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, Volume:
116, Issue:
23
Journal Article
Peer reviewed
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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.
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This paper studies the distribution of zeros of certain Epstein zeta functions, associated to positive definite binary quadratic forms with class number 2, in the region of absolute convergence of ...their Dirichlet series. In particular, one obtains upper and lower bounds for the rate of approach of zeros to the boundary of the zero-free half-plane for such functions. The proof for the lower bound depends on Bohr's method for studying simultaneous diophantine approximations. The upper bound uses instead a deep result on the diophantine type of the ratio of the logarithms of two rational numbers. 2000 Mathematics Subject Classification: 11E45; 11M41.
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