Page 430 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 430
NUMBER THEORY (MS-63)
Uniformity for the Number of Rational Points on a Curve
Philipp Habegger, philipp.habegger@unibas.ch
University of Basel, Switzerland
Coauthors: Ziyang Gao, Vesselin Dimitrov
In 1983, Faltings proved the Mordell Conjecture: a smooth projective curve of genus at least 2
that is defined over a number field K has at most finitely many K-rational points. Several years
later Votja gave a new proof. Neither proof provides a procedure to determine the set of rational
points, they are ineffective. But the number of rational points can be bounded from above ef-
fectively with bounds given by Bombieri, David-Philippon, de Diego, Parshin, Rémond, Vojta,
and others. I discuss a result where we show that the number of K-rational points is bounded
from above as a function of K, the genus, and the rank of the Mordell-Weil group of the curve’s
Jacobian. This is joint work with Vesselin Dimitrov and Ziyang Gao and our proof is based on
Vojta’s approach. Thanks to earlier work by other authors mentioned above, we may reduce to
bounding the number of points in a certain height range. For this we develop an inequality for
the Néron-Tate height in a family of abelian varieties and use a recent functional transcendence
result of Gao.
Bounding the Iwasawa invariants of Selmer groups
Sören Kleine, soeren.kleine@unibw.de
Universität der Bundeswehr München, Germany
Let p be a rational prime. After recalling some basic notation from Iwasawa theory, we study
the growth of p-primary Selmer groups of abelian varieties with good and ordinary reduction at
p in Zp-extensions of a fixed number field K. Proving that in many situations the knowledge of
the Selmer groups in a sufficiently large number of finite layers of a Zp-extension of K suffices
for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different
Zp-extensions of K.
Zeros of Fekete polynomials
Marc Munsch, munsch@math.tugraz.at
TU Graz, Austria
The study of the location of zeros of polynomials with coefficients constrained in different sets
has a very rich history. The case of random polynomials has been studied intensively and the
asymptotic number of real zeros has been computed in various cases (Gaussian, Bernoulli etc).
We investigate related questions in the deterministic family of Fekete polynomials. These are
constructed with coefficients being Legendre symbols and are related to the study of zeros of
real Dirichlet L- functions. We discuss previous results, conjectures and the progress we made
towards the understanding of real zeros in this family of polynomials.
This is a joint work with O. Klurman and Y. Lamzouri.
428
Uniformity for the Number of Rational Points on a Curve
Philipp Habegger, philipp.habegger@unibas.ch
University of Basel, Switzerland
Coauthors: Ziyang Gao, Vesselin Dimitrov
In 1983, Faltings proved the Mordell Conjecture: a smooth projective curve of genus at least 2
that is defined over a number field K has at most finitely many K-rational points. Several years
later Votja gave a new proof. Neither proof provides a procedure to determine the set of rational
points, they are ineffective. But the number of rational points can be bounded from above ef-
fectively with bounds given by Bombieri, David-Philippon, de Diego, Parshin, Rémond, Vojta,
and others. I discuss a result where we show that the number of K-rational points is bounded
from above as a function of K, the genus, and the rank of the Mordell-Weil group of the curve’s
Jacobian. This is joint work with Vesselin Dimitrov and Ziyang Gao and our proof is based on
Vojta’s approach. Thanks to earlier work by other authors mentioned above, we may reduce to
bounding the number of points in a certain height range. For this we develop an inequality for
the Néron-Tate height in a family of abelian varieties and use a recent functional transcendence
result of Gao.
Bounding the Iwasawa invariants of Selmer groups
Sören Kleine, soeren.kleine@unibw.de
Universität der Bundeswehr München, Germany
Let p be a rational prime. After recalling some basic notation from Iwasawa theory, we study
the growth of p-primary Selmer groups of abelian varieties with good and ordinary reduction at
p in Zp-extensions of a fixed number field K. Proving that in many situations the knowledge of
the Selmer groups in a sufficiently large number of finite layers of a Zp-extension of K suffices
for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different
Zp-extensions of K.
Zeros of Fekete polynomials
Marc Munsch, munsch@math.tugraz.at
TU Graz, Austria
The study of the location of zeros of polynomials with coefficients constrained in different sets
has a very rich history. The case of random polynomials has been studied intensively and the
asymptotic number of real zeros has been computed in various cases (Gaussian, Bernoulli etc).
We investigate related questions in the deterministic family of Fekete polynomials. These are
constructed with coefficients being Legendre symbols and are related to the study of zeros of
real Dirichlet L- functions. We discuss previous results, conjectures and the progress we made
towards the understanding of real zeros in this family of polynomials.
This is a joint work with O. Klurman and Y. Lamzouri.
428
