Geometrization of continuous characters of \(\mathbb Z_p^\times\) (Q1951184)

Let \(p\) and \(\ell\) be distinct primes and let \(q\) be a power of \(p.\) Let \(G\) be a connected commutative algebraic group over \({\mathbb F}_q.\) For any character \(\psi\, :\, G({\mathbb F}_q) \to \overline{\mathbb Q}^\times_\ell,\) one can push forward the central extension \(0 \longrightarrow G({\mathbb F}_q) \longrightarrow G \displaystyle\buildrel{\text{Frob}-1}\over \longrightarrow G \longrightarrow 0\) by \(\psi^{-1}\) to obtain a local system \({\mathfrak L}_\psi\) on \(G,\) such that the trace of Frobenius on \({\mathfrak L}_\psi\) equals \(\psi\). \textit{P. Deligne} [Seminaire de géométrie algébrique du Bois-Marie SGA 4 1/2 par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. Cohomologie étale. Lecture Notes in Mathematics. 569. Berlin-Heidelberg-New York: Springer-Verlag. (1977; Zbl 0345.00010)] used these local systems \({\mathfrak L}_\psi\) to prove bounds on the trigonometric sums over finite fields, using Grothendieck's trace formula as a key tool. An analogue of this trace formula is missing over \(p\)-adic fields. Here the authors show that Sekiguchi and Suwa's unification of Kummer and Artin-Schreier-Witt theories allows to define a \(p\)-adic trace which yields an isomorphism between the abelian group on rank-one local systems on \({\mathbb G}_{m, \overline{\mathbb Q}_p}\) whose orders divide \((p-1) p^n\) and the abelian group of characters of \({\mathbb Z}^\times_p\) of depth \(\leq n,\) for any \(n \in {\mathbb N}\). In the language of character sheaves, this means that the \(p\)-adic trace (defined above) of every \({\mathbb Q}_p (\mu_{p^\infty})\)-rational character sheaf on \({\mathbb G}_{m, \overline{\mathbb Q}_p}\) is a continuous character \({\mathbb Z}^\times_p \to {\mathbb Q}^\times_\ell\) and, moreover, every continuous \(\ell\)-adic character of \({\mathbb Z}^\times_p\) is obtained in this manner, each one from a unique sheaf of \({\mathbb G}_{m, \overline{\mathbb Q}_p}\). Let \(\ell\) and \(p\) be two distinct primes, \(p \not= 2.\) Let \({\mathbb B}_\infty = \bigcup_{n \geq 0} {\mathbb B}_n\) be the cyclotomic \({\mathbb Z}_p\)-extension of \({\mathbb Q}.\) The authors aim to study the triviality of the \(\ell\)-class group of \({\mathbb B}_\infty\). In a series of previous papers (see e.g. \textit{K. Horie} and \textit{M. Horie} [Tohoku Math. J. (2) 61, No. 4, 551--570 (2009; Zbl 1238.11101)]), mainly the case where \(\ell\) varies with \(p\) fixed was considered. Here they attack the opposite case (\(\ell\) fixed and \(p\) variable) and give sufficient conditions (too technical to be stated here) for the triviality of the \(\ell\)-class group of \({\mathbb B}_\infty,\) as well as for the non divisibility by \(\ell\) of the quotient \(h_n/h_{n-1}\) for a given \(n,\) where \(h_n\) denotes the class number of \({\mathbb B}_n.\) Among the main ingredients of the proofs are Leopoldt's reflection theorem on \(\ell\)-class groups together with some results previously obtained [loc. cit.] through algebraic study of the analytic class number formula.