GAUSSIAN LAWS ON DRINFELD MODULES

Abstract

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.