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Heights in Diophantine Geometry
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  • Cited by 18
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    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Ghioca, Dragos and Nguyen, Khoa 2018. A dynamical variant of the Pink–Zilber conjecture. Algebra & Number Theory, Vol. 12, Issue. 7, p. 1749.

    Grizzard, Robert and Gunther, Joseph 2017. Slicing the stars: counting algebraic numbers, integers, and units by degree and height. Algebra & Number Theory, Vol. 11, Issue. 6, p. 1385.

    Spatzier, Ralf and Yang, Lei 2017. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, Vol. 11, Issue. 03, p. 263.

    Binyamini, Gal and Novikov, Dmitry 2017. Wilkie's conjecture for restricted elementary functions. Annals of Mathematics, Vol. 186, Issue. 1, p. 237.

    D’Andrea, Carlos Narváez-Clauss, Marta and Sombra, Martín 2017. Quantitative equidistribution of Galois orbits of small points in the N-dimensional torus. Algebra & Number Theory, Vol. 11, Issue. 7, p. 1627.

    Hsia, Liang-Chung and Tucker, Thomas 2017. Greatest common divisors of iterates of polynomials. Algebra & Number Theory, Vol. 11, Issue. 6, p. 1437.

    Lee, Tim Weng 2017. Fundamental domains of arithmetic quotients of reductive groups over number fields. Pacific Journal of Mathematics, Vol. 290, Issue. 1, p. 139.

    Müller, Jan and Stoll, Michael 2016. Canonical heights on genus-2 Jacobians. Algebra & Number Theory, Vol. 10, Issue. 10, p. 2153.

    Habegger, Philipp 2015. Singular moduli that are algebraic units. Algebra & Number Theory, Vol. 9, Issue. 7, p. 1515.

    Watanabe, Takao 2014. Ryshkov domains of reductive algebraic groups. Pacific Journal of Mathematics, Vol. 270, Issue. 1, p. 237.

    Ghioca, Dragos Hsia, Liang-Chung and Tucker, Thomas 2013. Preperiodic points for families of polynomials. Algebra & Number Theory, Vol. 7, Issue. 3, p. 701.

    Kawaguchi, Shu 2013. Local and global canonical height functions for affine space regular automorphisms. Algebra & Number Theory, Vol. 7, Issue. 5, p. 1225.

    Pila, Jonathan 2011. O-minimality and the André-Oort conjecture for C^n. Annals of Mathematics, Vol. 173, Issue. 3, p. 1779.

    Breuillard, Emmanuel 2011. A height gap theorem for finite subsets of GL_d(Q) and nonamenable subgroups. Annals of Mathematics, Vol. 174, Issue. 2, p. 1057.

    Bourgain, Jean Gamburd, Alex and Sarnak, Peter 2010. Affine linear sieve, expanders, and sum-product. Inventiones mathematicae, Vol. 179, Issue. 3, p. 559.

    Templier, Nicolas 2010. On the sup-norm of Maass cusp forms of large level. Selecta Mathematica, Vol. 16, Issue. 3, p. 501.

    Berman, Robert and Boucksom, Sébastien 2010. Growth of balls of holomorphic sections and energy at equilibrium. Inventiones mathematicae, Vol. 181, Issue. 2, p. 337.

    Petsche, Clayton 2010. A criterion for weak convergence on Berkovich projective space. Mathematische Annalen, Vol. 348, Issue. 2, p. 449.

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Book description

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.

Reviews

'This monograph is a bridge between the classical theory and a modern approach via arithmetic geometry. The authors aim to 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.’

Source: L'enseignement mathematique

‘The quality of exposition is exemplary, which is not surprising, given the brilliant expository style of the elder author.’

Yuri Bilu Source: Mathematical Review

‘Bombieri and Gubler have written an excellent introduction to some exciting mathematics … written with an excellent combination of clarity and rigor, with the authors highlighting which parts can be skipped on a first reading and which parts are particularly important for later material. The book also contains a glossary of notation, a good index, and a nice bibliography collecting many of the primary sources in this field.’

Source: MAA Reviews

'…a fundamental and pioneering standard text in the field, which will undoubtedly serve as a basic source for the future development of number theory and arithmetic geometry as a whole.'

Werner Kleinert Source: Zentralblatt MATH

'… remarkable …'

Source: European Mathematical Society Newsletter

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