arXiv subjects

Francesco Maria Saettone

Publications and source records attributed to Francesco Maria Saettone.

11 recordsLinked to original sources

arXiv2026

Procounting measures and the Bateman--Horn conjecture

Let $D$ be the ring of $S$-integers in a global field and $\widehat{D}$ its profinite completion. We propose a profinite version of the Bateman--Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations. Our approach is based on the new notion of procounting measure: a distribution on $\widehat{D}$ which should be seen as a profinite analogue of the counting function for a subset of $\mathbb{R}$. This allows us to deal with subsets of $\widehat{D}$ having Haar measure $0$ (corresponding to density zero in $\mathbb{R}$).

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arXiv2026

Ax-Schanuel for the Drinfeld $j$-function

We prove an analog of the Ax-Schanuel theorem for the Drinfeld $j$-function in odd characteristic. Roughly speaking, if the graph of $\boldsymbol{j}\colonΩ^n\rightarrow\mathbb{A}^n_{\mathbb{C}_\infty}$ and its derivatives has an atypical intersection $\mathcal{V}$ with an algebraic variety, then $\mathcal{V}$ projects to a weakly-special subvariety in $\mathbb{A}^n_{\mathbb{C}_\infty}$. More generally, we prove a positive characteristic analog of the differential Galois theoretic Ax-Schanuel theorem of Blázquez-Sanz, Casale, Freitag and Nagloo. Our main theorem for $\boldsymbol{j}$ follows by applying this result to a suitable foliation. A theorem of Pink allows us to conclude that the special varieties in the sense of Blázquez-Sanz et al. in this context agree with the classical weakly-special subvarieties in the Drinfeld sense.

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arXiv2026

The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves

We extend the Pila-Zannier strategy to Drinfeld modules: we prove analogues of the Manin-Mumford theorem for a product of two Drinfeld modules of equal rank, and of the André-Oort theorem for a product of two Drinfeld modular curves. In characteristic zero, several steps of this strategy rest on $o$-minimality, which has no counterpart over a function field; we replace the counting step by the rigid analytic Pila-Wilkie theorem of Binyamini-Kato, and this appears to be its first arithmetic application. The functional transcendence input, namely an analogue of the Ax-Lindemann theorem in both settings, is established here by an independent point counting argument.

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arXiv2026

The fundamental group of surfaces parametrizing cuboids

We prove that an irreducible projective complete intersection of dimension at least two with isolated singularities has trivial fundamental group. As an application, the surface $Υ$ parametrizing cuboids and its minimal resolution of singularities are simply connected. By an independent argument we also show that the surface $V$ parametrizing face cuboids and its resolution are simply connected as well. We then introduce two smooth open subvarieties $S_{1}$ and $S_{2}$ of the surface parametrizing face cuboids, show that each has fundamental group isomorphic to $\mathbb{F}_{3}\ltimes \mathbb{Z}^{2}$, and prove that their Malcev completions reduce to the free pro-unipotent group on three generators. In an appendix we treat the corresponding real loci, whose fundamental groups, in contrast, are far from trivial.

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arXiv2026

Duke for Drinfeld

We prove a function field analogue of Duke's equidistribution theorem for CM points, in the setting of Drinfeld--Stuhler modular curves. Our results thus extend, to the Drinfeld setting, both Duke's theorem on the modular curve and S.-W. Zhang's equidistribution in the case of Shimura curves. Equidistribution is reduced via a Weyl criterion to the decay of toric periods, which Waldspurger's formula expresses through central values of automorphic $L$-functions, bounded in Lindelöf-strength form by the Riemann Hypothesis over function fields. We work at arbitrary level structures and in every positive characteristic.

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arXiv2026

A note on equidistribution on a product of Shimura curves and André--Oort

In this short note we show that Galois orbits of CM points equidistribute on a product of $r\ge 2$ non-isomorphic Shimura curves by applying the adelic toral-packet equidistribution theorem of Aka--Luethi--Michel--Wieser. As a consequence, we deduce André--Oort for the product of those curves, previously studied by Edixhoven and Yafaev, replacing GRH by a Linnik-type splitting condition at two auxiliary primes.

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arXiv2026

Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants

Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach.

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arXiv2025

Equidistribution of CM points on Shimura Curves and ternary theta series

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field, considering both the split and the ramified case. The main novelty of the ramified case consists in the use of the moduli interpretation of the Cerednik--Drinfeld uniformisation. Our result is achieved by associating to the reduction of CM points certain Hilbert modular forms of weight $3/2$ and by analyzing their Fourier coefficients. Moreover, we also deduce the Shimura curves case of the integral version of the André--Oort conjecture.

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arXiv2025

Heights and transcendence of $p$--adic continued fractions

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin $p$--adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then, we focus on the heights of some $p$--adic numbers having a periodic $p$--adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with $p$--adic Roth-like results, in order to prove the transcendence of two families of $p$--adic continued fractions.

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arXiv2024

Equidistribution of CM points on a Shimura Curve modulo a ramified prime

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field attached to some ramified primes. To do so we study the reduction of CM points in the special fiber and we use Ratner's theorem to obtain the desired equidistribution.

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arXiv2022

Coset topologies on $\mathbb{Z}$ and arithmetic applications

We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on $\mathbb{Z}$, obtained from closed sets of the profinite completion $\hat{\mathbb{Z}}$. We also discuss various applications to number theory.

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