Dissertation
Converse theorems without root numbers
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2022
DOI: 10.17077/etd.006356
Abstract
A converse theorem characterizes automorphic representations in terms of analytic properties of associated L-functions. Converse theorems have historically played a key role in proving cases of modularity, and also to establish various instances of Langlands functoriality. Weil proved a converse theorem for modular forms for congruence subgroups of SL2(Z). In his work, he requires functional equations for L-functions twisted by a family of Dirichlet characters. In Weil's hypothesis, these functional equations must have precise values for the so called root numbers. Booker relaxes this condition by allowing arbitrary root numbers in the functional equations. This thesis explores to what extent can Booker's result be extended to an arbitrary global field. We answer the question in the positive for a rational function field. The method developed in this thesis should also be applicable to number fields. A result of this kind is of interest because computation of root numbers is in general a difficult problem. For instance, it is known that the root number in the functional equation for elliptic curves is +/-1, but computing it exactly is not easy. This work suggests that complete information about root numbers in functional equations may not be needed in converse theorems. This can be useful in showing L-functions are automorphic even if the theory of their ramified $\varepsilon$-factors is not well developed.
Details
- Title: Subtitle
- Converse theorems without root numbers
- Creators
- Shantanu Agarwal
- Contributors
- Muthu Krishnamurthy (Advisor)Frauke Bleher (Committee Member)Ionut Chifan (Committee Member)Miodrag Iovanov (Committee Member)Yangbo Ye (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2022
- DOI
- 10.17077/etd.006356
- Publisher
- University of Iowa
- Number of pages
- viii, 46 pages
- Copyright
- Copyright 2022 Shantanu Agarwal
- Language
- English
- Description bibliographic
- Includes bibliographical references (pages 45-46).
- Public Abstract (ETD)
- Automorphic forms are a generalization of trigonometric functions and elliptic functions, first discovered by Poincar´e. Rather than studying them individually, we can study spaces of automorphic forms along with certain group representations on such spaces. These are called automorphic representation. L-functions are meromorphic functions on the complex plane that can be associated to a large class of mathematical objects. In particular one can associate L-functions to automorphic representations. One of the goals of modern number theory is to show that L-functions have nice analytic properties. In particular, it is known that a large class of automorphic L-functions (i.e. L-functions associated to automorphic representations) have these nice analytic properties. A typical converse theorem proves that the other direction is true as well, i.e., given an L-function with nice analytic properties, it comes from an automorphic representation. Over the previous few decades, converse theorems have played a key role in an area of mathematics known as Langlands program, and several improvements have been made to converse theorems along the way. One way of improving a converse theorem comes in the form of weakening the hypothesis required when we say ‘nice analytic properties’. In this thesis we carry out one such improvement, in particular by not requiring a specific constant of proportionality in certain functional equations appearing in the hypothesis of converse theorems.
- Academic Unit
- Mathematics
- Record Identifier
- 9984270954102771
Metrics
15 File views/ downloads
56 Record Views