Presentation

Since the end of the 19th century, the analogy between number fields and function fields has played a crucial role in arithmetic geometry. The interpretation of this analogy in the geometric framework has led to the definition of arithmetic varieties over the ring of integers of a number field. To make this analogy more satisfactory, it is important to consider the Archimedean and ultrametric places of a number field in the same way. The work of Arakelov in the '70s has initialized the comprehension of the role the archimedean embeddings of a number field should play in order to compactify the arithmetic variety, giving rise to the theory of Arakelov geometry. These ideas have inspired many new results, including the proof by Faltings of the Mordell conjecture. The slope theory of Bost belongs to Arakelov geometry. This theory had a profound influence on Diophantine geometry, showing how to prove explicit results in an intrinsic and elegant way. It has shed lights on some new arithmetic invariants which are comparable to the successive minima of Minkowski but more relevant in a geometric point of view. Based on these results, the recent works of the French school have established a new geometry of numbers, which we may call absolute, on any algebraic extension of Q. Fruitful interactions of this theory with other domains, such as number theory (Siegel's lemmas in transcendence theory) or algebraic geometry (algebraicity of formal varieties) have led to many applications in the study of Diophantine problems. The birational arithmetic geometry, and in particular the study of the arithmetic volume function of Hermitian line bundles on projective arithmetic varieties, has also benefited from these advances. Several members of the project have already significantly contributed to these developments. Starting from the existing results, the aim of our project is to develop this absolute geometry of numbers, opening new directions of research (for example, making links with the theory of error correcting codes) and to explore further applications in Arakelov geometry (birational invariants, counting rational points) and Diophantine geometry (transcendence criteria, theory of linear forms in logarithms, Lehmer's problem). A particular attention will be paid to the participation of PhD students and young researchers as well as the diffusion of the results obtained in the project by communications in seminars and conferences.