An explicit family of twists of the Carlitz module with unbounded analytic rank

Abstract

Grishkov and Logachev have asked (Problem 1.4 of their recent survey) whether, for a fixed finite field F_q, the analytic ranks of the twists of the Carlitz module are bounded. We answer this in the negative for every q>= 3. For q = p^s and n>= 1 put P_{q,n} = (theta^(q^n) - theta)^(q-2). We show that the directed graph of the matrix M(P_{q,n},1,k) of Grishkov-Logachev has a completely explicit set of circuits: they are pairwise disjoint, none of them meets an entry involving t, and they are in bijection with the orbits of E^n under cyclic rotation, where E is the set of l in [0,q-2] with p not dividing l+1. Consequently L(C_P, t, T) does not depend on t, it factors as a product of one term 1 - wt(nu) T^per(nu) per orbit, and the analytic rank equals the sum of p^(v_p(per(nu))) over the orbits of weight 1. When n is a power of p this equals (q/p)^n (p-1)^(n-1), which is unbounded. The argument also answers questions 3.6.1 to 3.6.4 of the survey for this family.