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 using a random matrix model. This result is motivated by an equidistribution assumption; it is what one would expect if matrices with the same characteristic polynomial as that of the isogeny class were random and evenly distributed in the group $Gl_2$. While not the case, Gekeler shows that the product of these factors does give 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 using the Langlands-Kottwitz method of describing the size of an isogeny class via adelic orbital integrals. This talk focuses on the function field analog to this problem. Due to Laumon, one can express the size of an isogeny class of Drinfeld modules over finite fields via adelic orbital integrals. Gekeler has proven a product formula for rank two Drinfeld modules using a similar argument to that for elliptic curves. In this talk, we discuss the generalization of Gekeler's formula to higher rank Drinfeld modules by the direct comparison of Gekeler-style density functions with orbital integrals.
Isogeny Classes of Drinfeld Modules over Finite Fields via Frobenius Distributions
Amie Bray, Colorado State UniversityAuthors: Amie Bray
2023 AWM Research Symposium
Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]