“Can one hear the shape of a drum?” This seemingly innocent question spawned a lot of research in the early 20th century. Even though the answer is “No, we can't”, we can hear the volume. This is known as Weyl's Law.
In a more modern context, we can use new methods to study similar questions. More precisely, we can study locally symmetric spaces and the algebra of invariant differential operators. Generalizing the above, we can incorporate Hecke operators and find asymptotic formulas for their traces.
We study this problem in a global context, namely if the underlying group is the group of adelic points of a quasi-split, simple reductive group.
Our main tool is the Arthur-Selberg trace formula. The spectral side is dealt with, utilizing a condition on the normalizing factors of certain intertwining operators. The geometric side is more complicated and needs a more refined analysis. Most importantly, the test functions need to be specifically crafted to ensure compact support on the one hand, and sufficiently strong estimates on the other. The resulting geometric side can be split according to the Bruhat decomposition and treated separately, using various methods from reduction theory to algebraic and analytic number theory.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:80648 |
| Date | 15 September 2022 |
| Creators | Eikemeier, Christoph |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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