March 2026. Rational angles in plane lattices, at the university of Grenoble.
Generalising classical questions about regular polygons with vertices
on a plane lattice, we are interested in pairs of lattice points \(A, B\) such that the angle \(\widehat{AOB}\) is a rational multiple of
\(\pi\). This problem leads to
diophantine-trigonometric equations that in turn involve the study of
rational points on curves of genus 0,1,2,3,5. I will present the full
classification of plane lattices according to how many independent
rational angles they contain and in which configurations they appear.
This is a joint work with R. Dvornicich, D. Lombardo and U. Zannier
December 2025. On the disk of convergence of algebraic power series, during the December workshop in Number Theory, in Pisa.
In a recent work with U. Zannier we considered the disk of
convergence of a power series \(s(x)\)
representing an algebraic function of \(x\) and specifically the relation between
this disk and the branch points of the \(x\)-coordinate on the corresponding
algebraic curve. We focus especially on the \(p\)-adic case, answering some questions of
basic nature, seemingly absent from the existing literature. To
illustrate the issues, recall that in the complex case the open disk of
convergence cannot contain all the branch points of \(x\), except for the trivial case of a
rational function. In the p-adic case, we show that the analogous
assertion is not true in complete generality; but we also confirm it in
a number of cases, for instance under the assumption that \(p\) is not smaller than the degree of \(s(x)\) over the field of rational functions
of \(x\).
November 2025. The field of definition of the iterates of a rational function, during the conference Diophantine approximation and
transcendence, in Luminy.
In this joint work with Solomon Vishkautsan we study how the field of definition of a rational function changes under iteration. We provide a complete classification of polynomials with algebraic coefficients with the property that one of their iterates is defined over a smaller field than the polynomial itself. We show with families of examples originating from algebraic groups that this characterization does not hold for rational functions. Finally, we also classify fractional linear transformations with this property.
January 2025. The field of iterates of a rational function, at the university of Udine.
In this joint work with Solomon Vishkautsan we study how the field of definition of a rational function changes under iteration. We provide a complete classification of polynomials with the property that the field of definition of one of their iterates drops in degree (over a given base field). We show with families of examples that this characterization does not hold for rational functions. Finally, we also classify fractional linear transformations with this property.
August 2024. Rational angles in plane lattices, during the study week on arithmetics of the Scuola Normale Superiore, in Cetraro.
May 2024. Rational angles in plane lattices, at the university of Roma 3.
January 2024. Rational angles in plane lattices, during the 13th Swiss-French workshop in algebraic geometry, in Charmey.
September 2023. Angoli razionali tra punti in un reticolo piano, during the XXII congress of the Unione Matematica Italiana,
in Pisa.
January 2023. Rational angles in plane lattices: a complete classification, at the Scuola Normale Superiore, in Pisa.
October 2022. Angoli razionali tra punti in un reticolo piano, in the Algebra & Geometry seminar, in Genova.
July 2022. \(\beta\)-continued fractions over
quadratic number fields, during the study week on arithmetics of the Scuola Normale Superiore, in Cetraro.
May 2022. Rational angles between points in a plane lattice, during the Genova - Graz Number Theory Workshop, in Genova.
November 2021. Angoli razionali tra punti in un reticolo piano, at the Scuola Normale Superiore, in Pisa.
October 2020. Finiteness and periodicity of continued fractions over quadratic number fields, in the One World Numeration Seminar cycle, online.
We consider continued fractions with partial quotients in the ring of
integers of a quadratic number field \(K\); a particular example of these
continued fractions is the \(\beta\)-continued fraction introduced by
Bernat. We show that for any quadratic Perron number \(\beta\), the \(\beta\)-continued fraction expansion of
elements in \(\mathbb{Q}(\beta)\) is
either finite of eventually periodic. We also show that for certain four
quadratic Perron numbers \(\beta\), the
\(\beta\)-continued fraction represents
finitely all elements of the quadratic field \(\mathbb{Q}(\beta)\), thus answering
questions of Rosen and Bernat. Based on a joint work with Zuzana
Masáková and Tomáš Vávra.
July 2020. Rational angles between elements in a plane lattice, during the study week on arithmetic of the Scuola Normale
Superiore, in Cetraro.
October 2019. Rational points on explicit families of curves, at the Charles University of Prague.
October 2019. An effective criterion for periodicity of \(\ell\)-adic continued fractions, at the Czech Technical University of Prague.
The theory of continued fractions has been generalized to \(\ell\)-adic numbers by several authors and
presents many differences with respect to the real case. For example, in
the \(\ell\)-adic case, rational
numbers may have a periodic non-terminating expansion; moreover, for
quadratic irrational numbers, no analogue of Lagrange’s theorem holds,
and the problem of deciding whether the continued fraction expansion is
periodic was still open. In our paper (a joint work with Laura Capuano
and Umberto Zannier) we investigate the \(\ell\)-adic continued fraction expansions
of rationals and quadratic irrationals using the definition introduced
by Ruban. We give general explicit criteria to assess the periodicity of
the expansion in both the rational and the quadratic case.
October 2019. Finiteness of \(\beta\)-continued
fractions over quadratic number fields, during the 4th Number Theory Meeting, in Torino.
September 2019. An effective criterion for periodicity of \(\ell\)-adic continued fractions, during the GMT workshop, in Genova.
July 2019. Hyperelliptic continued fractions in the nonsingular case of genus zero, during the study week on arithmetic of the Scuola Normale
Superiore, in Cetraro.
October 2018. An effective criterion for periodicity of \(\ell\)-adic continued fractions, at the Université de Caen Normandie.
September 2018. Plane lattices whose elements form angles which are rational
multiples of \(\pi\), during the study week on arithmetic of the Scuola Normale
Superiore, in Cetraro.
July 2018. An effective criterion for periodicity of \(\ell\)-adic continued fractions, during the conference Dynamical Methods in Algebra, Geometry and
Topology, in Udine.
May 2018. An effective criterion for periodicity of \(\ell\)-adic continued fractions, during the conference Numeration 2018, in Paris.
August 2017. An effective criterion for periodicity of \(\ell\)-adic continued fractions, at the Universität Basel.
July 2017. An effective criterion for periodicity of \(\ell\)-adic continued fractions, in Cetraro.
July 2017. Rational points on curves in powers of elliptic curves, during the workshop Rational Points 2017 in Schney.
June 2017. Unlikely intersections and the effective Mordell–Lang conjecture, during the Journées Algophantiennes Bordelaises 2017 in
Bordeaux.
I will recall the link between the subject of Unlikely Intersections and the effective Mordell-Lang Conjecture. In this context, I will present an explicit height bound for rational points on some curves and show many concrete examples in which it is possible to apply this result and list all such points.
May 2017. Anomalous intersections and the effective Mordell-Lang conjecture, during the Basel-Dijon Seminar in Basel.
November 2016. The height of subvarieties, at the Universität Basel.
Heights are a fundamental tool in diophantine geometry. I will show how to extend the definition of the Weil height for points in the projective space to subvarieties of (multi)projective spaces and what properties can be proved about it.
September 2016. Rational points on some families of curves, during the conference Geometric and Analytic Number Theory,
at ETH Zürich.
February 2016. Rational points on explicit families of curves, during the conference Computational Aspects of Diophantine
Equations, in Salzburg.
January 2016. Rational points on explicit families of curves, at the Institut Fourier, in Grenoble.
January 2016. Rational points on explicit families of curves, during the 5th Swiss-French workshop on algebraic geometry,
in Charmey.
September 2015. Some explicit cases of the Torsion Anomalous Conjecture, during the Third Italian Number Theory Meeting, in Pisa.
March 2015. Torsion-anomalous Intersections, during the Winter seminar of the Darmstadt algebra group, in
Manigod.
Anomalous Intersections are a framework introduced by Bombieri,
Masser and Zannier, which comprises and generalises a vast body of
problems and conjectures in Arithmetic Geometry. Let \(V\) be a variety contained in a group
variety \(G\), which is usually taken
to be an abelian variety or a torus. When intersecting \(V\) with an algebraic subgroup \(B\), if the intersection \(V\cap B\) has a component of dimension
strictly greater than "expected", then such a component is said to be
torsion-anomalous. In analogy with many fundamental results in the
field, there are conjectures giving geometrical conditions for the
variety \(V\) to have only finitely
many (maximal) torsion-anomalous subvarieties. I will present partial
results in this direction. Joint work with S. Checcoli and E. Viada
October 2014. The first step in the direction of Bombieri-Vinogradov, at the Technische Universität Darmstadt.
July 2014. Formal groups, during the workshop Periods and heights of CM abelian
varieties, in Alpbach.
May 2014. Bounds for the greatest common divisor between values of certain exponential sequences, during the Workshop on diophantine problems in Graz.
February 2014. Torsion-anomalous Intersections, during the conference Unlikely Intersections in Luminy.
December 2013. Torsion-anomalous Intersections, during the workshop Heights in Diophantine Geometry, Group Theory
and Additive Combinatorics in Vienna.
November 2013. Torsion-anomalous Intersections, at Universität Salzburg.
October 2013. Torsion-anomalous Intersections, during the conference Méthodes diophantiennes, in
Besançon.
July 2013. Torsion-anomalous Intersections, at TU Graz.
June 2013. About Dirichlet's theorem on primes in arithmetic progressions, at Georg-August-Universität Göttingen.
March 2013. Torsion-anomalous Intersections, at Adam Mickiewicz University Poznań.
February 2013. Anomalous intersections and the effective Mordell-Lang conjecture, at Georg-August-Universität Göttingen.
January 2013. Anomalous intersections in diophantine geometry, at Georg-August-Universität Göttingen.
June 2012. A primer on anomalous intersections, at Georg-August-Universität Göttingen.
January 2012. Arithmetic geometry, heights, and anomalous intersections, at Georg-August-Universität Göttingen.
October 2011. Bounds on the number of points ``with small denominator'' on algebraic curves, at ETH Zürich.
September 2011. Punti interi quadratici su curve di genere 1 definite da una doppia equazione di Pell, during the XIX congress of the Unione Matematica Italiana in
Bologna.
May 2011. An effective version of Siegel's theorem on integral points in the function field case, at Universität Basel.
The finiteness of (\(S\)-)integral
points on an elliptic curve over a number field follows from Siegel’s
theorem, but Siegel’s methods do not allow to effectively compute those
points. Working on an elliptic curve defined over a function field, one
can obtain a similar finiteness result with explicit bounds on the
height of the \(S\)-integral points and
good dependence on places of \(S\).
This is done along the lines of the proof of Siegel’s result, using an
effective theorem by Julie Wang where in the number field case the
ineffective Roth’s theorem is used.
September 2010. Quadratic integral solutions to double Pell equations, during the workshop Approximation diophantienne et
transcendance in Luminy.
I will discuss quadratic integral points on an elliptic curve defined by a system of two Pell equations. It is known that all but finitely many such points belong to a finite number of families given by geometric maps. Exploiting the structure of the curve one can give a direct proof of this fact which allows to bound the number of exceptions.
June 2010. An effective Siegel theorem for function fields, at the Scuola Normale Superiore di Pisa.
December 2009. Graduate students Seminar, at the Scuola Normale Superiore di Pisa.