A New Perspective on Zeta Functions Under the Number Field Function Field Analogy
School Name
Heathwood Hall Episcopal School
Grade Level
10th Grade
Presentation Topic
Mathematics
Presentation Type
Non-Mentored
Oral Presentation Award
1st Place
Abstract
In this paper we present striking similarities between the zeta function of an elliptic differential operator and the Hasse-Weil Zeta Function, showing they both give rise to self-intersection numbers. This observation supports a more rigorous formulation of the function field analogy. Repercussions of this result on such a theory are discussed. Proofs are given relating the zeta function of an operator to the Selberg Zeta Function, which connects the Selberg Zeta Function to the Hasse-Weil Zeta Function. Finally, both functions are connected to Selberg's "relative trace formula". This connection lays the groundwork for a geometric theory of zeta functions as discussed in Brown (2009).
Recommended Citation
Adamo, Nico, "A New Perspective on Zeta Functions Under the Number Field Function Field Analogy" (2019). South Carolina Junior Academy of Science. 34.
https://scholarexchange.furman.edu/scjas/2019/all/34
Location
Founders Hall 140 B
Start Date
3-30-2019 9:00 AM
Presentation Format
Oral and Written
Group Project
No
A New Perspective on Zeta Functions Under the Number Field Function Field Analogy
Founders Hall 140 B
In this paper we present striking similarities between the zeta function of an elliptic differential operator and the Hasse-Weil Zeta Function, showing they both give rise to self-intersection numbers. This observation supports a more rigorous formulation of the function field analogy. Repercussions of this result on such a theory are discussed. Proofs are given relating the zeta function of an operator to the Selberg Zeta Function, which connects the Selberg Zeta Function to the Hasse-Weil Zeta Function. Finally, both functions are connected to Selberg's "relative trace formula". This connection lays the groundwork for a geometric theory of zeta functions as discussed in Brown (2009).
