# Program

Number Theory concerns the study of properties of the integers, rational numbers, and other structures that share similar features. It is a central branch of mathematics with a well-known feature: it is often the case that easy-to-state problems in number theory turn out to be exceedingly difficult (e.g. Fermat’s Last Theorem), and their study leads to groundbreaking discoveries in other fields of mathematics.
A fundamental theme in number theory concerns the study of integer and rational solutions to Diophantine equations. This topic originated at least 3,700 years ago (as documented in babylonian clay tablets) and it has evolved into the highly sophisticated field of Diophantine Geometry. There are deep and fruitful interactions between Diophantine Geometry and seemingly distant fields such as representation theory, algebraic geometry, topology, complex analysis, and mathematical logic, to mention a few. In recent years, these connections have led to a large number of new results and, specially, to the partial or complete resolution of important conjectures in the field.
While the study of rational solutions of diophantine equations initiated thousands of years ago, our knowledge on this subject has dramatically improved in recent years. Especially, we have witnessed spectacular progress in aspects such as height formulas and height bounds for algebraic points, automorphic methods, unlikely intersection problems, and non-abelian and p-adic approaches to algebraic degeneracy of rational points. All these groundbreaking advances in the study of rational and algebraic points in varieties will be the central theme of the semester program “Diophantine Geometry” at MSRI. The main purpose of this program is to bring together experts as well as enthusiastic young researchers to learn from each other, to initiate and continue collaborations, to update on recent breakthroughs, and to further advance the field by making progress on fundamental open problems and by developing further connections with other branches of mathematics. We trust that younger mathematicians will greatly contribute to the success of the program with their new ideas. It is our hope that this program will provide a unique opportunity for women and underrepresented groups to make outstanding contributions to the field, and we strongly encourage their participation.
Bibliography

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

Arithmetic geometry

Arakelov theory

unlikely intersections

rational points

integral points

CM points

Colmez’s conjecture

CM abelian varieties

Vojta’s conjecture

Grothendieck-Katz p-curvature conjecture

Faltings height

Chabauty’s method

non-abelian Chabauty

period mappings

Bombieri-Lang conjecture

Lang’s conjecture

Effectivity

Diophantine equations

**Primary Mathematics Subject Classification**

11G15 - Complex multiplication and moduli of abelian varieties [See also 14K22]

11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35]

14G40 - Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

**Secondary Mathematics Subject Classification**

February 02, 2023 - February 03, 2023 | Connections Workshop: Diophantine Geometry |

February 06, 2023 - February 10, 2023 | Introductory Workshop: Diophantine Geometry |

April 24, 2023 - April 28, 2023 | Degeneracy of Algebraic Points |