I just received a hand-written letter from a very famous mathematician, who in his email used the notation $\widehat{\mathbb{Z}}(1)$. For the longest time I thought that he meant $\widehat{\mathbb{Z}}^\times$ (the group of units in the ring of profinite integers), which would have been very confusing to me (though you never know if it's because you interpreted his notation wrong or if there's something I didn't understand).
However, I have just realized that it's possible that maybe $\widehat{\mathbb{Z}}(1)$ is a special case of $\widehat{\mathbb{Z}}(m)$ which would maybe denote the maximal pro-$m$ quotient of $\widehat{\mathbb{Z}}$. In other words, it's possible that $\widehat{\mathbb{Z}}(1)$ is just the full profinite completion of $\mathbb{Z}$, so according to the notation I'm used to, maybe his $\widehat{\mathbb{Z}}(1)$ is just $\widehat{\mathbb{Z}}$.
Is this possible? Does anyone know of any references which uses the notation $\widehat{\mathbb{Z}}(1)$ in either of these ways?
Let G be the absolute Galois group of Q (= the Galois group over Q of an algebraic closure of Q). If W is the group of all the roots of 1 contained in this algebraic closure, then G acts on W via the so-called cyclotomic character Kappa, defined as follows. Decompose W into the direct sum of its p-components W_p, where W_p is the group of all the p-primary roots of 1 (p running through all primes). Since W_p is an inductive limit (=union here) of groups of p^n-th roots of 1, choose a coherent sequence w_n of such roots of 1, so that w_n=(w_n+1)^p. Then, for any g of G, for any n, g(w_n) is of the form (w_n)^a_n, where a_n is invertible mod p^n, and Kappa_p(g) is defined as the projective limit of the a_n. This Kappa_p is obviously a group homomorphism of G to Z_p*, the group of invertibles of the ring Z_p of p-adic integers, which is also the pro-p-completion of Z. If Z^ is the profinite completion of Z, then Kappa is the homomorphism from G to Z^ defined by its p-primary components Kappa_p. The 1-st Tate twist of a G-module M, denoted M(1), is then the new Z^G-module which is M as a set, but with a modified Galois action defined (in additive notation) by g*(m)=Kappa(g).g(m),where * (resp. .) denotes the new (resp. old) action.
Similarly, for any positive integer n, define the n-th twist M(n) by replacing Kappa by Kappa^n, and for n negative, M(n)=Hom(M(-n), Z^) .