Advisor:  Dr. Jeff Achter

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

Title: Counting the Size of an Isogeny Class of Drinfeld Modules over Finite Fields via Frobenius Distribution

Abstract: Drinfeld modules are the function field analogue to elliptic curves. Two Drinfeld modules are isogenous if and only if they have the same characteristic polynomial of Frobenius. In 2008, Gekeler proved a product formula giving the size of an isogeny class of a rank two Drinfeld module over a “prime” field $L = \mathbb{F}_q[T]/\mathfrak{p}$ for some prime $\mathfrak{p}$ of the polynomial ring $\mathbb{F}_q[T]$ using a random matrix model. The factors in Gekeler’s formula are expressed in terms of how often a given two-by-two matrix over a finite ring has a given characteristic polynomial. The product formula is the result of relating the weighted cardinality of an isogeny class with class numbers of (the function field analog to) imaginary quadratic orders over the polynomial ring $\mathbb{F}_q[T]$ and expressing them through the analytic class number formula. This gives the size of an isogeny class in terms of an $L$-function, which through Euler expansion gives a product of local terms for each prime of $\mathbb{F}_q[T]$.

Gekeler then goes on to interpret each factor as a local density function to arrive at his final product formula. On the other hand, a key tool in the function field analog to Langlands’ program is that there is a formula (due to Drinfeld and Laumon) for the size of an isogeny class of any rank Drinfeld module over extensions of $L$ via ad\'{e}lic orbital integrals.

The purpose of this project is to recover and generalize Gekeler’s formula to Drinfeld modules of arbitrary rank over extensions of $L$ by direct comparison to orbital integrals. The steps are as follows:

1. Define local density functions for each prime of the polynomial

ring which generalize Gekeler’s factors to higher rank Drinfeld modules  2. Show that each factor can be expressed in terms of a geometric

orbital integral

3. Compare the geometric and canonical measures to arrive at a formula

in terms of canonical orbital integrals  4. Describe the relationship between standard and twisted orbital

integrals for the for the factor at the prime $\mathfrak{p}$  5. Show that the factor corresponding to the place at infinity in

Gekeler’s formula when multiplied by the factors arising in each

previous step recovers the constant factor of Laumon’s formula  6. Arrive at the final Gekeler-type product formula by gluing together

each step

So far, the author has completed steps one and two. In this preliminary exam, we will discuss the historic motivation of the project, discuss known results due to Achter, Altug, Garcia, and Gordon in the case of abelian varieties over finite fields, review the definition of Drinfeld modules and classification by isogeny, review Gekeler’s and Laumon’s formulas, and explicitly state current progress. Finally, we will describe the remaining work and the methods we expect to use.