Result: a sato tate law for drinfeld modules: A Sato-Tate law for Drinfeld modules

If \(E\) is an elliptic curve without complex multiplication over a global field \(L\), the Sato-Tate law describes conjecturally the distribution of the characteristic polynomials of the Frobenius \(\text{ Fr}_v\) acting on a suitable Tate module as \(v\) varies over all places of \(L\) where \(E\) has good reduction. It has been proved by Deligne in the case where \(L\) is a function field, but it is not proved for a single elliptic curve over a number field. The same question can be asked for Drinfeld modules. Let \(K\) be a global function field, \(\infty\) a fixed place of \(K\), and \(A\) the ring of elements of \(K\) that are integral outside \(\infty\). Moreover, let \(L\) be a finite extension of \(K\) and \(\phi\) a Drinfeld \(A\)-module of rank \(r\) on \(L\). For every place \(v\) of \(L\) where \(\phi\) has good reduction we obtain the Frobenius characteristic polynomial \[ P_v(X):=\det(1-X\text{ Fr}_v| T_{\ell}(\phi\bmod v)) =\sum_{i=0}^r c_{v,i}X^i\in A[X] \] with \(| c_{v,i}| _\infty\leq q_v^{\frac{i}{r}}\) where \(q_v\) is the cardinality of the residue field. Partial results, notably on the distribution of the coefficients \(c_{v,1}\) and \(c_{v,r}\), have been obtained by \textit{L.-C. Hsia} and \textit{J. Yu} [Compos. Math. 122, 261--280 (2000; Zbl 0965.11026)]. (Attention: Their polynomials are scaled differently.) The present paper formulates and proves a Sato-Tate law in this context. The main point (which the author attributes to Drinfeld) is the construction of a homomorphism \(\rho\) from a certain subgroup of \(\text{Gal}(\overline{L}/L)\) to \(D^\times\) where \(D\) is the central division algebra over \(K_\infty\) with Brauer invariant \(\frac{1}{r}\). This homomorphism is unramified at almost all places \(v\) and maps \(\text{ Fr}_v\) to an element of \(D\) whose characteristic polynomial over \(K_\infty\) is \(P_v(X)\). The distribution comes from the Haar measure on the profinite completion of the image of \(\rho\). The author also explains why such a construction cannot exist in the case of elliptic curves and how the results of Hsia and Yu follow from his. From these results (compare e.g. Theorem 6.1 in loc. cit.) it is also clear that the uniformly but inexplicitly described Sato-Tate law, when made explicit in a specific situation, will give different distributions, depending for example on whether \(r\) is divisible by the characteristic \(p\) or not.