AbstractThe theme of this thesis is to apply ultrametric analysis to classical problems in analytic number theory. This allows one to handle situations featuring high ramiﬁcation at ﬁnite places. While this strategy works in many cases, the main focus of this work is the supnorm problem for automorphic forms on GL2. Our treatment of the problem is spread over two main parts.
First, we have to develop the necessary local theory which splits into archimedean and p-adic cases. The results needed in the archimedean cases are mostly classical, but recalling them in some detail will provide some guidance and intuition for the nonarchimedean cases. The ultrametric situation is far less developed. Here we compute explicit expressions for the p-adic Whittaker function associated to a newform. These expressions are new in most cases and lead to tight bounds for the Whittaker function in question.
Second, we use the adelic framework and the theory of automorphic representations to put the local pieces together and treat the sup-norm of automorphic forms over number ﬁelds. We establish lower bounds far up in the cusp coming from the transition region, archimedean and non-archimedean, of the global Whittaker new vector. Furthermore, we prove hybrid upper bounds, in other words estimates that are explicit in all major aspects of the automorphic form under investigation. We allow a wide variety of representations at the archimedean places and make no restrictions at the ﬁnite one. In that sense we go beyond the existing work.
|Date of Award||26 Jun 2019|
|Supervisor||Andrew R Booker (Supervisor) & Abhishek Saha (Supervisor)|