Complex multiplication and lifting problems (Q2871895)

The three authors of this book consider the at first sight straightforward question of whether an abelian variety over a finite field has a lift to characteristic zero that is a CM abelian variety (a CM lift). Immediately one realises that the problem needs to be more carefully formulated, and even a first attempt to do that reveals subtleties. Nevertheless, in 2006 the authors [\textit{C.-L. Chai} and \textit{F. Oort}, Pure Appl. Math. Q. 2, No. 1, 1--27 (2006; Zbl 1156.14320)] had given two versions ((I) and (IN), below) of the problem, and soon afterwards counterexamples were found to one of the versions. It eventually turned out that these counterexamples were in some way central to understanding the problems and, in many cases, answering them. This book explains how.NEWLINENEWLINEThe experts may begin at the beginning, but the inexpert (but necessarily skilled) reader is advised to turn first to page~87, near the end of Chapter~1, where the authors ask us to consider five (actually six) assertions concerning existence of a CM lift of an abelian variety \(B\) over a finite field \({\mathbb F}_q\) or of its extension of scalars \(B_\kappa\) to a finite extension \(\kappa:{\mathbb F}_q\). One should first assume, to avoid trivialities, that \(B\) is isotypic over \({\mathbb F}_q\), i.e.\ has up to isomorphism a unique simple factor. Then the assertions areNEWLINENEWLINE(CML) \textit{CM lifting}: There is a characteristic zero local domain \(R\) with residue field \({\mathbb F}_q\), and an abelian scheme \(A\) over \(R\) of relative dimension \(g=\dim B\), together with a CM field \(L\subset {\text{End}}^0(A)={\mathbb Q}\otimes_{\mathbb Z}{\text{End}}(A)\) of degree \(2g\) over \({\mathbb Q}\), and an isomorphism \(\phi: A_{{\mathbb F}_q}\to B\) of abelian varieties over \({\mathbb F}_q\).NEWLINENEWLINE(R) \textit{CM lifting after finite residue field extension}: \(R\) now has residue field \(\kappa\) and \(\phi: A_{\kappa} \to B\times_{\text{Spec}}{({\mathbb F}_q)}{\text{Spec}}(\kappa)\).NEWLINENEWLINE(I) \textit{CM lifting up to isogeny}: as (CML), but now \(\phi\) is an isogeny, not an isomorphism.NEWLINENEWLINE(IN) \textit{CM lifting to normal domains up to isogeny}: as (I), t but \(R\) is required to be a normal local domain.NEWLINENEWLINE(RIN) as (IN), but with the residue field of \(R\) being \(\kappa\) as in (R).NEWLINENEWLINE(sCML) \textit{Strong CM lifting}: for every CM field \(L\subset {\text{End}}^0(B)\) of degree \(2g\) with \({\mathcal O}_L\subset {\text{End}}(B)\), (CML) holds for \(B\) in such a way that \(L\subset {\text{End}}^0(A)\).NEWLINENEWLINEAlready in 1992 \textit{F. Oort} [Lect. Notes Math. 1566, 117--122 (1993; Zbl 0811.14025)] had shown that (R), and hence (CML), fails in general. In other words, an isogeny is necessary, and one should turn one's attention to (I) and (IN). The examples given there are of a restricted type, however, and a substantial part of this book -- Chapter~3, on CM lifting of \(p\)-divisible groups -- is partly motivated by the authors' desire to understand the true extent of this failure.NEWLINENEWLINEThe results of Chapter~3 are then used in Chapter~4 to show the main result of the book: that assertion (I), on the other hand, is always true. Thus, to get a CM lifting, allowing isogeny is enough but allowing field extensions is not. This also implies that (RIN) is true, because the normality condition can be achieved if one allows field extension, as the authors show.NEWLINENEWLINEOn the other hand, (IN) is false in general. In Chapter~2 of the book, specific examples are given, but more than that: the authors give necessary and sufficient conditions for a lifting in this sense to exist. The condition is quite subtle, in that it does not depend merely on \(B\), or even on \(B\) and \(L\).NEWLINENEWLINEThe essential notion is that of a CM type \((L,\Phi)\). This consists of a CM field \(L\) of degree \(2g\) over \({\mathbb Q}\), and a collection \(\Phi\) of \(g\) maps from \(L\) into some algebraically closed field \(K\) of characteristic zero, in this context usually \(\overline{{\mathbb Q}}_p\), which are inequivalent under the action of complex conjugation. In other words. \({\text{Hom}}(L,K)\) consists precisely of elements of \(\Phi\) and their composition with complex conjugation in \(L\). In this situation, the authors define a field \(E\subset \overline{\mathbb Q}\), called the reflex field, by the condition that \({\text{Gal}}(E,{\mathbb Q})\) is the subgroup of \({\text{Gal}}(\overline{\mathbb Q}/{\mathbb Q})\) that stabilises the set~\(\Phi\). The (main) obstruction to (IN) is then that if (IN) holds, the residue field \(\kappa_v\) of \({\mathcal O}_{E,v}\) for the \(p\)-adic valuation of \(E\subset \overline{\mathbb Q}_p\) must be realisable as a subfield of \({\mathbb F}_q\). This really can depend on \(\Phi\) as well as \(B\) and \(L\).NEWLINENEWLINEThe authors give, in Chapter~2, two classes of counterexample to (IN), one supersingular and the other absolutely simple. The second of these leads to the identification of the condition above as the essentially only obstruction to (IN). The first example is of less importance when one considers (IN) but proves illuminating in Chapter~4, when we are given the complete unconditional proof of (I).NEWLINENEWLINEIn fact, (I) is proved in a more precise form, with bounds on the isogeny degree needed. The basic example is a case of the ``toy model'', a class of CM abelian surfaces \(C\) over \(k\) for which \(L\) is a quartic field, \(p\) is inert in \(L\) and the subset \(\overline\Phi\) of \({\mathbb F}_p\)-homomorphisms from \(\kappa={\mathcal O}_L/p{\mathcal O}_L\cong{\mathbb F}_{p^4}\) that determine the action of \(\kappa\) on the \(\bar k\) tangent space to \(C\) is invariant under the unique involution of \(\kappa\). In fact the authors classify such toy models, but the real progress comes from looking at the corresponding local condition, i.e.\ for \(p\)-divisible groups. It turns out that (very roughly) the algebraisation problem that is the core difficulty in the proof of (I) can be reduced to the toy model case, after which it is sufficient to solve the \(p\)-divisible group version of the problem. That is precisely the problem that has been considered in Chapter~3. Both the reduction steps and the results for \(p\)-divisible groups are very delicate and many incidental results of independent interest are proved along the way.NEWLINENEWLINEBefore the lifting problems are stated in full, Chapter~1 gives a thorough and readable account of large parts of the theory of abelian varieties over finite fields in general, including for example a review of the basics of Dieudonné theory and \(p\)-divisible groups. Additionally, a substantial part of the book (almost a third) is given over to two appendices containing further results that are either required for the main part or sufficiently interesting digressions to merit the space (or both). These parts make the book into a valuable resource even for purposes not directly connected with CM lifting.NEWLINENEWLINEIn particular, in Appendix~A, Section~1 gives a proof of the \(\ell=p\) case of Tate's isogeny theorem over finite fields (that \({\text{Hom}}(A,B)\otimes {\mathbb Z}_\ell\simeq {\text{Hom}}(A[\ell^\infty],B[\ell^\infty])\)). The usual proof, due to Tate, Milne and Waterhouse, is heavily algebraic but the one given here, based on notes by Eisenträger, avoids that. Even more useful, perhaps, is Section~2 of Appendix~A, which reproves the Main Theorem of Complex Multiplication in a scheme-theoretic way which may be more accessible to modern readers than the famous but now stylistically dated book of Shimura. The other parts of Appendix~A are a converse to the Main Theorem, needed in Chapter~2 of the book, and a result (due, though in a slightly less general form, to Shimura) about the existence of algebraic Hecke characters extending a given map between the Weil restriction of tori, which is needed in Chapter~3 in order to realise given CM types via \(p\)-divisible groups. Appendix~B is more specific, being devoted to constructions of CM lifting via \(p\)-adic Hodge theory, but it also includes a comparison of various Dieudonné theories (classical, crystalline etc.)NEWLINENEWLINEThe book as a whole is very attractive but, because of the technical difficulties of the subject matter, it is not easy reading. Specialists will need no encouragement to study it, but it contains much information of wider interest too.