Abstract

As a function field analogue of singular moduli for elliptic curves estimated by Gross-Zagier, we estimate the valuation at certain places of singular moduli for prime rank Drinfeld modules. Our estimation can be viewed as a generalization of rank-2 case proved by Dorman. This talk consists of three parts:

Firstly, we compare the valuation of singular moduli with the number of isomorphisms between “Drinfeld module with CM by the ring of integer of an imaginary extension over F_q(T)” and “a specific prime rank Drinfeld module with CM by a constant extension of the polynomial ring F_q[T]”. Due to the difference between the structure of moduli scheme for Drinfeld modules of rank > 2 and that for Drinfeld modules of rank 2 (which is similar to the elliptic curve case), this comparison can only result into an inequality relation.

Secondly, we reduce counting number of isomorphisms into counting number of certain endomorphism on a Drinfeld module whose reduced characteristic polynomial is of certain form. This reduction step makes the counting process more concrete and computable. Lastly, we compute some examples on singular moduli estimation for rank-3 Drinfeld modules.