Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most ...recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.
New progress on the zeta function Bombieri, Enrico
Proceedings of the National Academy of Sciences - PNAS,
06/2019, Letnik:
116, Številka:
23
Journal Article
We give here a full account of Markoff's celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey ...Cohn's method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers.
ARMAND BOREL Bombieri, Enrico
Proceedings of the American Philosophical Society,
09/2010, Letnik:
154, Številka:
3
Journal Article
Topology at its simplest is the study of geometric shapes under continuous deformations, namely without jumps or breaks; a Lie group (and I hope not to raise the disapproval of my mathematical ...colleagues in my gross oversimplification) can be seen as formed by continuous transformations of a highly symmetric object. ... everyone was happy.
Cet article est un exposé de plusieurs résultats sur la distribution des points algébriques sur les variétés algébriques. This is a survey paper on the distribution of algebraic points on algebraic ...varieties.
Designed Pseudo-Laplacians Bombieri, Enrico; Garrett, Paul
arXiv (Cornell University),
02/2020
Paper, Journal Article
Odprti dostop
We elaborate and make rigorous various speculations about the implications of spectral properties of self-adjoint operators on spaces of automorphic forms for location of zeros of \(L\)-functions. ...Some of these ideas arose in work of Colin de Verdière, Lax-Phillips, and Hejhal, from the late 1970s and early 1980s, not to mention semi-apocryphal attributions to Pólya and Hilbert. For example, given a complex quadratic extension \(k\) of \(\mathbb Q\), we give a natural self-adjoint extension of a restriction of the invariant Laplacian on the modular curve whose discrete spectrum, if any, consists of values \(s(s-1)\) for zeros \(s\) of \(\zeta_k(s)\). Unfortunately, there seems to be no reason for this discrete spectrum to be large. In fact, Montgomery's pair correlation, and the behavior of \(\zeta(1+it)\), imply that at most \(94\%\) of zeros of \(\zeta(s)\) can appear in this discrete spectrum. Less naively, some preliminary positive results about the dynamics of zeros do follow from these considerations.
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