Schefer, Gerold. Logarithmic equidistribution and other problems involving torsion points of the algebraic torus. 2024, Doctoral Thesis, University of Basel, Faculty of Science.
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Abstract
One main result of this thesis is the proof of a conjecture of Ih for a certain class of polynomials. Let $S$ be a finite set of primes. The conjecture states that, if $P$ is a Lautrent polynomial over a number field $K$ in $n$ variables, then the set of torsion points $\boldsymbol\zeta$ of the $n$-dimensional algebraic torus $\mathbb G_{m, \overline{K}}^n$, such that $P(\boldsymbol\zeta)$ is an $S$-unit is not dense in $\mathbb G_{m, K}^n$, unless the zero set of $P$ is a finite union of torsion cosets.
As main tools we use equidistribution theorems in the archimedean setting and for all places above primes in $S$. In the first case we use a recent result of Dimitrov and Habegger and with the same method we prove the analog statements in the $p$-adic case. Since the Theorem of Tate and Voloch is crucial here, we tried to understand how the size of the constant depends on the height of the coefficients if all of them are algebraic.
We also had a look at other problems of logarithmic equidistribution. A very specific one is considered for $\mathbb G_a$-extensions $G$ of elliptic curves which admit a maximal compact subgroup $\Gamma$. The function $f$ we look at is the logarithm of the absolute value of a coordinate of the embedding of $G$ into $\mathbb P^4$ found by Masser using results of Serre. We show that the average over the Galois orbit of a torsion point of $G$ the evaluation of $f$ tends to the integral over $\Gamma$ with respect to the Haar measure as the order of the torsion point tends to infinity.
Further we consider the function $\log|z-\kappa|$ on the complex plane, where $\kappa$ is an algebraic number on the unit circle which is not a root of unity. Instead of considering roots of unity, we work with strict sequences of algebraic numbers whose height tends to zero. The question is, whether the conclusion of Bilu's equidistribution theorem is true in this case. We can show that the answer is no using results about simultaneous approximation. The question has a natural $p$-adic analogue, which is proved in a similar way.
In another chapter we describe all sums of at most four roots of unity which add to an algebraic unit. The main trick is due to Dimitrov and uses Kronecker's theorem to translate the condition of being a unit to a necessary condition given by an equation.
Finally, we count torsion points in algebraic subvarieties of the algebraic torus. We estimate the growth rate of the number of torsion points of order bounded by $T$ as $T$ tends to infinity. There is a general bound which is sharp. If the algebraic subvariety does not contain a torsion coset of maximal dimension we get a power saving improvement of the general bound. The main tools are the theorems of Minkowski.
As main tools we use equidistribution theorems in the archimedean setting and for all places above primes in $S$. In the first case we use a recent result of Dimitrov and Habegger and with the same method we prove the analog statements in the $p$-adic case. Since the Theorem of Tate and Voloch is crucial here, we tried to understand how the size of the constant depends on the height of the coefficients if all of them are algebraic.
We also had a look at other problems of logarithmic equidistribution. A very specific one is considered for $\mathbb G_a$-extensions $G$ of elliptic curves which admit a maximal compact subgroup $\Gamma$. The function $f$ we look at is the logarithm of the absolute value of a coordinate of the embedding of $G$ into $\mathbb P^4$ found by Masser using results of Serre. We show that the average over the Galois orbit of a torsion point of $G$ the evaluation of $f$ tends to the integral over $\Gamma$ with respect to the Haar measure as the order of the torsion point tends to infinity.
Further we consider the function $\log|z-\kappa|$ on the complex plane, where $\kappa$ is an algebraic number on the unit circle which is not a root of unity. Instead of considering roots of unity, we work with strict sequences of algebraic numbers whose height tends to zero. The question is, whether the conclusion of Bilu's equidistribution theorem is true in this case. We can show that the answer is no using results about simultaneous approximation. The question has a natural $p$-adic analogue, which is proved in a similar way.
In another chapter we describe all sums of at most four roots of unity which add to an algebraic unit. The main trick is due to Dimitrov and uses Kronecker's theorem to translate the condition of being a unit to a necessary condition given by an equation.
Finally, we count torsion points in algebraic subvarieties of the algebraic torus. We estimate the growth rate of the number of torsion points of order bounded by $T$ as $T$ tends to infinity. There is a general bound which is sharp. If the algebraic subvariety does not contain a torsion coset of maximal dimension we get a power saving improvement of the general bound. The main tools are the theorems of Minkowski.
Advisors: | Habegger, Philipp |
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Committee Members: | Blanc, Jérémy and Sombra, Martín |
Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Zahlentheorie (Habegger) |
UniBasel Contributors: | Habegger, Philipp and Blanc, Jérémy |
Item Type: | Thesis |
Thesis Subtype: | Doctoral Thesis |
Thesis no: | 15588 |
Thesis status: | Complete |
Number of Pages: | 135 |
Language: | English |
Identification Number: |
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edoc DOI: | |
Last Modified: | 24 Jan 2025 05:30 |
Deposited On: | 23 Jan 2025 11:01 |
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