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Counting isogeny classes of Drinfeld modules over finite fields via Frobenius distributions

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dc.contributor.authorBray, Amie M., author
dc.contributor.authorAchter, Jeffrey, advisor
dc.contributor.authorGillespie, Maria, committee member
dc.contributor.authorHulpke, Alexander, committee member
dc.contributor.authorPallickara, Shrideep, committee member
dc.contributor.authorPries, Rachel, committee member
dc.date.accessioned2024-05-27T10:32:48Z
dc.date.available2024-05-27T10:32:48Z
dc.date.issued2024
dc.description.abstractClassically, the size of an isogeny class of an elliptic curve -- or more generally, a principally polarized abelian variety -- over a finite field is given by a suitable class number. Gekeler expressed the size of an isogeny class of an elliptic curve over a prime field in terms of a product over all primes of local density functions. These local density functions are what one might expect given a random matrix heuristic. In his proof, Gekeler shows that the product of these factors gives the size of an isogeny class by appealing to class numbers of imaginary quadratic orders. Achter, Altug, Garcia, and Gordon generalized Gekeler's product formula to higher dimensional abelian varieties over prime power fields without the calculation of class numbers. Their proof uses the formula of Langlands and Kottwitz that expresses the size of an isogeny class in terms of adelic orbital integrals. This dissertation focuses on the function field analog of the same problem. Due to Laumon, one can express the size of an isogeny class of Drinfeld modules over finite fields via adelic orbital integrals. Meanwhile, Gekeler proved a product formula for rank two Drinfeld modules using a similar argument to that for elliptic curves. We generalize Gekeler's formula to higher rank Drinfeld modules by the direct comparison of Gekeler-style density functions with orbital integrals
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierBray_colostate_0053A_18221.pdf
dc.identifier.urihttps://hdl.handle.net/10217/238476
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2020-
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectfunction fields
dc.subjectTamagawa numbers
dc.subjectisogeny classes
dc.subjectDrinfeld modules
dc.titleCounting isogeny classes of Drinfeld modules over finite fields via Frobenius distributions
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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