Arithmetic equivalence of modular curves and Shimura varieties
Lead Research Organisation:
University of Reading
Department Name: Mathematics and Statistics
Abstract
This project is in number theory and algebraic geometry, and the interface at which these topics meet. At the centre of the project are Shimura varieties. Shimura varieties are geometric objects that encode a great deal of arithmetic and geometric information. They are parameter spaces for central objects in mathematics, namely, abelian varieties, and the setting for many important aspects of number theory, not least the Langlands Programme.
We will be interested in several natural questions pertaining the geometry and arithmetic of Shimura varieties.
The first of these questions relates to zeta functions of Shimura varieties. Zeta functions are analytic objects that can be attached to many different mathematical structures and encode a variety of properties. It is known that non-isomorphic Shimura varieties may possess the same zeta function, and explicit examples have been found. We intend to explore these examples and construct new families of non-isomorphic Shimura varieties sharing the same zeta function.
The second of these questions relates to so-called "unlikely intersections" in Shimura varieties. There is now a well-established area of research centred around the distribution of so-called special subvarieties in a Shimura variety, in other words, how smaller Shimura varieties sit inside a larger ambient Shimura variety. New results from model theory have given rise to new methods through which to investigate these questions, and we intend to apply these methods to unresolved problems of unlikely intersections that now appear tractable.
The third of these questions relates to geometric and arithmetic properties of Shimura varieties, many of which have relevance for the second question above. We intend to investigate various problems relating to the degrees of so-called Hecke correspondences, the fields of definition of special subvarieties, and the complexities of special subvarities, all of which are technical ingredients in unlikely intersections, but also interesting pieces of mathematics in their own right.
The methodology for studying the above questions is centred in the development and application of new tools, from model theory, number theory, and algebraic geometry, as well as pursuing new connections between different branches of mathematics. The project will require the student to learn a substantial amount of new mathematics, as well as state of the art methodologies in these specific areas of research, in order to enable him to obtain new results.
We will be interested in several natural questions pertaining the geometry and arithmetic of Shimura varieties.
The first of these questions relates to zeta functions of Shimura varieties. Zeta functions are analytic objects that can be attached to many different mathematical structures and encode a variety of properties. It is known that non-isomorphic Shimura varieties may possess the same zeta function, and explicit examples have been found. We intend to explore these examples and construct new families of non-isomorphic Shimura varieties sharing the same zeta function.
The second of these questions relates to so-called "unlikely intersections" in Shimura varieties. There is now a well-established area of research centred around the distribution of so-called special subvarieties in a Shimura variety, in other words, how smaller Shimura varieties sit inside a larger ambient Shimura variety. New results from model theory have given rise to new methods through which to investigate these questions, and we intend to apply these methods to unresolved problems of unlikely intersections that now appear tractable.
The third of these questions relates to geometric and arithmetic properties of Shimura varieties, many of which have relevance for the second question above. We intend to investigate various problems relating to the degrees of so-called Hecke correspondences, the fields of definition of special subvarieties, and the complexities of special subvarities, all of which are technical ingredients in unlikely intersections, but also interesting pieces of mathematics in their own right.
The methodology for studying the above questions is centred in the development and application of new tools, from model theory, number theory, and algebraic geometry, as well as pursuing new connections between different branches of mathematics. The project will require the student to learn a substantial amount of new mathematics, as well as state of the art methodologies in these specific areas of research, in order to enable him to obtain new results.
Organisations
People |
ORCID iD |
| Christopher Daw (Primary Supervisor) | |
| Zachary Youell (Student) |