Categories of abelian varieties over finite fields. II: Abelian varieties over \(\mathbb{F}_q\) and Morita equivalence (Q6584653)

Let \({\mathbb F}_{q}\) be the finite field of order \(q\) which is a power of a prime number \(p\), and denote by \(W_{q}\) the set of conjugacy classes of Weil \(q\)-numbers defined as algebraic integers whose all conjugate numbers have the absolute value \(\sqrt{q}\). Then by the classical theorem of Honda and Tate, the set of \({\mathbb F}_{q}\)-isogeny classes of simple abelian varieties over \({\mathbb F}_{q}\) is in a one-to-one correspondence with \(W_{q}\) by taking roots of the characteristic polynomials of the \(q\)-Frobenius endomorphisms. Therefore, if \(B_{\pi}\) is a simple abelian variety over \({\mathbb F}_{q}\) corresponding to \(\pi \in W_{q}\), then for an abelian variety \(X\) over \({\mathbb F}_{q}\), \N\[\N\mathrm{Hom}_{{\mathbb F}_{q}} (X, B_{\pi}) \otimes {\mathbb Q} \cong \mathrm{End}_{{\mathbb F}_{q}} (B_{\pi})^{\oplus n} \otimes {\mathbb Q}, \N\]\Nwhere \(n\) is the number of simple factors of \(X\) (up to \({\mathbb F}_{q}\)-isogeny) which are \({\mathbb F}_{q}\)-isogenous to \(B_{\pi}\).\N\NIn the present paper, as a categorification of an integral version of Honda-Tate theory, the authors construct a pro-ring \({\mathcal S}_{q}\) in the following way such that the category of abelian varieties over \({\mathbb F}_{q}\) can be expressed as a category of certain left \({\mathcal S}_{q}\)-modules. For a finite subset \(w\) of \(W_{q}\), they show that there exists an abelian variety \(A_{w}\) over \({\mathbb F}_{q}\) which is \({\mathbb F}_{q}\)-isogenous to \(\prod_{\pi \in w} B_{\pi}^{m_{\pi}}\), where \N\[\Nm_{\pi} = \frac{\log_{p}(q) [{\mathbb Q}(\pi): {\mathbb Q}]}{\dim(B_{\pi})} \N\]\Nsuch that the \(l\)-adic \((l \neq p)\) Tate modules and the \(p\)-adic Dieudonné module of \(A_{w}\) are explicitly described in terms of \(w\). They also construct an ind-abelian variety \({\mathcal A} = (A_{w}, \varphi_{w, w'})\) over \({\mathbb F}_{q}\) consisting of such varieties \(A_{w}\)'s for all finite subsets \(w, w'\) of \(W_{q}\) satisfying \(w \subset w'\). Then the main result of the present paper states that by putting \({\mathcal S}_{q} =\mathrm{End}_{{\mathbb F}_{q}} ({\mathcal A})\), \(X \mapsto \mathrm{Hom}_{{\mathbb F}_{q}} (X, {\mathcal A})\) gives an anti-equivalence between the category of abelian varieties over \({\mathbb F}_{q}\) and the category of left \({\mathcal S}_{q}\)-modules which are free of finite rank over \({\mathbb Z}\). The case that \(q = p\) is the main result of the former paper by the authors [Algebra Number Theory 9, No. 1, 225--265 (2015; Zbl 1395.11102)].