Page 431 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 431
NUMBER THEORY (MS-63)
Regulators of elliptic curves over global fields
Fabien Pazuki, fpazuki@math.ku.dk
University of Copenhagen, Denmark
In a recent collaboration with Pascal Autissier and Marc Hindry, we prove that up to isomor-
phisms, there are at most finitely many elliptic curves defined over a fixed number field, with
Mordell-Weil rank and regulator bounded from above, and rank at least 4. We will explain how
to obtain an even stronger result in the case of elliptic curves defined over a function field of pos-
itive characteristic, in particular removing the conditions on the rank (while adding a necessary
assumption on the inseparability degree). This opens up interesting questions about surfaces.
Kummer theory for number fields
Antonella Perucca, antonella.perucca@uni.lu
University of Luxembourg, Luxembourg
Let K be a number field and let G be a finitely generated and torsion-free subgroup of K×
of rank r. I will present various results (which are joint work with Hörm√ann, Perissinotto,
Sgobba, and Tronto) concerning the cyclotomic-Kummer extensions K(ζN , n G) where n | N .
For example there is an explicit finite procedure to compute a positive integer C (depending
only on√G and K) such that the ratio between nr and the degree of the Kummer extension
K(ζN , n G)/K(ζN ) divides C. For some families of number fields I will also present concrete
strategies to compute all of the above degrees.
Schinzel’s Hypothesis (H) with probability 1 and random Diophantine
equations
Alexei Skorobogatov, a.skorobogatov@imperial.ac.uk
Imperial College London, United Kingdom
In a joint work with Efthymios Sofos it was proved that Schinzel’s Hypothesis (H) holds for
100% of polynomials of any fixed degree. In the talk I will discuss applications of this analytic
result to proving that among surfaces in specific families over Q, a positive proportion have
rational points. The main examples are diagonal conic bundles of any fixed degree over PQ1 and
generalised Châtelet equations.
GCD results on semiabelian varieties and a conjecture of Silverman
Amos Turchet, amos.turchet@uniroma3.it
Roma Tre University, Italy
Coauthors: Fabrizio Barroero, Laura Capuano
A divisibility sequence of polynomials is a sequence dn such that whenever m divides n one
has that dm divides dn. Results of Ailon and Rudnick, among others, have shown that pairs of
divisibility sequences corresponding to subgroups of the multiplicative group have only limited
429
Regulators of elliptic curves over global fields
Fabien Pazuki, fpazuki@math.ku.dk
University of Copenhagen, Denmark
In a recent collaboration with Pascal Autissier and Marc Hindry, we prove that up to isomor-
phisms, there are at most finitely many elliptic curves defined over a fixed number field, with
Mordell-Weil rank and regulator bounded from above, and rank at least 4. We will explain how
to obtain an even stronger result in the case of elliptic curves defined over a function field of pos-
itive characteristic, in particular removing the conditions on the rank (while adding a necessary
assumption on the inseparability degree). This opens up interesting questions about surfaces.
Kummer theory for number fields
Antonella Perucca, antonella.perucca@uni.lu
University of Luxembourg, Luxembourg
Let K be a number field and let G be a finitely generated and torsion-free subgroup of K×
of rank r. I will present various results (which are joint work with Hörm√ann, Perissinotto,
Sgobba, and Tronto) concerning the cyclotomic-Kummer extensions K(ζN , n G) where n | N .
For example there is an explicit finite procedure to compute a positive integer C (depending
only on√G and K) such that the ratio between nr and the degree of the Kummer extension
K(ζN , n G)/K(ζN ) divides C. For some families of number fields I will also present concrete
strategies to compute all of the above degrees.
Schinzel’s Hypothesis (H) with probability 1 and random Diophantine
equations
Alexei Skorobogatov, a.skorobogatov@imperial.ac.uk
Imperial College London, United Kingdom
In a joint work with Efthymios Sofos it was proved that Schinzel’s Hypothesis (H) holds for
100% of polynomials of any fixed degree. In the talk I will discuss applications of this analytic
result to proving that among surfaces in specific families over Q, a positive proportion have
rational points. The main examples are diagonal conic bundles of any fixed degree over PQ1 and
generalised Châtelet equations.
GCD results on semiabelian varieties and a conjecture of Silverman
Amos Turchet, amos.turchet@uniroma3.it
Roma Tre University, Italy
Coauthors: Fabrizio Barroero, Laura Capuano
A divisibility sequence of polynomials is a sequence dn such that whenever m divides n one
has that dm divides dn. Results of Ailon and Rudnick, among others, have shown that pairs of
divisibility sequences corresponding to subgroups of the multiplicative group have only limited
429
