplicity in Random and Perfect Profinite Groups

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 Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu--Wood and Sawin--Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with $d$ generators and $d$ relations, and with no fewer relations on $d$ generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.

math.GR↗
arXiv2026

Abelian dynamical Galois groups over global function fields

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, whenever the degree of the polynomial is smaller than $p$, the characteristic of the field. The proof of this characterization is achieved in four independent steps as follows: $$\text{Abelian} \implies \text{Finite ramification} \implies \text{PCF map} \implies \text{Isotrivial map} \implies \text{Isotrivial pair}, $$ and works more generally for maps with a superattracting fixed point. We observe that this chain of implications is sharp in the degree: as soon as one reaches $p$ there are new non-isotrivial examples coming from Drinfeld modules. We propose a full conjectural classification in all degrees, taking into account all of the new exotic examples coming from Drinfeld modules and their associated Lattès maps.

math.NT↗
arXiv2026

$\Kab$ has the Bogomolov property for canonical heights

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 $K^{\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. We also classify the pairs $(f,α)$ whose backward orbit contains infinitely many abelian points. This settles the Andrews--Petsche conjecture to rational maps over a number field and to infinite abelian subsets of backward orbits. The authors were led to the main idea of the proof in conversation with \emph{Astra}. The key insight consists of applying the equidistribution results \cite{Yua08}, followed by a classification of $(f,f)$-preperiodic curves \cite{Pak23,Pak20, Bea25} followed by an additional application of equidistribution on a parametrized preperiodic curve.

math.NT↗
arXiv2026

Strongly Polynomial Time Complexity of Policy Iteration for $L_\infty$ Robust MDPs

Markov decision processes (MDPs) are a fundamental model in sequential decision making. Robust MDPs (RMDPs) extend this framework by allowing uncertainty in transition probabilities and optimizing against the worst-case realization of that uncertainty. In particular, $(s, a)$-rectangular RMDPs with $L_\infty$ uncertainty sets form a fundamental and expressive model: they subsume classical MDPs and turn-based stochastic games. We consider this model with discounted payoffs. The existence of polynomial and strongly-polynomial time algorithms is a fundamental problem for these optimization models. For MDPs, linear programming yields polynomial-time algorithms for any arbitrary discount factor, and the seminal work of Ye established strongly--polynomial time for a fixed discount factor. The generalization of such results to RMDPs has remained an important open problem. In this work, we show that a robust policy iteration algorithm runs in strongly-polynomial time for $(s, a)$-rectangular $L_\infty$ RMDPs with a constant (fixed) discount factor, resolving an important algorithmic question.

cs.AI↗
arXiv2026

Aletheia tackles FirstProof autonomously

We report the performance of Aletheia (Feng et al., 2026b), a mathematics research agent powered by Gemini 3 Deep Think, on the inaugural FirstProof challenge. Within the allowed timeframe of the challenge, Aletheia autonomously solved 6 problems (2, 5, 7, 8, 9, 10) out of 10 according to majority expert assessments; we note that experts were not unanimous on Problem 8 (only). For full transparency, we explain our interpretation of FirstProof and disclose details about our experiments as well as our evaluation. Raw prompts and outputs are available at https://github.com/google-deepmind/superhuman/tree/main/aletheia.

cs.AI↗
arXiv2026

Towards Autonomous Mathematics Research

Recent advances in foundational models have yielded reasoning systems capable of achieving a gold-medal standard at the International Mathematical Olympiad. The transition from competition-level problem-solving to professional research, however, requires navigating vast literature and constructing long-horizon proofs. In this work, we introduce Aletheia, a math research agent that iteratively generates, verifies, and revises solutions end-to-end in natural language. Specifically, Aletheia is powered by an advanced version of Gemini Deep Think for challenging reasoning problems, a novel inference-time scaling law that extends beyond Olympiad-level problems, and intensive tool use to navigate the complexities of mathematical research. We demonstrate the capability of Aletheia from Olympiad problems to PhD-level exercises and most notably, through several distinct milestones in AI-assisted mathematics research: (a) a research paper (Feng26) generated by AI without any human intervention in calculating certain structure constants in arithmetic geometry called eigenweights; (b) a research paper (LeeSeo26) demonstrating human-AI collaboration in proving bounds on systems of interacting particles called independent sets; and (c) an extensive semi-autonomous evaluation (Feng et al., 2026a) of 700 open problems on Bloom's Erdos Conjectures database, including autonomous solutions to four open questions. In order to help the public better understand the developments pertaining to AI and mathematics, we suggest quantifying standard levels of autonomy and novelty of AI-assisted results, as well as propose a novel concept of human-AI interaction cards for transparency. We conclude with reflections on human-AI collaboration in mathematics and share all prompts as well as model outputs at https://github.com/google-deepmind/superhuman/tree/main/aletheia.

cs.LG↗
arXiv2026

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

cs.AI↗
arXiv2026

Hilbert's tenth problem for finitely generated rings

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 problem over the integers, which was resolved negatively by Matiyasevich--Robinson--Davis--Putnam in 1970. In order to pass from $\mathbb{Z}$ to the finitely generated setting, we explain a criterion of Poonen that connects this to a problem in the theory of elliptic curves. Finally, we outline the main ideas behind the recent resolution of this elliptic curve problem by the authors.

math.NT↗
arXiv2025

Hilbert's tenth problem via additive combinatorics

For all infinite rings $R$ that are finitely generated over $\mathbb{Z}$, we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic curves $E$ without rank growth in certain quadratic extensions $L/K$. To achieve such a result unconditionally, our key innovation is to use elliptic curves $E$ with full rational $2$-torsion which allows us to combine techniques from additive combinatorics with $2$-descent.

math.NT↗
arXiv2025

A heuristic for ray class groups of quadratic number fields

We formulate a model for the average behaviour of ray class groups of real quadratic fields with respect to a fixed rational modulus, locally at a finite set $S$ of odd primes. To that end, we introduce Arakelov ray class groups of a number field, and postulate that, locally at $S$, the Arakelov ray class groups of real quadratic fields are distributed randomly with respect to a natural Cohen--Lenstra type probability measure. We show that our heuristics imply the Cohen--Lenstra heuristics on class groups of real quadratic fields, as well as equidistribution results on the fundamental unit of a real quadratic field modulo an integer, and are consistent with Varma's results on average of sizes of $3$-torsion subgroups of ray class groups of quardratic fields.

math.NT↗
arXiv2024

Diophantine stability and second order terms

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 stable with probability $1$. Our asymptotic formula for the second order term exhibits strong bias towards instability.

math.NT↗
arXiv2024

Elliptic fibrations and $3 \cdot 2^k$

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 quadratic fields.

math.NT↗
arXiv2024

Higher Rédei reciprocity and integral points on conics

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 $\mathbb{Q}(\sqrt{d})$. Building on a breakthrough of Alexander Smith, we give an asymptotic formula for the solubility of $N_d(x, y) = l$ in integers $x$ and $y$ as $d$ varies among squarefree integers divisible by $l$. As a corollary we give, in case $l > 0$, an asymptotic formula for the event that the Hasse Unit Index of the field $\mathbb{Q}(\sqrt{-l}, \sqrt{d})$ is $2$ as $d$ varies over all positive squarefree integers. We also improve the results of Fouvry and Klüners and recent results of Chan, Milovic and the authors on the solubility of the negative Pell equation. Our main new tool is a generalization of a classical reciprocity law due to Rédei.

math.NT↗