The Sato-Tate law for Drinfeld modules (Q2787969)

Let \(F\) be a global function field with finite field \(k\) of constants, \(q=\#(k)\), \(\infty\) a fixed place of \(F\), and \(A\) the Dedekind subring of elements of \(F\) regular off \(\infty\). Write \(F_{\infty}\) for the \(\infty\)-adic completion of \(F\). Let further \(L\) be a field with a structure of \(A\)-algebra and \(L[\tau]\) the twisted polynomial ring over \(L\) with commutation rule \(\tau a = a^q\tau\) for constants \(a\).NEWLINENEWLINEGiven a Drinfeld module \(\phi: A \longrightarrow L[\tau]\) of rank \(n \in \mathbb N\), the central division algebra \(D\) over \(F_{\infty}\) with invariant \(-\frac{1}{n}\) may be described geometrically through \(\phi\), as follows. Let \(L^{\text{ perf}}\) be the perfect hull of \(L\) and \(L^{\text{ perf}}((\tau^{-1}))\) the skew field of formal Laurent series in \(\tau^{-1}\) with coefficients in \(L^{\text{ perf}}\). There is a unique continuous extension, also labelled \(\phi\), to a ring homomorphism \(\phi: F_{\infty} \longrightarrow L^{\text{ perf}}((\tau^{-1}))\), and \(D = D_{\phi}\) may be realized as the centralizer of \(\phi(F_{\infty})\).NEWLINENEWLINENow suppose that \(L\) is finitely generated and of general \(A\)-characteristic (i.e., \(A \hookrightarrow L\)). Suppose further that \(\phi\) has no extra endomorphisms, so that \({\text{ End}}(\phi) = A\).NEWLINENEWLINEIn this important paper, the author constructs a natural group homomorphism \(\rho_{\infty}\) from the Weil group \(W_L\) of \(L\) to \(D_{\phi}^{\ast}\), the multiplicative group of \(D_{\phi}\), characterized through comparison identities on Frobenius elements. The central result is Theorem 1.1, which states that \(\rho_{\infty}(W_L)\) is an open subgroup of finite index in \(D_{\phi}^{\ast}\). A similar result (Theorem 1.2, which is more technical to state) holds if we suppress the condition \({\text{ End}}(\phi) = A\).NEWLINENEWLINEIn the introduction the author explains to what extent Theorems 1.1 and 1.2 may be seen as ``Sato-Tate laws'', and how one gets asymptotic properties of and bounds for the number of places \(x\) of \(L\) (\(L\) supposed to be a finite extension of \(F\)) with a given Frobenius trace, cf. Theorems 1.5 and 1.7. In the case where \(A\) is a polynomial ring \(\mathbb F_q[t]\) and the rank \(n\) of \(\phi\) equals 2, a necessary and sufficient condition for the surjectivity of \(\rho_{\infty}\) is given in Theorem 1.12.