Mathematics faculty and graduate students are invited to attend the PhD Defense of Amie Bray.

Join Zoom Meeting:  https://zoom.us/j/91425675584?pwd=Zms5bXNFR3U2YXgzKytFRXZVMnA1Zz09

Meeting ID: 914 2567 5584

Passcode: 581375

Advisor: Dr. Jeff Achter

Committee: Dr. Rachel Pries, Dr. Alexander Hulpke, Dr. Maria Gillespie, Dr. Shrideep Pallickara

Title: Counting Isogeny Classes of Drinfeld Modules over Finite Fields via Frobenius Distributions

Abstract:   Classically, 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.