Small Height and Infinite Nonabelian Extensions
This paper by P. Habegger explores the interplay between the absolute logarithmic Weil height and infinite nonabelian Galois extensions generated by torsion points of elliptic curves over the rationals. The main focus is on proving that such fields possess the Bogomolov property, implying that zero is isolated among their height values. The study also extends to the Néron-Tate height on these elliptic curves, leading to a comprehensive analysis of their structure.
Core Contributions and Theorems
Bogomolov Property for Nonabelian Extensions:
Habegger establishes that the field ( Q(E) ), generated by torsion points of an elliptic curve ( E ) over ( Q ), satisfies the Bogomolov property. This involves showing that any element in ( Q(E) ) either has height zero or is bounded below by a positive constant, contingent only on the elliptic curve ( E ). This property holds despite the infinite, nonabelian and wildly ramified nature of ( Q(E) ).
Lower Bound on Néron-Tate Height:
The paper also asserts an analogous theorem for the Néron-Tate height on elliptic curves, confirming a positive lower bound for non-torsion points within ( E(Q(E)) ). This gives affirmative answers to conjectures and questions posed by other researchers like Baker, highlighting intrinsic gaps in height values that support structural assessments of ( E(Q(E)) ).
Implications and Further Questions
The nonabelian nature of ( Q(E) ): Habegger provides a detailed group-theoretic argument to affirm that the Galois group ( Q(E)/Q ) is anabelian, as shown by its failure to contain significant abelian subgroups. This insight extends prior understanding, primarily related to abelian extensions and their properties.
Free Abelian Structure of ( E(Q(E)) ): The paper confirms that ( E(Q(E))/{E} ) forms a free abelian group of countable infinite rank. This connects the research with classical results about the rank of Mordell-Weil groups over finite fields and opens further discussions about the structural composition of such groups over infinite or nonabelian field extensions.
Open problems: Habegger concludes by posing several open questions regarding the explicit computation of the minimal constant ( \epsilon ) in terms of ( E ), the potential dependency of such bounds on broader classes of elliptic curves, and the extension of these results to other algebraic structures or field combinations.
Methodological Approach
The complexity of this study features deep interactions between number theory and algebraic geometry. Fundamental concepts and results employed include:
Theorem of Elkies and Serre: Utilized to select primes where the supersingular reduction of the elliptic curve and Galois properties assure the nonabelian nature of the field extensions.
Equidistribution Theorems: Both Bilu's Theorem for algebraic numbers and equidistribution results in the elliptic curves setting underpin crucial arguments regarding the distribution and isolation of height values.
Techniques from Local Class Field Theory and Lubin-Tate Theory: These are brilliantly intertwined to handle the infinitude and complex ramification that characterize the extension fields ( Q(E) ).
Conclusion
Philipp Habegger’s paper delivers a robust exploration of nonabelian extensions linked to elliptic curves, advancing both theoretical understanding and practical implications, especially in the dynamics of height functions. The results challenge existing limitations posed by abelian frameworks and contribute fresh perspectives to the ongoing discourse in number theory and algebraic dynamics. Future research as envisaged by the author might unlock broader applications and deeper insights into the intricate architecture of field extensions associated with elliptic curves.
