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# Arakelov geometry and Diophantine geometry – Gardio

## GARDIO

Arakelov geometry and Diophantine geometry

### Development of effectives aspects of Diophantine geometry by means of Arakelov geometry

Since the end of the 19th century, the analogy between number fields<br />and function fields has played a crucial role in arithmetic geometry.<br />The interpretation of this analogy in the geometric framework has led to<br />the definition of arithmetic varieties over the ring of integers of a number<br />field. To make this analogy more satisfactory, it is important to consider <br />the Archimedean and ultrametric places of a number field in the same way. <br />The work of Arakelov in the '70s has initialized the comprehension of the role the archimedean embeddings<br />of a number field should play in order to compactify the arithmetic variety,<br />giving rise to the theory of Arakelov geometry. These ideas have<br />inspired many new results, including the proof by Faltings of the Mordell<br />conjecture.<br /><br />The slope theory of Bost belongs to Arakelov geometry. This theory<br />had a profound influence on Diophantine geometry, showing how to prove explicit<br />results in an intrinsic and elegant way. It has shed lights on some new <br />arithmetic invariants which are comparable to the successive minima of <br />Minkowski but more relevant in a geometric point of view. Based on<br />these results, the recent works of the French school have established<br />a new geometry of numbers, which we may call absolute, on any algebraic<br />extension of the rational field Q. Fruitful interactions of this<br />theory with other domains, such as number theory (Siegel's lemmas in <br />transcendence theory) or algebraic geometry (algebraicity of formal<br />varieties) have led to many applications in the study of<br />Diophantine problems. The birational arithmetic geometry, and in<br />particular the study of the arithmetic volume function of Hermitian<br />line bundles on projective arithmetic varieties, has also benefited<br />from these advances.

Birational geometry, arithmetic volume

Geometry of numbers, rigid adelic space, Arakelovian successive minima

Theory of linear forms in logarithms

1) Proof of an explicit version of Mordell's conjecture for quite general families of curves of increasing rank in the sqaure of an elliptic curve of Mordell-Weil of rank one.

2) Bring to light of relationships between special varieties in order to interpret the asymptotic behaviour of rational points of bounded height

3) Development of a new geometry of numbers

June 2017: Summer school (three weeks) at Grenoble entitled «Arakelov geometry and diophantine applications«

May 2018: Conference at CIRM «Diophantine geometry«

For conferences, see the webpage of the project

math.univ-bpclermont.fr/~gaudron/Gardio/gardio.html

After 18 months, members of the project published about thirty articles.

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 the rational field 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). To carry over this program, we plan to organize several workshops (at least one workshop per year), a summer school and an international conference at the end of the project. 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 and also through an internet website to increase the visibility of our activities.

## Project coordination

Eric Gaudron (Laboratoire de mathématiques)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

Université Blaise Pascal Laboratoire de mathématiques
IF - UJF Institut Fourier - Université Joseph Fourier Grenoble 1

Help of the ANR 182,728 euros
Beginning and duration of the scientific project: September 2014 - 48 Months