Explicit twists of the Carlitz module and of its tensor powers 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 = {l in [0,q-2] : p does not divide l+1}. Consequently L(C_{P_{q,n}},t,T) does not depend on t, 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 3.6.1-3.6.4 of the survey for this family. We then compute the L-function for arbitrary nonzero coefficients on supports of the same shape: taking all coefficients equal to -1 multiplies the rank by p-1, and the full support gives twists of degree (q-2)q^n and analytic rank (q-1)^n, the largest rank attainable on such supports. The same method applies to the tensor powers of the Carlitz module: for 1 <= e <= q-2 we obtain twists of the e-th tensor power of analytic rank (q-e)^n, and by a formula of Grishkov and Logachev the same holds for every exponent e = c q^j with 1 <= c <= q-2, with rank (q-c)^n; in particular these analytic ranks are unbounded as well.